### quantitative_practice

```Problem solving practice
This file contains over 600 problem solving and data
sufficiency questions for the GMAT with full answers and
explanations.
1. 15 Java programmers, working in a constant pace, finish a web page in 3 days. If
after one day, 9 programmers quit, how many more days are needed to finish the
remainder of the job?
(a) 5.
(b) 2.
(c) 8.
(d) 4.
(e) 6.
The total working days for finishing a web page are (15 x 3) 45. If after one day 9
programmers quit, only 15 working days are done and the rest of the programmers (6)
Need to finish (45 – 15) 30 days of work. It will take them 5 more days.
2. Two carpenters, working in the same pace, can build 2 desks in two hours and a
half. How many desks can 4 carpenters build in 4 hours?
(a) 2.4.
(b) 3.6.
(c) 4.2.
(d) 5.5.
(e) 6.4
2 carpenters build 2 desks in 2.5 hours ---> 4 carpenters build 4 desks in 2.5 hours ---> In 4 hours there are (4/2.5 = 1.6) time units. And (4 x 1.6) is 6.4 desks.
6. There are 40 students in a classroom, 9/20 of them are boys and 4/5 of them are
right-handed. How many right-handed boys are there in the classroom?
(a) Between 10 and 32.
(b) Between 14 and 32.
(c) Between 10 and 18.
(d) Between 14 and 18.
(e) Between 18 and 36.
There are (9/20 x 40 = 18) boys in the class. 80% of them are right-handed, meaning
that (4/5 x 18 = 14.4). Answer C is the best answer.
7. In Jonathan’s pen there are 300 sheep’s. 5/6 of the sheep’s are white, 2/3 of the
sheep’s have soft wool. What can’t be the number of white sheep’s that also have soft
wool in the pen?
(a) 100.
(b) 200.
(c) 190.
(d) 180.
(e) 160.
There are (5/6 x 300 = 250) white sheep’s.
There are (2/3 x 300 = 200) soft woolen sheep’s.
The maximum overlap is the size of the smallest among the groups, thus 200. The
minimum overlap is (250 + 200 – 300 = 150).
Therefore the number of sheep’s can be somewhere between 150 and 200.
8. Ross has 40 shirts, ¾ of the shirts are green and 1/10 is without buttons.
Therefore Ross has between ___ and ___ shirts with buttons that are not green.
(a) 6 ; 10.
(b) 4 ; 25.
(c) 4 ; 10.
(d) 5 ; 25.
(e) 3 ; 10.
Notice that the groups that we are looking for a overlapping are the not-green shirts
and the buttoned ones. The not-green shirts are a quarter of 40, 10 shirts.
The shirts with buttons are (9/10 x 40 = 36).
The maximum overlapping is the size of the smallest group: 10.
The minimum overlapping is: 36 + 10 – 40 = 6.
9. In the Kan film festival, 50 movies were presented. 3/5 of the movies are action
movies and 4/5 is science fiction movies. How many of the movies were science
fiction action movies?
(a) 10.
(b) 15.
(c) 20.
(d) 30.
(e) 35.
There were (3/5 x 50 = 30) action movies.
There were (4/5 x 50 = 40) science fiction movies.
Exact overlapping is calculated by minimum overlapping method.
Therefore there are (40 + 30 – 50 = 20) movies that belong to both categories.
10. There are 200 cats in Cat-City. Out of the 200, 70 are street cats and the rest are
domestic cats. 110 cats are gray, 30 out of the gray cats are domestic ones. How many
domestic cats are there which are not gray in Cat-City?
(a) 90.
(b) 80.
(c) 50.
(d) 40.
(e) 25.
Out of 200 cats, 130 are domestic ones. Out of 110 gray cats, 30 are street cats
therefore 80 are grey and domestic ones.
Altogether there are 130 domestic cats, 80 are grey so (130 – 80) = 50 are not grey.
11. Chandler is building a fence in the following method: He grounds 10 poles, each
10 Cm thick, in 1 meter spaces from each other. He then connects the poles with a
barbed wire. What is the total length of the fence?
(a) 11.
(b) 12.
(c) 9.9.
(d) 10.
(e) 13.
The total width of the poles is (10 x 0.1 = 1) meter.
There are 9 spaces between the poles, each 1 meter, so it’s another 9 meters.
The total length is (1 + 9 = 10) meters.
12. In a psychology school the grade of the students is determined by the following
method: At the end of the first year the grade equals to twice the age of the student.
From then on, the grade is determined by twice the age of the student plus half of his
grade from the previous year. If Joey’s grade at the end of the first year is 40, what
will be his grade at the end of the third year?
(a) 75.
(b) 62.
(c) 80.
(d) 44.
(e) 56.
From the grade 40 at the end of the first year we learn that his age is 20.
At the end of the second year, he will be 21 and his grade will be
(21 x 2 + ½ x 40 = 62).
At the end of the third year, he will be 22 and his grade will be (22 x 2 + ½ x 62 =
75).
13. What is the sum of all the even numbers bigger than (-10) and smaller than 12?
(a) 2.
(b) 10.
(c) 0.
(d) 8.
(e) 4.
This is a series of numbers with a constant spacing between them.
The first number is (-8) and the last is (10), there are 10 numbers altogether.
The formula for such a series is: ((-8 + 10) x 10)/2 = 10.
The second way to answer such a question is to write the numbers and add them.
14. The value of an “Aerosoul” stock changes according to the following method:
At the end of each month her value is doubled but due to commission the stock’s
value is decreases by \$10. If the value at the beginning of January is \$A, what would
be her value at the end of February?
(a) 4A – 10.
(b) 4A – 20.
(c) 4A – 30.
(d) 4A – 40.
(e) 4A – 50 .
At the end of January her value is 2A – 10.
At the end of February her value is (2 x (2A – 10) – 10 = 4A – 30).
15. An Ameba is an organic life form that divides into two Amebas each round hour.
If at a certain round hour, two Amebas were placed in a jar, how many Amebas will
be in the jar in N hours?
(a) 2N
(b) 22N
(c) 2N+1
(d) 2N-1
(e) 2N
Let’s find the number of Amebas in the first hours.
After one hour (N=1) there will be 4 Amebas.
After two hours (N=2) there will be 8 Amebas.
After three hours (N=3) there will be 16 amebas.
Therefore the formula that fits this series is 2N+1.
16. Alfa, Beta and Gamma are inner angles in a triangle. If Alfa = Beta + Gamma,
what can’t be the size of Beta?
(a) 44 degrees.
(b) 45 degrees.
(c) 89 degrees.
(d) 90 degrees.
(e) There isn’t enough data to determine.
If Beta is 90 degrees than Alfa is bigger than 90 and the sum of the angles in the
triangle will be bigger than 180 degrees.
18. In a triangle, one side is 6 Cm and another side is 9 Cm. which of the following
can be the perimeter of the triangle?
(a) 18.
(b) 25.
(c) 30.
(d) 32.
(e) 34.
The third side of the triangle is larger than 3 (The difference between the other two)
and smaller than 15 ( The sum of the other two).
The perimeter is between (6+9+3 = 18) and (6+9+15 = 30). The only answer that is in
this range is B.
19. To which of the following shapes the area can’t be calculated if the perimeter is
given?
(a) Circle.
(b) An isoceles right triangle.
(c) Rectangle.
(d) A regular Hexagon.
(e) Square.
The perimeter of a rectangle is 2a + 2b. In order to calculate the area we need to know
the multiplication of a x b.
20. A and B are two circles. The radius of A is twice as large as the diameter of B.
What is the ratio between the areas of the circles?
(a) 1:8.
(b) 1:2.
(c) 1:4.
(d) 1:16.
(e) 1:6.
The radius of circle A is 4 times larger than the radius of circle B. The area of a circle
is a function of the radius squared, therefore the area of radius A is 16 times bigger.
21. A, B, C, D and E are 5 consecutive points on a straight line. If BC = 2CD, DE = 4,
AB = 5 and AC = 11, what is the length of AE?
(a) 21.
(b) 26.
(c) 30.
(d) 18.
(e) 16.
First, draw the line and the points.
In order to find the length of AE, find the length of CD and BC first.
BC = AC – AB = 11 – 5 = 6.
BC = 2CD CD = 3.
AE = 5 + 6 + 3 + 4 = 18.
22. In a rectangular axis system, what is the distance between the following
points: A(3,2) and B(7,5) ?
(a) 5.
(b) 7.
(c) 6.
(d) 4.
(e) 3.
First, draw a rectangular axis system and mark the two points.
The easiest way to find the distance between them is to draw a triangle, where the line
AB is the hypotenuse. You can see that the length of one side of the triangle is (52=3) and the other side is (7-3=4). The length of the line AB is received with the help
Of the Pythagoras principle: AB = 3 2 + 4 2 = 5.
23. In a rectangular axis system, what is approximate distance between the following
points: C(1,2.5) and D(6.5,5.5) ?
(a) 5.5.
(b) 7.2.
(c) 6.3.
(d) 4.1.
(e) 3.8.
First, draw a rectangular axis system and mark the two points.
The easiest way to find the distance between them is to draw a triangle, where the line
CD is the hypotenuse. You can see that the length of one side of the triangle is
(5.5 - 2.5 = 3) and the other side is (6.5 – 1 = 5.5). The length of the line CD is
received with the help Of the Pythagoras principle: CD = 3 2 + 5.5 2 = 39.25 ≅ 6.3 .
24. In a rectangular axis system, what is the distance between the following
points: A(24.4,30) and B(34.4,42.49) ?
(a) 5.
(b) 7.
(c) 8.
(d) 12.
(e) 16.
First, draw a rectangular axis system and mark the two points.
The easiest way to find the distance between them is to draw a triangle, where the line
AB is the hypotenuse. You can see that the length of one side of the triangle is
(34.4 – 24.4 = 10) and the other side is (42.49 – 30 = 12.49). The length of the line
AB is received with the help
Of the Pythagoras principle: AB = 10 2 + 12.49 2 = 256 = 16 .
25. In a rectangular axis system, what is the area of a parallelogram with the
coordinates: (5,7), (12,7), (2,3), (9,3) ?
(a) 21.
(b) 28.
(c) 35.
(d) 49.
(e) 52.
First, draw the axis system and mark the 4 points. Connect the points to get a
parallelogram. The area is calculated by the multiplication of one on of the bases and
the height. The height is (7 – 3 = 4), the length of the base is (9 – 2 = 7).
The area is 4 x 7 = 28.
29. If the radius of a cylinder is doubled and so is the height, what is the new volume
of the cylinder divided by the old one?
(a) 8.
(b) 2.
(c) 6.
(d) 4.
(e) 10.
The volume of a cylinder is (pie x R2) x (height of cylinder).
The new volume is (4 x 2 = 8) bigger.
30. If the radius of a cylinder is doubled and so is the height, how much bigger is the
new lateral surface area (with out the bases)?
(a) 8.
(b) 2.
(c) 6.
(d) 4.
(e) 10.
The lateral surface area of a cylinder is (2 x pie x R) x (height of cylinder).
The new lateral surface area is (2 x pie x 2R) x (double the height) = 4 times bigger.
1. If X ~ Y = X2 + XY, then what is the value of -1 ~ 2 ?
(a) 1.
(b) -1.
(c) 3.
(d) 4.
(e) 2.
-1 ~ 2 = (-1)2 + (-1)2 = -1.
2. If X Y = XY2, then what is the value of 3 (t-1) ?
(a) 3t2 – 2t + 2.
(b) 3t2 – 2t + 4.
(c) 3t2 – 6t +3.
(d) 3t2 – 6t – 3.
(e) 3t2 – 6 + 3.
3 (t-1) = 3(t-1)2 = 3(t2-2t+1) = 3t2 – 6t +3.
3. If Q
(a) 7.
= Q + 2, then what is the value of (3 )
?
(b) 5.
(c) 6.
(d) 4.
(e) 8.
(3r) = (3 + 2) = 5 = 5 + 2 = 7.
4. If (3 ) = 9, then which of the following expressions can x
(a) x2.
(b) 3x – 5.
(c) 2x – 1.
(d) 2x + 1.
(e) none of the answers above.
be equal to?
Check the answers by replacing the x with 3 and try to see if it works out.
Answer (a): (3 ) = (32)2 = 81. Not good.
Answer (b): (3 ) = (3 x 3 – 5) = (4) = (12 – 5) = 7. Not good either.
Answer (c): (3 ) = (3 x 2 -1) = (5) = (10 – 1) = 9. Good enough.
5. If (4 2 = 14) and (2
(a) ab.
(b) (a+3)b
(c) a2 – b.
(d) ab – 2.
(e) ba + 1.
3 = 6), what can replace (a
b) ?
Check every answer until you hit the jackpot.
(a) (4 2) = 8. The answer should be 14.
(b) (2 3) = (2 + 3)3 = 15. The answer should be 6.
(c) (2 3) = (22 – 3) = 1. The answer should be 6.
(d) (4 2) = (42 – 2) = 14. This is the right answer, check (2
6. If
(a,b) =
3a
b
, what is the value of
(a) 1.
(b) 4.
(c) 6.
(d) 9.
(e) 18.
[ (4,4), (1,9)] ?
3) also.
3⋅ 4
=6.
4
3 ⋅1
(1,9) =
=1 .
9
3⋅ 6
= 18 . Therefore E is the best answer.
(6,1) =
1
(4,4) =
7. If 5 = 13, which of the following can describe a ?
(a) 3a + 1.
(b) 2a + 3.
(c) 3a – 2.
(d) 3a – 1.
(a) 5 = 3 x 5 +1 = 16.
(b) 5 = 2 x 5 + 3 = 13.
(c) 5 = 3 x 5 – 2 = 13.
11. For every X, the action [X] is defined in the following matter: [X] is the greatest
integer that is smaller or equal to X. For example: [8.9] = 8.
What is the value of [6.5] x [2/3] + [2] x 7.2 + [8.4] – 6.6 ?
(a) 15.8.
(b) 16.2.
(c) 16.4.
(d) 14.4.
(e) 12.6.
[6.5] x [2/3] + [2] x 7.2 + [8.4] – 6.6 = 6 x 0 + 2 x 7.2 + 8 - 6.6 = 15.8.
15. If (1 < A < 3 < B), then which of the following expressions is the largest?
(a) (B+2)/(A-1).
(b) (B-2)/(A+1).
(c) A/B.
(d) (B-2)/(A-1).
(e) B/A.
Try some numbers and check the answers. A=2, B=4.
(a) 6/1 = 6.
(b) 2/3.
(c) 1/2.
(d) 2.
(e) 2.
16. Which of the following fractions is the smallest?
(a) 3/10.
(b) 6/19.
(c) 3/8.
(d) 11/30.
(e) 12/31.
Compare all of the answers to (a) 3/10.
(b) 3/10 x 2 = 6/20 which is smaller than 6/19.
(c) 3/10 is smaller.
(d) 3/10 = 9/30, and this is smaller than 11/30.
(e) 3/10 = 12/40 and that is smaller than 12/31.
The smallest fraction is A.
17. Which of the following fractions is the largest?
(a) 2/7.
(b) 2/3.
(c) 7/9.
(d) 7/12.
(e) 3/5.
Lets compare all the answers to 2/7, unless we find a larger fraction.
(b) 2/3 is larger than 2/7. For now, this is the right answer.
(c) 2/3 is also 6/9 and that is smaller than 7/9. For now this is the right answer.
(d) 7/9 is bigger than 7/12.
(e) Bring this answer and (c) to a common denominator.
7/9 = 35/45 and 3/5 = 27/45.
7/9 is the largest fraction.
19. If A2 + B2 = 15 and AB = 10, what is the value of the expression
(A – B)2 + (A + B)2 ?
(a) 10.
(b) 20.
(c) 30.
(d) 60.
(e) 70.
(A – B)2 + (A + B)2 = A2 – 2AB + B2 + A2 + 2AB + B2 = 2(A2 + B2) = 30.
20. If A and B are positive integers, which of the following expressions is not an
integer for certain?
(a) (2A2 – 2B2)/(A+B).
(b) (6B + 8A)/(3B + 4A).
(c) (3A – B)/(B - 3A).
(d) (A + B)/(A2 + B2 + 2AB).
(e) (A2 – B2)/(A - B).
All the answers besides D are numbers after some simplification.
Answer D = (A + B)/(A+B)2 = 1/(A+B), and this is a fraction of a number.
21. In the “Big-Reds” parking lot there are 56 vehicles, 18 of them are buses and the
rest are private cars. The color of 32 vehicles is red, from which 17 are buses. How
many private cars can be found in the parking lot, which are not colored red?
1.
23.
17.
15.
20.
Out of 56 vehicles, 32 are colored red, therefore 24 are in different color.
17 of the red vehicles are buses, therefore (18 – 17 = 1) are in different color.
(24 – 1 = 23) private cars are in the parking lot with a different color than red.
22. In Sam’s hanger there are 23 boxes, 16 out of the boxes are filled with toys and
the rest are filled with electrical appliances. 8 boxes are for sale, 5 of them are filled
with toys. How many boxes with electrical appliances are in Sam’s hanger that are
not for sale?
1.
2.
3.
4.
5.
8 boxes are for sale, 5 of them are with toys, and therefore 3 of them are with
electrical appliances.
Out of 23 boxes, 16 are with toys, therefore, and therefore 7 of them are with
electrical appliances.
(7 – 3 = 4) is the number of electrical appliances boxes, which are not for sale.
1. In the fifth grade at Parkway elementary school there are 420 students. 312 students
are boys and 250 students are playing soccer. 86% of the students that play soccer are
obviously boys. How many girl student are in Parkway that are not playing soccer?
69.
73.
81.
91.
108.
There are (420 – 312 = 108) girls in Parkway.
86% of 250 are boys, therefore 14% of 250 are girls that play soccer, which is 35
girls.
The number of girls that do not play soccer is (108 – 35 = 73).
2. In the quiet town of “Nothintodo” there are 600 inhabitants, 400 are unemployed
and 300 are somnambulists. If half of the somnambulists are unemployed, how many
are employed and are not somnambulists?
50.
100.
150.
200.
300.
There are 300 people that are not somnambulists. There are (600 – 400 = 200) people
that are employed in the town, half of the somnambulists are employed (150),
therefore (200 – 150 = 50) is the number of people that are employed which are also
not somnambulists.
3. In the youth summer village there are 150 people, 75 of them are not working, 50
of them have families and 100 of them like to sing in the shower. What is the largest
possible number of people in the village, which are working, that doesn’t have
families and that are singing in the shower?
25.
50.
75.
100.
150.
The number of people that work is 75.
The number of people that doesn’t have families is (150 – 50 =100).
100 of the people like to sing in the shower.
The largest possible number of people that belong to all three groups is the smallest
among them, Meaning 75.
4. In the junior basketball league there are 18 teams, 2/3 of them are bad and ½ are
rich. What can’t be the number of teams that are rich and bad?
4.
6.
10.
7.
8.
(2/3 x 18 = 12) teams are bad and 9 are rich.
The number of teams which are rich and that are bad must be between 9 and
(9+12-18 = 3).
The only answer, which is not in that range, is C.
5. In the third grade of Windblow School there are 108 students, one third of them
failed the math test and 1/6 failed that literature test. At least how many students
failed both tests?
0.
6.
8.
10.
12.
(1/3 x 108 = 36) failed the math test.
(1/6 x 108 = 18) failed that literature test.
The least amount of people that failed both tests is (18 + 36 –108 = -54), there cant be
an negative Overlapping between the groups so the least amount of people who failed
both tests is zero.
6. If 1/X = 2.5, then what is the value of 1/(X – 2/3)?
2.25.
–3.5.
–3.75.
1.75.
3.75.
If 1/X is 2.5 or 5/2 then X = 2/5.
1/(2/5 – 2/3) is 1/(6/15 – 10/15) = -15/4 = -3.75.
8. Travis is working as a programmer of IBW. Travis earns \$3,500 annually.
If Travis pays 2.5% of that amount quarterly to support groups and he paid \$525 so
far, for how many years now has Travis been paying?
2.
2.5.
4.
5.5.
6.
Travis pays 2.5% of 3500, which is \$87.5 every 3 months (quarterly).
(525/87.5 = 6), therefore Travis has been paying for (6 x 3 = 18) months now, that is
2.5 years.
9. Dana borrows 5500 pounds annually for her college education. If Dana gives her
parents 3% of that amount back each month, how much will she still owe her parents
after four years of college?
12,430.
13,640.
14,000.
14,080.
15,020.
Dana takes 5500 each year and returns (0.03 x 5500 = 165) each month, which is (165
x 12 = 1980) each passing year. That means that each year Dana owes her parents
(5500 – 1980 = 3520) pounds.
After 4 years in college she will owe them (4 x 3520 = 14,080) pounds.
10. Mr. Rusty owes the bank \$1,040,000, he returns \$40,000 quarterly to the bank. If
the tax on the money Rusty owes is compounded quarterly by 0.25% starting before
Rusty paid the first payment, how months would it take poor Rusty to reach a point
where he owes the bank no more than 1 million dollars?
3.
6.
9.
12.
15.
Every three months Rusty gives the bank \$40,000.
After the first quarter the bank took (0.0025 x 1040000 = 2600) and Rusty paid
\$40,000 so the new
Debt is now (1,040,000 - 40,000 + 2,600 = 1,002,600).
After the second quarter the bank took (0.0025 x 1002600 = 2506.5) and Rusty paid
again \$40,000 so the new Debt is now (1,002,600 – 40,000 + 2506.5 < 1 million
dollars).
11. Simba borrowed \$12,000 from his brothers so he can buy a new sports car. If
Simba returns 4.5% of that amount every 2 weeks, after how many months Simba
wouldn’t owe his brothers any more money?
8.
12.
15.
18.
20.
Simba gives (0.045 x 12,000 = 540) to his brothers every 2 weeks, in a month he
gives (540 x 2 = 1080). (12,000/1,080 is a little over 11), therefore after 12 months he
won’t owe any more money.
12. If A and B are two roots of the equation X2 –6.5X – 17, then what is the value of
A x B?
15.
–18.
16.5.
–17.
22.
The roots of the equation are 8.5 and (-2).
The multiplication of the roots is equal to (-17).
13. If A,B and C are roots of the equation X3 – 16X2 +48X, what is the sum of the
roots?
16.
14.
17.
18.5.
22.5.
The equation can be written as: X(X2 – 16X +48) = X(X – 12)(X – 4).
The roots of the equation are: 0,4 and 12. The sum of the roots is 16.
14. If R is a root of the equation X2 +3X – 54, than which of the following
equations have also the root R ?
X2 – 12X +27.
X2 – 6X – 16.
X2 – 10X – 31.25.
X2 – 15X + 54.
X2 + 10X + 16.
The original equation is X2 + 3X – 54, it can be written as (X – 6)(X + 9). The roots
are 6 and (-9).
We are looking for an equation that has one of the same roots.
Answer D: X2 – 15X +54 = (X – 6)(X – 9) This equation has the root 6.
All the other answers have different roots than the original equation.
15. If P is a root of the equation X3 +10X2 + 16X, than which of the following
equations have also the root P ?
X2 – 10X +16.
X + 8.
X2 +3X – 54.
X2 – 6X – 187.
X2 + 8X - 20.
The original equation is X3 +10X2 + 16X, it can be written as X(X + 8)(X + 2). The
roots are
(-8),0 and (-2).
We are looking for an equation that has one of the same roots.
Answer B: X + 8 This equation has the root (-8).
All the other answers have different roots than the original equation.
16. If X is a root of the equation a3 +8a2 – 20a, than which of the following equations
Don’t have the root X as one of their roots?
X3 + 4X2 – 32X.
X2 + 18X + 80.
X2 – 12X + 20.
X2 + 5X – 14.
X2 + 10X + 16.
The original equation is a3 +8a2 – 20a, it can be written as a(a – 2)(X + 10). The roots
are 2,0 and (-10).
We are looking for an equation that has none of the same roots.
Answer E: X2 – 10X +16 = (X + 2)(X + 8) This equation has none of the original
roots. All the other answers have one or more of the same original roots.
17. Gwen has to divide her money between her three sons. If the older brother
received 65% of the total amount and the other two received the same amount of
money, how much money did the median brother get?
(1) The combined amount of money of the older brother and the small one is \$45,000.
(2) The older brother received \$35,454.5.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
The data gave us the ratio of the amounts each one got (65 : 17.5 : 17.5), therefore all
we need is one number to know how much each of the brothers received. Each of the
statements above gives us enough information to solve the problem.
18. Little Timmy spends half of his allowance on his favorite pet Din and the other
half on candies. How much money did Timmy spend on Din?
(1) Din eats 1.5Kg of food every day.
(2) Timmy buys 110 gr. Of candies each day. One Kg of candies costs \$7.5.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From the question we know the ratio of the money that Timmy is spending on Din
and on candies.
In order to know how much Timmy spends on each, we need to know one of the
expanses in real
Amount of money and not in percent terms. The first statement doesn’t provide us any
sufficient information but the second one gives us the exact amount of money that
Timmy spends on Candies, which is equal to the amount that he spends on Din.
19. A, B, C and D are four consecutive points on a straight line. What is the distance
between A to D?
(1) AC = 6.
(2) BD = 8.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
First, draw the line with the points marked.
We know AC and BD but it’s not sufficient to know the length of AD.
If the question said the points are evenly spaced than the answer would be solvable.
20. A, B, C, D and E are five consecutive points with equal spacing on a straight line.
What is the distance between A to E?
(1) AB = 3.
(2) BE = 9.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
First, draw the line with the points marked.
Because the points are evenly spaced on the straight line, only one measurement is
needed to answer the question. Both statements give us a measurement of some kind
therefore each of them, by itself is sufficient.
21. A, B and C are 3 consecutive points on an arc with a constant radius. What is the
distance between A and C?
(1) The radius of the arc is 25 Cm.
(2) The length of AB is 5 Cm.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
In order to know the distance between two points on an arc you need to know the
angle that the points make and the radius of the arc.
Statement (1) gives us the radius.
Statement (2) gives us the length of AB, but the question didn’t mention that there is
equal spacing and therefore the length of BC can’t be found with both of the
statements taken together.
23. If X and Y are positive integers, is X greater than Y?
(1) X > Y – 2.
(2) X > 2.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Take some numbers for example.
Y= 8 from statement (1) we know that X > 6 and from statement (2) we know that
X >2, but X can be 7 or even 24 and he will still fit the equation properly, therefore
both statements, taken together are not sufficient.
24. If X and Y are positive integers, is X greater than Y?
(1) X > 2.
(2) Y < 3.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) we learn that X is 3 or bigger and from statement (2) we learn that
Y is 2 or smaller. Therefore both statements are sufficient to answer the question.
25. If X, Y and Z are positive integers, is X greater than Z – Y?
(1) X – Z – Y > 0.
(2) Z2 = X2 + Y2.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) we learn that X > Z + Y therefore X must be bigger than Z – Y
(positive integers).
From statement (2) we learn that X2 = Z2 – Y2 and that tells us nothing relevant.
26. (x, y) are the coordinates of the intersection of the following lines:
(3x – 2y = 8) and (3y + x = 10). What is the value of (x/y)?
1.
2.
3.
4.
5.
There is no need to draw the lines. There are two equations with two variable that you
have to solve.
Take the second equation and multiply it by (-3) to get: -9y –3x = -30 add this
equation to the first and
You’ll get: -11y = -22 y=2 and x=4. (x/y) is 2.
27. A(a, b) is the coordinates of the intersection between the lines:
(x + y –1 = 0) and (4x – 2y = 5). What is the shortest distance between
A(a, b) and the coordinate B(25/6, 23/6)?
1.
2.
3.
4.
5.
There is no need to draw the two lines. Multiply equation (1) by 2 and then add the
equations to get:
6x = 7 x = 7/6, y = -1/6.
Draw a rectangular axis system and mark the point A and B.
Complete the two points to a triangle so one of sides is 3 and the other is 4, the
hypotenuse, which is also the requested length is 5.
28. P(x, y) is the intersection point between the circle (x2 + y2 = 4) and the line (y = x
+2). Which of the following can be the point P?
(1, 2).
(2, 0).
(0, -2).
(-2, 0).
(2, 2).
First, draw the circle and the line. The circle is centered at (0, 0) with a radius of 2.
You can see that the line and the circle intersect at two points: (-2, 0) and
(0, 2). Another way is to insert y = x+2 into the equation of the circle and solve it.
29. Is the intersection of the two lines: (x + y = 8) and (4y – 4x = 16) inside the circle:
x2 + y2 = r2?
(1) r = 81.
(2) The center of the circle is at the coordinate (-99, -99).
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
The intersection point of the two lines is easy to find, its (2, 6).
In order for us to know if the point is inside the circle we need to know the exact
location of the circle. Statement (1) clears the problem by giving us the radius so all
the sufficient data is know.
Statement (2) is not sufficient because it tells us nothing about the radius of the circle.
30. Is there an intersection between the line (Y = aX - b) and the parabola
(Y = X2 + b)?
(1) a < 0.
(2) 0 > b.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
First, draw the parabola and the line.
We can see that the thing that determines if the two intersect is b and not a.
If b > 0 there is no intersection but if b<0 there is an intersection no matter what the
value of a is.
Therefore statement (2) is sufficient and (1) is not.
31. Is there a point of intersection between the circle (X2 + Y2 = 4)and the
Line ( Y = aX + b) ?
(1) a = b2.
(2) The line intersects the X-axis at (40, 0).
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) we learn that the equation of the line can be written as Y = b2X +
b.
From statement (2) we learn that the line goes threw the point (40, 0), from that we
can find the equation of the line by posting the coordinate in the equation: 0 = b240 +
b.
There is no need to solve it, both statements are sufficient to solve the problem.
32. Zigfield bought his car using M% of his bank savings. He also bought a house that
costs 4 times the price of the car. What is the price of the house?
(1) M = 12.
(2) The price of the car and the house was \$140,000.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
With statement (1) we know that the car cost 12% of his life savings.
From statement (1) and (2) we know that (12 x 4 = 48) percent of his life savings
went to the house. 60% of his life savings is \$140,000. From here we know how much
were his life savings and so we know that 48% of that went to the house. Both
statements are needed.
33. Eddy gave Q% of the money he earned last year to his first wife Sandra, W% of
the money he earned last year went to his second wife Tawana. How much money did
Eddy earn this year?
(1) Q = 20, W = 2Q.
(2) All the money Eddy earned last year went to his two wives.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) and (2) taken together we know how much Eddy earned last year
but we know
statements are not sufficient to answer the question.
34. Of the 10,000 people that went to the state-fair, how many men ate at the fair?
(1) The percentage of men who ate at the state-fair was twice as those who didn’t eat.
(2) 3,500 women ate at the state-fair.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) we know the ratio between the men who ate to those who didn’t,
but we don’t know how many men were at the fair. Statement (2) doesn’t reveal the
number of woman that went to the fair, only the number of woman that ate there.
Therefore, more data is needed to answer the question.
35. Out of the 100 kids that went to the party, how many girls danced there?
(1) 25 girls don’t like to dance and so they didn’t.
(2) The number of boys that danced is twice the number that didn’t dance.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) we know that 25 out of X girls didn’t dance. We need to know
how many girls in total were in the party. Statement (2) doesn’t tell us anything about
the number of boys or girls that went to the party but only the ratio between those
who danced to those who didn’t.
Therefore, more sufficient data is needed to solve the problem.
36. 990 people went to the GMAT exam, how many boys didn’t pass the test?
(1) 321 girls didn’t pass the test, which is the number of boys that did.
(2) One fifth of the people that went to the GMAT exam were boys who eventually
didn’tpass the test.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Statement (1) gives us information about the number of boys that passed the test but
no useful information about the other part of the boys.
Statement (2) by itself gives us the answer to the question (1/5 x 990).
37. What is the area of the rectangle with the following coordinates: (x, y), (10, y),
(10, 5), (x, 5)?
6.
8.
12.
32.
It cannot be determined from the information given.
First of all, draw the rectangle with the given coordinates.
You can see that only one side of the rectangle is given and not the second, therefore
there isn’t enough data to answer the question.
38. What is the area of the square with the following coordinates: (x, y),
(20, 20), (20, 5), (x, 5)?
60.
85.
125.
225.
It cannot be determined from the information given.
First of all, draw the square with the given coordinates.
We know only one of the square’s sides but it’s enough because it is a square and
both sides are equal. The area, therefore, is (15 x 15 = 225).
39. Is the largest of 7 consecutive numbers odd?
(1) The product of the seven numbers is zero.
(2) The sum of the seven numbers is zero.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) we learn that there is a 0 among the seven numbers, yet the largest
number can be odd or even. (0, 1, 2, 3, 4, 5, 6 or -1, 0, 1, 2, 3, 4, 5).
From statement (2) we know that the numbers are located symmetrically around the
zero, therefore the largest number is even.
40. Is the sum of X consecutive numbers zero?
(1) The largest number is 5.
(2) The median number is zero.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Statement (1) is not sufficient because the series of numbers is not blocked from the
smaller numbers.
Statement (2) is sufficient by itself because we know that if the median number is 0,
then the sum of the numbers must be even.
2. If X and Y are positive integers, what is the ratio between Y and X?
(1) XY = 150.
(2) Y is 22% of X.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
The question actually asks what is Y/X or X/Y.
Statement (1) is not sufficient because from the product of the two variables we can’t
make out the ratio. Statement (2) is sufficient by itself, Y = 22X/100 Y/X = 11/50.
3. If x and y are positive integers (x>y), what is the units’ digit of (10x – 9y)2 ?
9.
7.
5.
3.
1.
Try some numbers, x=2, y=1.
(106 – 92)2 = 81. And it will work with any given number under the conditions given.
4. What is the value of A + B ?
(1) A = 8 – B.
(2) (A + B)2 – 64 = 0.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) we know right away that A + B = 8.
From statement (2) we don’t know if A + B = 8 or –8.
Therefore only statement one is sufficient to answer the question.
5. What is the value of (A – B)?
(1) A = 8 – B.
(2) A2 – B2 – 64 = 0.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) we know the value of A + B.
From statement (2) we know the value of A2 – B2 = (A – B)(A + B) (A – B)8 = 64
the answer is equal to 8, therefore both statements are needed on order to answer
the question.
6. What is the value of (A + B) ?
(1) B = 12 – 3B.
(2) A2 + 4A – 16 = 0.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) we can find the exact value of B.
From statement (2), we can find two answers for variable A, therefore the answer is
not unequivocally and both statements taken together are not enough, more sufficient
data is needed.
7. What is the value of (X2 + Y2)?
(1) (X – Y)2 = 36.
(2) (X + Y)2 = 48.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Statement (1) can be written as X2 – 2XY +Y2 = 36.
Statement (2) can be written as X2 + 2XY +Y2 = 48.
Adding both equations will give: 2X2 + 2Y2 = 84 X2 + Y2 = 42.
Therefore, both statements are needed in order to solve the question.
8. There are X dogs in the dog hound, each dog eats Y Kg of food every day. What
percent of the total food weight does each dog eat?
(1) If there were 3 dogs less then each dog could eat 1.2 Kg more than he is does now.
(2) If there were half the dogs, each dog could eat 3 Kg more than he is does now.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
In order to know the answer we need two equations:
From statement (1) we can write: XY = (X – 3)(Y + 1.2).
From statement (2) we can write: XY = (X/2)(Y + 3).
You don’t need to solve the equations, the answer is C, both equations are needed to
solve the question.
9. If (R, R2 + 1) is the (x, y) coordinate of a point located on the line: Y = 2X + 4,
what Can be the value of the parameter R?
–3.
2.
4.
3.
1.
If the point is on the line then you can plug the coordinate into the equation.
Y = 2X + 4 R2+1 = 2R + 4 R= 3 or R= -1.
Therefore the best answer is D.
10. A(5, w3) is the (x, y) coordinate of point located on the parabola Y = X2 + 2.
What is the value of w?
3.
4.
5.
6.
9.
Plug into the equation the coordinate to get: w3 = 52 + 2 = 27
w = 3.
11. If x and y are positive integers, is 5x(1/4)y < 1 ?
(1) y = 3x.
(2) x = 2.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Use statement (1) to write: 5x(1/4)3x = (5/64)x. Because x is a positive integer only, the
expression will always be smaller than 1. This statement alone provides us the
Use statement (2) to write: 52(1/4)y the answer here is dependent on y, a different
combinations of the variable y will give different results.
12. If x and y are integers, is 3x(0.5)y < 1 ?
(1) y = 2x.
(2) x = 8.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Use statement (1) to write the expression: 3x(0.5)2x = (0.75)x the value of this
expression can be either smaller or larger than 1, if x was only a positive integer the
Use statement (2) alone to write the expression: 38(0.5)y this expression is either
bigger or smaller than 1.
Use both statements together: (0.75)8 < 1. Therefore both statements are needed to
13. A and B are integers, is (0.5)AB > 1 ?
(1) A is positive integer and B is negative integer.
(2) A and B are two consecutive numbers.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) we know that one is positive and the other is negative, therefore
their product is negative. (0.5)negative = a number bigger than 1. This statement is
From statement (2) we know the answer also. This is a tricky part.
Try all the options: (-2 and –1), (-1 and 0), (0 and 1), (1 and 2).
All of these options give out AB that is positive or equal to zero, in both cases (0.5)AB
will be either smaller than 1 or equal to 1 but never bigger. Therefore each statement
by itself is sufficient.
14. A jar of 264 marbles is divided equally among a group of marble-players. If 2
people join the group, each one would receive 1 marble less. How many people are
there in the group today?
20.
21.
22.
23.
24.
You can back-solve it. 264 marbles divided by 22 (answer C) is 12 marbles per
person.
If two people join, there will be 24 people, 264/24 is 11, which is 1 marble less.
15. A basket of 1430 apples is divided equally among a group of apple lovers. If 45
people join the group, each apple lover would receive 9 apples less. How many apples
did each person get before 45 people joined the feast?
20.
21.
22.
23.
24.
Try to back-solve the problem. (1430/22 = 65) people, if 45 came then there are 110
people.
(1430/110 = 13) apples, which is 9 apples less per person.
16. A confectioner decides to sell all of his pastry due to the coming holiday. His
pastry goods are equally divided among a group of 28 regular customers. If only 49
customers come to the bakery, each one will receive 6 less pastry goods. How much
pastry does the confectioner needs to sell?
392.
412.
432.
502.
522.
You can use the answers to back-solve the question or you could write the equations.
Take 392 pastry goods and divide them by 28 customers, each one will receive 14
products.
If there were 49 customers, each one would receive (392/49 = 8), which is 6 less.
17. In the equation 4Y – 3kX = 18, k is a constant and Y equals 42 when X equals 12.
What is the approximate value of X when Y equals 36?
5.
10.
15.
20.
25.
First, find the constant k. Plug in the numbers for X and Y, to receive 4 x 42 – 3k x 12
= 18
k = (18 – 168)/36 = -25/6.
Now, plug in the value of Y to receive: 4 x 36 – 3kX = 18 after a little math, X is
equal to 10.08, therefore the approximate answer is 10.
18. In the equation (X + Y = k), k is a constant and X equals 13 when Y equals 23.5.
What can be the value of X2 when Y2 is equal to 36?
456.5.
673.25.
830.75.
890.35.
930.25.
First, find the constant k. Plug in the numbers to get k = 13 + 23.5 = 36.5.
Now, Y2 = 36 Y = 6 or (-6). Plug both numbers to get X = 30.5 or X = 42.5.
The best answer is E, (30.5)2 = 930.25.
19. Did the owner of the garage sale made more than \$130 last Saturday?
(1) There were 15 products at the garage sale, each one cost \$25.
(2) All the products were sold.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Statement (1) tells us how many products were in the sale and how much did each
cost.
Statement (2) tells us that all the products were sold, therefore the owner made 15 x
\$25 = \$375.
Both statements are required to answer the question.
20. What is the total amount of Jellybeans that Benjamin ate last week?
(1) This week Benjamin ate 20% more Jellybeans than two weeks ago.
(2) Two weeks ago Benjamin ate 65 Jellybeans.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
also doesn’t give us any sufficient data on last week, but on two weeks ago.
Therefore, more sufficient data is required.
21. How many hamburgers did “Wacdonalds” sell last year?
(1) Two years ago “Wacdonalds” sold 422,000,000 hamburgers.
(2) The average amount of hamburgers sold by “Wacdonalds” each year is 5 million.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Statement (1) gives us data on the sales of two years ago.
Statement (2) gives us data on the average amount of sales each year.
We can’t assume that the sales two years ago + last years sales divided by two is the
average amount of sales, therefore more sufficient data is needed to answer the
question.
22. At the end of the year 2002, Monica and Chandler each purchased a certificate of
deposit that paid the same rate of interest, and each held the certificate of deposit
through the end of 2002. If Chandler invested X dollars and Monica invested
\$130,000, and if Chandler earned interest in 2002 totaling \$45,000, what was the
amount of interest that Monica earned on her \$130,000 investment?
(1) The rate of interest on the certificate of deposit that Chandler and Monica each
purchased was 8.5% annually.
(2) In 2002, Chandler invested \$529,412 in the certificate of deposit.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) we know the rate of interest, so we can easily calculate how much
Monica earned with her \$130,000 deposit.
From statement (2) we know how much Chandler invested and we already know from
the question how much he earned, we can calculate the interest and multiply it by the
Therefore, both statements, by themselves, are sufficient to answer the question.
23. Mickey made an X dollars loan at the beginning of 1996. Travis, who is Mickey’s
little brother also made a loan, only twice as large as Mickey’s but with the same
interest. If Travis pays \$10,000 interest on his loan each year, how big is Mickey’s
loan?
(1) The rate of interest on the loan that Travis took is 6% annually.
(2) The loan that Travis made was \$166,667.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) we know the rate of interest so we can find how much money
Travis loaned and multiply it by 2 to get Mickey’s loan.
From statement (2) we know the amount Travis loaned, which is doubled than that of
Mickey.
Therefore, both statements, by themselves, are sufficient to answer the question.
25. Concentrated orange juice comes inside a cylinder tube with a radius of 2.5 inches
and a height of 15 inches. The tubes are packed into wooden boxes, each with
dimensions of 11 inches by 10 inches by 31 inches. How many tubes of concentrated
orange juice, at the most, can fit into 3 wooden boxes?
24.
28.
36.
42.
48.
You want to waste as little amount of space as possible, therefore make the height of
the box 11 and fit 4 boxes at the bottom so you lose only 1 inch of margin at the top
and on one of the sides.
You can see that 8 tubes can fit into one box thus 24 tubes fit into 3 boxes.
1. A certain car’s price decreased by 2.5% (from the original price) each year from
1996 to 2002, during that time the owner of the car invested in a new carburetor and a
new audio system for the car, which increased her price by \$1,500. If the price of the
car in 1996 was \$22,000, what is the car’s price in 2002?
\$18,400
\$19,500
\$20,200
\$20,400
\$21,100
The price of the car decreased by 2.5% every year on a course of 6 years. That means
that the price of the car in 2002 is 15% lower than the original + \$1500 of new
investments.
The new price is (\$22,000 x 0.85 = 18,700 + 1500 = \$20,200).
2. The average price of an antique car increases over the years. If from 1990 to 1996,
the price of the car increased by 13% and from 1996 to 2001 it increased by 20%,
what is the price of the car in 2001 if the price in 1990 was \$11,500?
\$15,594.
\$15,322.
\$14,786.
\$14,543.
\$12,988.
The price in 1990 was 11,500. In 1996 the price is (11,500 x 1.13 = 12,995).
The price we are looking for, in 2002, is (12,995 x 1.2 = \$15,594).
3. The apartment on King-Williams street is an asset that it’s value is tramping about.
From the year 1973 to 1983 it’s value decreased by 16% and from 1983 to 1993 it’s
value increased by 16%. What is the value of the asset in 1993 if in 1973 it was worth
\$40,000?
\$38,796.
\$40,000.
\$38,976.
\$39,679.
\$36,796.
Be careful, the value of the asset didn’t stay the same after the two changes in the
value.
In the first 10 years, the value decreased by 16% (40,000 x 0.84 = 33,600).
Then, in the next ten years the value increased by 16% (33,600 x 1.16 = 38,976).
4. The value of a “Tin-Rin” stock in the stock market decreased by 15% in the last
two years.
The economic experts believe that the value of the stock will increase by 7% during
the following year, which will make the value \$440. What was the approximate price
of the stock two years ago?
\$473.
\$464.
\$455.
\$445.
\$430.
Start from the top, after a 7% increase the price of the stock is \$440.
440 are 107% of the price this year (440/107 x 100 = 411.215).
Two years ago the price was 15% higher, therefore (411.215 x 1.15) is approximately
\$473.
5. Which of the following expressions is equivalent to X < 4 ?
X < 4.
X > 4.
X > -4.
4 < X < -4.
–4 < X < 4.
An absolute value means that the sign of the variable is insignificant, therefore X can
be between –4 and 4 and still he will fulfill the original equation.
6. Which of the following statements is equivalent to (8 + 2X < 18 – 6X < 23 + 2X) ?
4/5 < X < 2.5.
4/5 < X < 8/5.
-5/8 < X < 5/4.
2.5 < X < 3.5.
5 < 8X < 12.
Take the expression and simplify it: Take (8 + 2x) from each side to get: (0<10 –
8X<15).
Substitute 10, -10<-8X<5.
Divide all by (-8), 5/4 > X > -5/8. Therefore the answer is C.
7. At the faculty of Aerospace Engineering, 312 students study Random-processing
methods, 232 students study Scramjet rocket engines and 112 students study them
both. If every student in the faculty has to study one of the two subjects, how many
students are there in the faculty of Aerospace Engineering?
424.
428.
430.
432.
436.
Use the group formula.
Total = groupA + groupB – Both + Neither.
Total = 312 + 232 – 112 + 0 = 432 students.
8. In the faculty of Reverse-Engineering, 226 second year students study numeric
methods, 423 second year students study automatic control of airborne vehicles and
134 second year students study them both. How many students are there in the faculty
if the second year students are approximately 80% of the total?
515.
545.
618.
644.
666.
Use the group formula.
Total = groupA + groupB – Both + Neither.
Total = 226 + 423 – 134 + 0 = 515 second year students.
The second year students are 80% of the total amount, therefore (515/80 x 100 =
643.75).
9. In the Biotechnology class of 2000, there were X graduates. 32 of the graduates
found a job, 45 continued on to their second degree and 13 did both. If only 9 people
didn’t do both, What is X equal to?
69.
71.
73.
75.
76.
Use the group formula.
Total = groupA + groupB – Both + Neither.
Total = 32 + 45 – 13 + 9 = 73 graduates.
10. If a, b, c, d and e are distinct integers, which one is the median?
(1) a < b – c.
(2) d > e.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Statement (1) tells us nothing about e and d, you can eliminate answers (a) and (d).
Statement (2) tells us nothing about a, b and c, you can eliminate answer (b) .
Try to plug in some numbers, take: a=3, b=7, c=1, d=9 and e=8. The median in that
case is 7.
Try other numbers, a=8, b=15, c=6, d=10 and e=9. The median is 9.
First the median was b, then the median was e. More sufficient data is required to
11. a, b and c are three odd and different integers. Which one is the median?
(1) a, b and c are consecutive numbers.
(2) c > a and b < c.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2)
by itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) we can learn that if they are consecutive numbers, the median is
B.
From statement (2) we have a connection between c to a and b, but we don’t know if a
or b is the smallest among the three, therefore this statement, by itself, is not
sufficient.
12. What is the ratio between W and Q?
(1) Q + W = 23.
(2) W is 25% of Q.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
We are looking for Q/W. From statement (1) we know the sum of the two variables,
which is not helpful in our case. From statement (2) we know that W = (0.25)Q,
therefore we know the ratio between the two variables.
13. What is the product of X and Y?
(1) 2X + 2Y = 46.
(2) (X + Y)2 = (X – Y)2 + 8.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
The product of X and Y is XY.
Statement (1) implies only about their sum.
Statement (2) can be written as: X2 + 2XY +Y2 = X2 – 2XY + Y2 + 8
XY = 2.
Only statement (2) is sufficient.
4XY = 8
20. Kramer can pack X boxes of cigarettes per minute. If there are Y boxes of
cigarettes in one case, How many cases can Kramer pack in 2 hours?
60X/Y.
120X/Y.
60Y/X.
120Y/X.
(X + Y)/60.
Y/X is the time it takes Kramer to fill a case with boxes (in minutes).
In two hours there are 120 minutes, so 120/(Y/X) is 120X/Y, and that is the number of
cases that Kramer can fill in two hours.
21. George can fill Q cans of paint in 3 minutes. If there are R cans of paint in one
gallon, how many gallons can George fill in 45 minutes?
30R/Q.
15R/Q.
30Q/R.
5Q/R.
15Q/R.
George can fill Q/3 cans of paint in one minute There are R cans in one gallon, so
R/(Q/3) = 3R/Q
Is the time it takes George to fill one gallon (in minutes).
In 45 minutes George can fill up 45/(3R/Q) = 15Q/R.
23. The junior soccer team is one of the best teams in the state of Alabama. The
season is divided into two parts, each part is 4 months. In the first part of the season,
the junior soccer team won half of their 32 games. How many games did the team win
in the entire season?
(1) In the second part of the season, the team lost 9 games, tied 6 games and won 18
games.
(2) From the 32 games remaining the team won twice as much as she lost.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) we can complete the missing data, in the first part of the season
the team won 16 games and on the second part of the season, the team won 18 games.
This statement is sufficient enough to answer the question.
Statement (2) is not sufficient by it self, it doesn’t mention how many games were
tied, therefore only statement (1) is sufficient.
24. “Queens” is a game of cards that distinguishes the cards into three groups: reds,
blacks and jokers. Four packets of cards are shuffled and only 50 cards are drawn out.
How many red cards are in the stack of the 50 cards?
(1) The number of black cards is twice the number of red cards.
(2) There is at least one joker in the stack of cards.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Statement (1) implies that if we knew the number of jokers, the answer would be
clear: take the cards that are not jokers and divide them by 3 to get the number of red
cards.
Statement (2) is not clear enough, the number of jokers is not distinct, therefore more
data is needed and the two statements taken together are not sufficient.
25. Ron has three kinds of shirts in his closet, white shirts, black shirts and fancy
shirts.
What is the ratio between the shirts in the closet?
(1) The total number of shirts is 100.
(2) 30% of the shirts are black, which is twice as much as the fancy shirts.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From statement (1) we know the total amount of shirts in the closet.
Statement (2) gives us the ratio between the shirts.
30% of the shirts are black (which is 30 shirts), this number is twice as much as the
fancy shirts (15).
The remaining shirts must be white. We know the ratio; therefore both statements are
required in order to answer the question correctly.
28. The roof of an apartment building is rectangular and its length is 4 times longer
than its width. If the area of the roof is 784 feet squared, what is the difference
between the length and the width of the roof?
38.
40.
42.
44.
46.
The area of a rectangle is (length) x (width), let X be the width of the roof 4X2 =
784
X2 = 196 X = 14.
The width of the roof is 14 and the length is 56. The difference is (56-14 = 42).
29. The length of a cube is three times its width and half of its height. If the volume of
the
Cube is 13,122 Cm cubed. What is the height of the cube?
49.
50.
54.
68.
81.
Normalize each dimension to the width of the cube (W).
The length is 3 times the width, therefore its 3W, which is half of the height (6W).
The volume of the cube is 13,122 = 6W x 3W x W = 18W3 W3 = 729 W = 9.
The height of the cube is six times the width, therefore its 54 meters.
30. The width of a cube is half the length and one third of the height. If the length of
the cube is 4 meters, what is the volume of three identical cubes?
96.
88.
74.
68.
62.
Normalize all the dimensions to the width. Let the width be X.
The length is twice the width, thus 2X.
The height is 3 times the width, thus 3X.
The volume of the cube is X ⋅ 2 X ⋅ 3 X = 6X3.
The length is equal to 4
2X = 4 X = 2 Volume = 6 x 8 = 48.
The volume of two cubes will be 96.
1. Two brothers took the GMAT exam, the higher score is X and the lower one is Y.
If the difference between the two scores is equal to their average, what is the value of
Y/X ?
(a) 3.
(b) 2.
(c) ½.
(d) 1/3.
(e) There isn’t enough data to answer the question.
If the difference is equal to the average, then we could write the equation: X – Y = (X
+Y)/2.
X – 3Y = 0 Y/X = 1/3.
2. Two people measure each other’s height, the height of the taller person is H and the
height of the other person is L. If the differences in their height is equal to their
average height, what is the Value of H/L ?
1/3.
½.
2.
3.
6.
If the difference is equal to the average, then we could write the equation: H – L =
(H+L)/2.
H – 3L = 0 H/L = 3.
8. If building X is less than 40 store’s high, is building Y taller than X?
(1) Building Y is at least three times as high as building X.
(2) On the fortieth floor of the Y building there is a gift shop.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Statement (1) tells us clearly that the Y building is taller than the X one.
Statement (2) implies that there is a gift shop on the 40’Th floor; therefore there are at
least 40 floors on the Y building, which make it taller than X.
Both statements, by themselves, are sufficient enough to answer the question.
9. There are two major statues in Tasmanian County; the first is no more than 45
meters high. How tall is the second statue?
(1) The second statue is 10 meters higher than the first statue.
(2) Both statues together are 80 meters high.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
The information on the first statue in the question is confusing and irrelevant.
Statement (1) tells us that: B = A + 10 (A is the first and B is the second statue).
Statement (2) tells us that: A + B = 80, therefore we have two equations with two
variables and so we can solve the problem.
Therefore, both statements are required in order to answer the question.
10. Tower X is smaller than tower Z. Is tower Y bigger than tower X?
(1) Tower Z higher than tower Y.
(2) Tower Y is one of the tallest in the world.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
We can write the data that is given to us: X < Z.
From statement (1) we can learn that: Y < Z also, this is not enough.
From statement (2) we know that Y is very tall, one of the highest in the world, but X
can still be higher. Therefore, more sufficient data is required to answer the question.
11. How many steaks did the restaurant sell between 20:00 P.M and 21:00 P.M on
Wednesday?
(1) On Tuesday the restaurant sold 25 steaks between the hours of 20:00 P.M and
21:00 P.M.
(2) The average amount of steaks that are sold on Wednesdays is 25 steaks per hour.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Both statements do not provide us with any vital information about the specific
number of steaks that were sold on that specific hour. The average is not accurate
enough for the question and the sales of Tuesdays could be different than those in
Wednesdays. Therefore, more sufficient data is required.
12. How many bananas did Jerry the monkey eat today?
(1) Today, Jerry ate 30% more than his regular average.
(2) Yesterday, Jerry ate 32 bananas, which is 15% less than his regular average.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From both statements together we know the average amount of bananas eaten by Jerry
and that today Jerry ate 30% more than his regular average. Therefore, both
statements together supply sufficient information to answer the question.
13. A student is studying for a test from 11:00 A.M to 20:00 P.M on weekdays and
one third of that time on Saturdays, on Sundays he takes a break from school and goes
fishing. For what fractional part of the entire week is the student studying?
(a) 2/5.
(b) 3/7.
(c) 2/7.
(d) 3/8.
(e) 5/2.
On weekdays, the student studies for 9 hours, on Saturdays 3 hours.
In one week the student studies for (5 x 9 + 3 = 48).
The total number of hours in one week is (24 x 7 = 168).
48 hours / 168 total hours = 2/7.
14. An average teenager sleeps from 23:00 P.M till 7:30 in the morning of a schools
night and on the weekend, which is Friday and Saturdays he sleeps 50% more each
night. For what fraction part of the entire week is the teenager asleep?
(a) 48.75/168.
(b) 52.5/120.
(c) 17/42.
(d) 55.5/168.
(e) 15/38.
In one week there are (24 x 7 =168) hours.
The average teenager sleeps 8.5 hours on weekdays and (1.5 x 8.5 = 12.75) hours on
the weekend.
Altogether, the total sleeping time per one week is (8.5 x 5 + 12.75 x 2 = 68).
The fractional part of the sleep is (68/168 = 17/42.
15. The number 12 bus is working from 6:00 in the morning to 10:00 P.M on
weekdays only. On Saturdays and Sundays the bus goes to the garage 3 times a day
for upgrading, each time for 3 hours. For what fraction part of the entire week is the
(a) 5/12.
(b) 4/7.
(c) 7/9.
(d) 6/11.
(e) 7/12.
In one week there are (24 x 7 =168) hours.
The bus is on the road everyday, on weekdays he is 16 hours on the road and on
Saturdays and Sundays he drives for 9 hours each day. The total amount of hours is
(16 x 5 + 18 = 98). The fraction part of the week is (98/168 = 7/12).
16. Lilac has three times more Barbie dolls than Orly. If Lilac gives 6 dolls to Nirit,
she would have 21 dolls left. How many dolls does Orly have?
(a) 5.
(b) 7.
(c) 9.
(d) 11.
(e) 13.
The easiest way to solve such problems is by back-solving it.
Take answer C: If Orly has 9 dolls then Lilac has 27. If Lilac would give away 6 dolls
she would really have 21 dolls left.
17. "Tires R' us" have 4 times more tires for sale than any regular tire shop. If "Tires
R' Us" Sells 122 tires, they will have only three times more tires than the rest.
How many more tires do "Tires R' us" have than the regular tire shop?
(a) 488.
(b) 388.
(c) 366.
(d) 299.
(e) 188.
Let's write the equations that come from the data: define x as the number of tires of
"Tires R' us" and y as number of tires of the regular shop.
X = 4Y and X – 122 = 3Y X= 488, Y=122. X – Y = 366.
18. If 512 = (A + 48)3, then (A + 53) is equal to
13.
-10.
15.
-12.
5.
If 512 = (A + 48)3 A + 48 = 8
Therefore, A + 53 = 13.
A = -40.
20. If 529 = (Y – 7)2, then Y/3 is equal to
6.
8.
9.
10.
12.
529 = (Y – 7)2, take the root out of both sides to get 23 = Y – 7
Therefore Y/3 is equal to 10.
Y = 30.
21. The East-17 pre-school is upgrading all of his classrooms by buying 46
computers, 6 printers and 5 fax machines. If a computer costs 4 times more than a
printer and 2 times more than the fax machine, what percent of the cost of the entire
purchase was the cost of one computer, 2 printers and 1 fax machine?
1%.
2%.
3%.
4%.
5%.
Let's define the price of a printer as X, the computer costs 4X and the fax costs 2X.
The total price of all the merchandise is (46 x 4)X + 6X + 10X = 200X.
The specific group that was asked upon is worth 4X + 2X + 2X = 8X.
The percentage of the price is (8/200) 4%.
22. A newly wed couple is designing their new house by purchasing 10 chairs, 3
desks, 3 televisions and 4 closets. If the price ratio between the new merchandise is
1 : 3 : 4 : 4, what fraction of the cost of the entire purchase was the cost of a desk and
a closet?
6/31.
7/47.
5/28.
9/43.
11/45.
Let X be the price of a single chair, normalize all the prices to the price of a chair.
The total price of the entire purchase is: 10X + 9X + 12X + 16X = 47X.
The specific purchase costs: 3X + 4X = 7X.
The percent required is 7/47.
23. A grocery store ordered a delivery of fresh milk products that contained 45 milk
bottles, 24 cheese packs and 23 cartons of chocolate milk. If the chocolate milk carton
costs like a bottle of milk, which is three times the price of a cheese pack, what
fraction of the cost of the entire purchase was the cost of 20 bottles of milk, 1 pack of
cheese and 5 chocolate milk cartons?
1/6.
2/5.
1/4.
1/3.
3/7.
Let X be the price of a pack of cheese.
The price of the entire purchase is (45 x 3)X + 24X + (23 x 3)X = 228X.
The specific required purchase is 60X + X + 15X = 76X.
The fraction of the specific purchase and the entire purchase is 76/228 = 1/3.
26. If A and B are two prime numbers bigger than 2, which of the following can't be
true?
(1) A + B is an even number.
(2) A x B is also a prime number.
(3) AB is also a prime number.
1 only.
2 only.
3 only.
2 and 3 only.
1, 2 and 3.
Try to find opposing examples to the statements.
A+B = 20, this is an even number.
1. Pick A=7, B=13
2. Try A=3, B=7 AB=21, and 21 is not a prime number.
3. Try A=3, B=5 AB = 35= 3 x 3 x 3 x 3 x 3 = a number which is dividable by 3
and 9 and therefore is not a prime number. Statements 2 and 3 can't be true, the
27. If X and Y are consecutive numbers (Y>X), which of the following statements
could be false?
1. The multiplication XY is an even number.
2. (Y/X) > 1.
3. (X + Y) is always an odd number.
1 only.
2 only.
1 and 2 only.
1 and 3 only.
1, 2 and 3.
Let's go over each statesman's at a time.
1, the multiplication of two consecutive numbers is always an even number.
In this case 3/2 is bigger than 1 but if you choose x=-3, y=-2, y/x
2, Try x=2, y=3
is 2/3 and that’s smaller than 1. Therefore this statement is not always true.
3, the sum of two consecutive numbers is always odd, try some numbers.
Therefore, only statement 2 is false.
28. X and Y are integers, X is even and negative, Y is odd and positive. Which of the
following could be false?
1. (X + Y) is an odd number.
2. Y(X + Y) is an integer.
3. XY is a positive number.
2 only.
3 only.
1 and 3 only.
2 and 3 only.
1, 2 and 3.
Find opposing examples for each of the statements.
1: An odd number + an even number are always an odd number.
2: Try the numbers: X=-4, Y=3 3(-1) = 1/3 which is not an integer.
XY = (-2)3 = -8, which is not positive.
3: Try the numbers: X=-2, Y=3
Statements 2 and 3 are not necessarily true.
If W is between (-1) and (0), which of the following is least?
W.
W2.
1/W.
(1/W)2.
0.
Try a number between –1 and 0, for example –1/2.
–1/2.
¼.
–2.
4.
0.
The smallest number is –2.
2. If R is between (-2) and (2), which of the following can be the greatest?
(1/R).
R2.
R3.
R4.
32.
If we can choose any number between (-2) and (2), choose a really small number, for
example
(1/100) plug it in answer (a), (1/1/100) = 100 none of the other answers can be
larger.
3. If all values of X such as (0 < X < 1), which can be the least?
0.
X.
X2.
(X – 1)2.
X3.
Since X is a fraction between 0 and 1, the smallest answer will be that of the highest
power, since all the answers are positive. Therefore (e) is the smallest possible.
Pay attention to answer (d), it’s the same as (c); it’s a fraction between 0 and 1
squared.
4. By what percent did the price of a pound of plum increase?
(1) Each pound of plum costs 28 cents more.
(2) The original price per pound was 52 cents.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
By using both statements together we know what was the original price and by how
much it increased.
Each statement alone is not sufficient, but the combination of the two is sufficient.
Remember, you don’t have to solve the problem; you only need to make sure you can.
5. What is the new price for a pound of persimmon?
(1) The old price is 45 cents per pound.
(2) The new price is more than half of the old price.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
We need to find the exact price of the persimmon.
If the old price is 45 cents and the new price is more than half of the old price than the
exact price of the new persimmon is not an exact price but a price bigger than 22.5
cents.
There isn’t enough sufficient data to solve the question.
6. How many days would it take two carpenters, working together, to build 5 desks?
(1) Each carpenter can build 4 desks in two days.
(2) Two carpenters, working together, work twice as fast as one carpenter working
alone.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
In order to solve the question, we need to know the output of one carpenter.
From statement (1) we can learn that one carpenter has a certain output, and using the
output formula we can calculate the desired time. Statement (2) is not useful; it tells
us something that we can already assume by ourselves.
7. How many diamond rings can a goldsmith refine in two weeks?
(1) There are 4 diamonds in each ring.
(2) One goldsmith can refine 2 diamonds in 4 hours.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Both statements combined are sufficient to answer the question.
Statement (1) tells us how many diamonds are there in one ring.
Statement (2) tells us how much time is needed to complete one ring; therefore both
data’s are sufficient to answer the question.
8. How long will it take 5 chambermaids to arrange the beds of the entire hotel?
(1) There are 4 floors in the hotel.
(2) Each floor contains 6 rooms.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
In order to answer the question, we need to know how many beds are there in the
hotel.
Both statements, taken together, don’t supply that kind of information.
Therefore more sufficient data is required.
9. How many patients can a group of dermatologists diagnose in one day?
(1) Two dermatologists can diagnose 3 patients in 1.5 hours.
(2) Dermatologists work for 8 hours a day.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Both statements are not sufficient, you don’t know what the size of the group of
dermatologists is; it can be 3 doctors or even 45. Therefore more sufficient data is
required.
11. How long will it take Jimmy to organize his stamp collection?
(1) Jimmy can organize 45 stamps in 2 hours.
(2) In the collection there are 30 regular stamps and 90 special ones.
a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Both statement combined are sufficient.
Statement (1) tells us that what is the “output” of organizing stamps and statement (2)
tells us how many apples are there in the collection. Pay attention, it doesn’t matter if
the stamps are regular or special ones because statement (1) states that (any) 45
stamps can be arranged in 2 hours.
15. A train traveled for three hours. In the first hour the train traveled 86 miles, which
was 25% farther than it traveled in the first hour. In the third hour the train traveled at
a speed of 120 miles per hour for 20 minutes. What is the total distance that the train
traveled?
190.6.
194.8.
198.2.
204.5.
212.8.
In the first hour it traveled 86 miles.
In the second hour it traveled x miles, x + 0.25x = 86
In the third hour it traveled (120 x 1/3 = 40) miles.
The total distance is (86 + 68.6 + 40 = 194.8) miles.
x= 68.8 miles.
16. A cruise ship traveled for 3 hours. In the first hour, the ship sailed at a speed of 25
Km/h, which was 25% faster than the speed in the third hour. In the middle hour the
ship sailed at the average speed of the first and third hours. What was the total istance
of the ship during the 3 hours of sailing?
65.
66.5.
67.5.
70.
72.5.
The distance in the first hour is 25 Km.
The speed in the third hour is (25/1.25 = 20 Km/h) and therefore the distance is 20
Km.
The average speed is 22.2 Km/h and that is also the distance.
The total distance is 25 + 20 + 22.5 = 67.5 Km.
17. Two cars are driving towards one another. The first car is traveling at a speed of
120 Km/h, which is 28% faster than the second car. If the distance between the cars is
855 Km, how long will it takes the cars to meet (in hours)?
2.5.
3.
3.5.
4.
4.5.
The speed of the second car is X, (X + 0.28X = 120) X = 93.75 Km/h.
In order to find the time it will take the cars to meet, you should divide the total
distance by the sum of the car's speeds: (855 / (120+93.75) = 4). Therefore the answer
is D, four hours.
18. Three cars are starting to drive from three corners of a huge axi-cimetrical triangle
towards the middle. Car A can travel at a speed of 110 miles per hour, car B can travel
10% less and car C can travel the average speed of the first two cars. If all cars leave
at the same time and it took car A 30 minutes to get to the middle, how long
approximately after car C reached the middle, did car B reached it?
They reached the middle at the same time.
Car A can travel at 110 mp/h.
Car B can travel at a speed of (110 x 0.9) 99 mp/h.
Car C can travel at a speed of (110 + 99)/2 = 104.5 mp/h.
If it took car A 30 minutes, the length towards the middle of the triangle is 55 miles.
Calculate how long it takes car B and car C travel 55 miles:
Car B It will take her (55/99) hours which is 33 minutes and a third.
Car C It will take her (55/104.5) hours which is approximately 31.6 minutes.
Therefore the differences in the time will be approximately 1.5 minutes.
19. Danny and Steve are running towards each other, each one from his own house.
Danny can reach Steve's house in 25 minutes of running, which is half the time it
takes Steve to reach Danny's house. If the two started to run at the same time, how
much time longer will it take Steve to reach the middle than Danny?
12.5 minutes.
25 minutes.
35 minutes.
50 minutes.
75 minutes.
If it takes Danny 25 minutes to travel the full way, it would take him 12.5 minutes to
reach the middle.
If it takes Steve 50 minutes to travel the full way, it would take him 25 minutes to
reach the middle.
The difference in the time it takes them to reach the middle is 12.5 minutes.
20. A green lizard can travel from the green cave to the blue cave in 108 minutes; the
blue lizard can travel from the blue cave to the green cave in 25% less time. If the
green lizard started to travel 7.5 minutes before the blue lizard, how many minutes
after the blue lizard, will the green lizard pass the middle line?
2.
3.5.
4.
5.5.
6.
It takes the green lizard (108/2 = 54) minutes to reach the middle.
It takes the blue lizard (108 x 0.75 = 81)/2 = 40.5 minutes to reach the middle.
The green lizard started to travel 7.5 minutes ahead of the blue lizard and therefore the
differences in the time they reach the middle is (54 – 7.5 – 40.5 = 6) minutes.
21. When an integer Q is divided by 6, the remainder is 4.
Which of the following is not a multiple of 6?
Q + 2.
Q – 4.
2Q + 6.
3Q.
4Q + 2.
Pick a number that fits into the question, for example 10.
The only answer that is not a multiple of 6 is C, 2Q + 6 = 26. And 26 are not divisible
by 6.
22. When the integer Y is divided by 11, the remainder is 3.
Which of the following can't be a multiple of (Y+1)?
2Y + 2.
1.5Y + 9.
2.5Y – 5.
3Y + 3.
3Y – 5.
Plug in a number that will give a remainder of 3 when divided by 11, for example 14.
We are looking for a number that is not a factor of (Y+1 = 15).
The only answer that is not a factor of 15 is E. 3Y – 5 = 37.
23. When the integer X is divided by 5, the remainder is 2.
Which of the following can be a multiple of (X+3)?
8X + 7.
9X – 3.
11X – 4.
13X.
13X + 1.
Plug in a number that will give a remainder of 2 when divided by 5, for example 7.
We are looking for a number that is a factor of 10.
The only possible answer is B 9 x 7 – 3 = 60, and that is a factor of 10.
25. What is the value of X + Y?
(1) X = 3Z.
(2) Y = 8Z.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
According to the statements, the value of X + Y is 3Z + 8Z = 11Z, but we don’t know
what's the value of Z. In other words, more sufficient data is required to answer the
question.
26. What is the value of (Q + W)?
W = 3R.
(2) Q = -3R.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Combine both statements to get, W + Q = 3R – 3R = 0.
In other words the statements tell us that W = (-Q) or the opposite.
27. What is the value of (XY)?
(1) X = 2/(9Q).
(2) Y = 4.5Q.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
Combine both statements to get, XY = 4.5Q x 2/(9Q) = 1.
Therefore both statements taken together are sufficient.
28. Harris invested \$45,000 in two different ventures, a car-cleaning machine and a
video mat. The yearly return on the video mat was 12% and the yearly return on the
car-cleaning machine was 8%. If the total return was \$4,000, how much did Harris
invest in the video mat?
\$8,000.
\$10,000.
\$14,000.
\$22,000.
\$35,000.
The easiest way is to back solve the question.
Take answer B, if that is the amount Harris invested in the video mat; the annual
return was \$1,200.
Therefore there are \$35,000 left to invest in the car-cleaning machine, 8% of 35,000 is
\$2,800.
Sum them up; the total return is like the question asked- \$4,000.
29. Jennifer bought two apartments in order to rent them to other people with
\$300,000. The monthly return on the first apartment is 1.5% of its value and on the
second apartment the return is 2% of its value. If the total returns of the entire year
were \$61,200, how much did Jennifer spent on the second apartment?
\$100,000
\$120,000
\$150,000
\$180,000
\$210,000
The easiest way is to back solve the question.
Take answer B, if that is the amount Jennifer invested in the second apartment; the
annual return from that apartment was (120,000 x 0.24 = 28,800). Therefore there are
\$180,000 left to invest in the first apartment, 18% of \$180,000 is \$32,400.
Sum them up; the total return is like the question asked- \$61,200.
3. Two cars are traveling on the same road towards each other. If car A is traveling at
a speed of 120 Km/h and car B is traveling 15% slower, how much time will it take
the cars to meet if the initial distance between the two is 668.4 Km and car A started
to drive one hour and a half before car B started?
(a) one hours and 30 minutes.
(b) two hours.
(c) two hours and 12 minutes.
(d) three hours and 15 minutes.
(e) three hours and 18 minutes.
Car B is traveling at a speed of 0.85 x 120 = 102 Km/h.
Car A travels alone a distance of 120 x 1.5 = 180 Km. The remaining distance should
be divided among the sum of the cars speeds: (668.4 – 180 = 488.4 Km) / (102 + 120)
= 2.2 = Two hours and 12 minutes.
5. Water has been poured into an empty rectangular tank at the rate of 8 cubic feet per
minute for 2.5 minutes. The length of the tank is 3 feet and the width is one half of the
length. Approximately how deep is the water in the tank?
(a) 3.23 feet
(b) 3.86 feet
(c) 4 feet
(d) 4.25 feet
(e) 4.44 feet
First calculate the volume of water that has been poured into the tank. If it has been
poured at a rate of 8 cubic feet per minute for 2.5 minutes, 8 × 2.5 = 20 cubic feet.
The tank is rectangular, so its volume is length × width × height (or depth), with the
answer in cubic units. We are given the length, and can calculate the width, since we
are told that the width is 1/2 the length, or 1/2 of 3 feet, or 1.5 feet. The volume we
have already calculated to be 20 cubic feet. Therefore, 20 = length × width × height,
or 20 = 3 feet × 1.5 feet × the height. Solving for the height, we get 40/9 , or
approximately 4.44 feet.
15. Roy is now 4 years older than Erik and half of that amount older than Iris. If in 2
years, Roy will be twice as old as Erik, then in 2 years what would be Erik’s age
multiplied by Iris’s age?
(a) 8
(b) 28
(c) 48
(d) 50
(e) 52
Translate piece by piece into numbers. R (Roy) = Erik (E) + 4.
The second equation: R = I (Iris)+ 2.
The third equation: R +7 = 2(E + 7). We have three equations with three variables.
Roy is 6, Iris is 4 and Erik is 2. In four years Erik would be 6 and Iris 8, the answer
is 48.
20. An investment yields an interest payment of \$228 each month. If the simple
annual interest rate is 9%, what is the amount of the investment?
(a) \$28,300
(b) \$30,400
(c) \$31,300
(d) \$32,500
(e) \$35,100
Principal × percent interest = interest earned
Principle × (0.09)× 1/12 = \$228.
Solve for the principal (228 × 12)/.09= \$30,400.
x, y, z, and w are integers. The expression x-y-z is even and the
Expression y-z-w is odd. If x is even what must be true?
y-z must be odd.
w must be even.
w must be odd.
z must be even.
Z must be odd
The first expression is even and the second is odd, the differences between the two
expressions is x instead of w. (remember, there is no difference in odd/even numbers
if the number is positive or negative so y-z is like z-y). Therefore if x is even w must
be odd.
5. X is a prime number bigger than 10. Also, Y = X+X3+X5+X7 .
What is definitely true about Y?
Y is a prime number.
Y is odd.
Y is even.
Y can be divided equally by 3.
Y can be divided equally by 7.
Because X is a prime number bigger than 10, he must be odd. Ignoring the powers of
X in the expression of Y, you’ll see that Y is a sum of 4 odd numbers therefore it must
be even.
12. In Tukitu village, one forth of the people are raising flowers, one ninth are
growing wheat and one eleventh are going bankrupt. What could be the number of
people in the village?
792.
540.
198.
132.
346.
The answer must be a number that is divisible equally by 4, 9 and 11.
The only possible answer is A.
17. Q is a prime number bigger than 10. What is the smallest positive number (except
1) that 3Q can be divided by equally?
3Q.
Q.
3.
Q+3.
2Q.
3Q is a prime number so it can be divide equally by 3Q, by 1 and by the components
3 and Q. The smallest number therefore is 3.
27. 352 - 342 =?
35 – 34.
35 + 34.
352.
2 x 35 x 34.
34.
352 - 342 = (35 – 34)(35 +34) = 1(35 + 34). B is the answer.
18. Kelly used to get a 30% discount on movie tickets. When the price of the movie
ticket increased by 50%, the amount of discount in dollars remained the same. What is
Kelly's discount with the new Ticket price in percent terms?
(a) 10%
(b) 20%
(c) 25%
(d) 35%
(e) 38%
The price of the ticket is unknown. Take 100 as an exapmle.
30% discount of 100 is \$30, that amount stayed the same after the price of a ticket
grew by 50%.
The new price of a ticket is \$150, so 30/150 is 20%.
1. Tom divided his cards between Tim and Din so each one received an odd amount
of cards. The number of cards that Tim received multiplied by the number of cards
that Din received is a number larger than 49 and smaller than 121. How many cards
did Tom have in the first place?
16.
22.
18.
14.
32.
Answers A and E are disqualified immediately because those are even numbers that
cannot be divided into two odd numbers. 22 is 11 + 11 but
11 x 11 is bigger than 121, the same idea with 14, therefore the answer is 18. 18 = 9
+ 9. 9 x 9 = 81.
10. In the beginning of the season, the owner of a football team bought T players for
the price of 4R each. At the end of the season the owner sold the players in a total
profit of X. How much did the owner get for all the players?
X – 4TR.
4X + 4TR.
4TR + X.
4(TR – X).
4TR – X.
The owner bought T player that cost him altogether 4TR.
He had a profit of X so he sold them for 4TR + X.
18. A bird is flying from an oak tree to a pine tree in a speed of 6 Km/h.
On her way back, she flew at a speed of 4 Km/h, thus, the trip lasted 4 hours more.
What is the distance between the trees? (In Km)
12.
24.
36.
48.
52.
The distance to the pine tree is 6 x X, where X represents the time of the trip. The
distance back to the oak tree is 4(X+4), assuming the trip back is equal in length.
X = 8. The length of the trip is 8 x 6 = 48 Km.
Therefore 6X = 4(X+4)
13. In a chocolate store, there are vanilla and chocolate flavor bon-bons only.
10% of the bon-bons are chocolate flavored, 90% of the rest are squashed.
What percentage of the bon-bons is vanilla flavored that are not squashed?
(a) 1%
(b) 5%
(c) 9%
(d) 10%
(e) 2%
Pick a number of bon-bons, like 100 for example.
10 are chocolate, 90% of the rest (0.9 x 90 = 81) are squashed.
That means that only 9 are vanilla and are not squashed.
23. A credit card number has 6 digits (between 1 to 9). The first two digits are 12 in
that order, the third digit is bigger than 6, the forth one can be equally divided by 3
and the fifth digit is 3 times bigger than the sixth one. How many credit cards can be
(a) 27.
(b) 36.
(c) 72.
(d) 112.
(e) 422.
First digit is 1, the second is 2, the third can be (7,8,9), the forth can be (0,3,6,9), the
fifth and the sixth are dependent with one another. The fifth one is 3 times bigger than
the sixth one, therefore there are only 3 options there: (1,3), (2,6), (3,9).
All together there are: 1 x 1 x 3 x 4 x 3 = 36 options.
25. Out of a box that contains 4 black mice and 6 white ones, three are picked up.
What is the probability that all three will be black mice?
(a) 8/125.
(b) 1/30.
(c) 2/5.
(d) 1/720.
(e) 3/10.
The probability for the first one to be black is: 4/(4+6) = 2/5.
The probability for the second one to be black is: 3/(3+6) = 1/3.
The probability for the third one to be black is: 2/(2+6) = 1/4.
The probability for all three events is (2/5) x (1/3) x (1/4) = 1/30.
28. A car is driving at 60 Km/h for 20 minutes, and then drives at 90Km/h for another
40 minutes. What is the average speed of the car?
(a) 80.
(b) 75.
(c) 70.
(d) 65.
(e) 54.
The average speed is equal to: (Total distance)/(Total time) = (60 x 1/3 + 90 x 2/3)/1
= 80 Km/h.
36. Two grandfathers can nit a sweater in 6 days. Two grandfathers and one
grandmother can nit a sweater in 3 days. How many days will it take the grandmother
to nit a sweater all by her self?
(a) 4.5.
(b) 5.
(c) 5.5.
(d) 6.
(e) 6.5.
Two grandfathers and a grandmother can nit a sweater in 3 days, therefore they can
nit 2 sweaters in 6 days. Because two grandfathers can nit 1 in 6 days then the other
sweater is done by the grandmother, she can nit 1 sweater in 6 days.
12. In a psychology school the grade of the students is determined by the following
method: At the end of the first year the grade equals to twice the age of the student.
From then on, the grade is determined by twice the age of the student plus half of his
grade from the previous year. If Joey’s grade at the end of the first year is 40, what
will be his grade at the end of the third year?
(a) 75.
(b) 62.
(c) 80.
(d) 44.
(e) 56.
From the grade 40 at the end of the first year we learn that his age is 20.
At the end of the second year, he will be 21 and his grade will be
(21 x 2 + ½ x 40 = 62).
At the end of the third year, he will be 22 and his grade will be (22 x 2 + ½ x 62 =
75).
18. In a triangle, one side is 6 Cm and another side is 9 Cm. which of the following
can be the perimeter of the triangle?
(a) 18.
(b) 25.
(c) 30.
(d) 32.
(e) 34.
The third side of the triangle is larger than 3 (The difference between the other two)
and smaller than 15 ( The sum of the other two).
The perimeter is between (6+9+3 = 18) and (6+9+15 = 30). The only answer that is in
this range is B.
24. A long rope was divided to three different parts. What is the length of the smallest
piece?
(1) The sum of the two smaller pieces is 14 inch.
(2) The sum of the two larger pieces is 22 inch.
Translate the statements into variables: Let X, Y and Z be the thee pieces of the rope,
X<Y<Z.
Statement (1) can be written as: X + Y = 14.
Statement (2) can be written as: Y + Z = 22.
In order to find the length of the smallest piece, we need another equation or data.
More data is required.
26. Fuel tanker A can fill the underground reservoir in 12 minutes. How long will it
take fuel tanker A and fuel tanker B to fill up the same reservoir together?
(1) The reservoir contains 3000 liters of fuel.
(2) Fuel tanker A alone will require the same number of hours to fill the same
reservoir.
Statement (1) is insufficient since the size of the reservoir is irrelevant.
Statement (2) is sufficient since it tells us that the second tanker has the same output
as the first one and so it will take them both half of the time it took the first tanker
alone.
28. What is the ratio between A and B?
(1) A is the sum of X, Y and Z.
(2) B is the average (arithmetic mean) of X, Y and Z.
Statement (1) tells us that A = X + Y + Z.
Statement (2) tells us that B = (X + Y + Z)/3.
Using both statements together: A/B is 3.
Both statements together are sufficient.
30. If X and Y are both integers different from zero, what is the value of (X + 2Y)?
(1) X4 = Y4.
(2) X = 5.
Don’t rush the answer, pay attention to the question carefully.
Statement (1) tells us that X and Y are equal? No, they could have different signs.
Statement (2) gives us X, which is not sufficient.
Both statements together are also insufficient since Y can be –5 or 5.
More sufficient data is required.
3. Is the square root of A an integer?
(1) The last digit of A is 8
(2) A is divisible by 6
If you square each digit {0, 1, 2,..., 8, 9}, you will see that the only possible last digits
for a square are 0, 1, 4, 5, 6 and 9. Thus, if the last digit of A is 8, A cannot be a
square. So the square root of A is not an integer. So statement (1) by itself is
sufficient. Since 36 is divisible by 6 and its square is an integer, this statement is
insufficient by itself.
7. Is the average of X consecutive numbers odd?
(1) The first number in the series is odd.
(2) The sum of the numbers is odd.
Statement (1) is insufficient by itself, take X as 2: if the first number is odd, the sum
of the two numbers is odd. Take X as 3: if the first number is odd, the sum of the three
numbers is even.
Statement (2) tells us that the sum of the numbers is odd and therefore the median
must be odd.
If the median is odd the average of these numbers is also odd because that means that
there is an even amount of even numbers and an odd amount of odd numbers.
This statement is sufficient by itself.
14. If X and Y are integers, what is the value of XY?
(1) X3 – 3X2 – 2X – 8 = 0.
(2) 4 + 3Y = 2Y + 8.
Statement (1) can be written as (X – 4)(X2 + X + 2) = 0.
The roots of this equation are one integer and two complex numbers, which you
should pay no attention to since you were told that X is an integer.
Statement (2) is a simple equation, Y = 4.
The value of the expression XY is 16.
Both statements, taken together, are sufficient to answer the question.
21. Each of the 850 local villagers in Lucia owns either a Golden Retriever or a
Bernard. How many people own both?
(1) The number of villagers who own a Golden Retriever only is 300.
(2) The number of villagers who own a Bernard only is 280.
a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by
itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by
itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question
Each one of the villagers, according to the question, has to own at least one of the two
dogs.
Statement (1) is insufficient because it says nothing about the Bernard owners.
Statement (2) is insufficient because it says nothing about the Golden Retriever
owners.
Combine the statements, all the information we need is present,(800 – 300 – 280) is
equal to the number of people who own both races of dogs.
4. How much is Y percent of X?
(1) 400% of Y is 90.
(2) X percent of Y is 25.
Statement (1) gives us data on Y only regardless to X and therefore it’s insufficient.
Statement (2) tells us how much is X percent of Y. Make up numbers, X = 25 and Y =
100.
X percent of Y is 25 but we also know how much is Y percent of X, 100 is 400% of
25 and therefore this statement is sufficient by itself.
11. What is the sum of 11 consecutive integers?
(1) The median of the 11 integers is 6.
(2) The average of the 11 integers is 6.
Statement (1) provides us with the numbers themselves- 5 on each side of 6.
From Statement (2) we know that average = sum / amount sum = 66.
Therefore either statement is sufficient to answer the question.
6. X, Y and Z are three positive integers. If Z = 2, what is their sum?
(1) X – Y = 5.
(2) 3Y + 15 = 3X.
We need to find the value of X + Y since Z is already given to us.
Statement (1) is insufficient since we need the sum of X and Y.
X – Y = 5, you can see that both
Statement (2) can be written as: 3X – 3Y = 15
statements are the same and therefore more sufficient data is required.
10. Is Y even?
(1) 2Y is even.
(2) Y2 is even.
Statement (1) by itself is insufficient because every number that is multiplied by 2
will result in an even number. Statement (2) is also insufficient by itself since
numbers like 6 fulfills this statement although it’s not even. Combine the
statements and Y must be an even number.
15. If A is a prime number, what is the value of A?
(1) 0 < A < 10.
(2) (A – 2) is divisible by 3.
24. A long rope was divided to three different parts. What is the length of the smallest
piece?
(1) The sum of the two smaller pieces is 14 inch.
(2) The sum of the two larger pieces is 22 inch.
Translate the statements into variables: Let X, Y and Z be the thee pieces of the rope,
X<Y<Z.
Statement (1) can be written as: X + Y = 14.
Statement (2) can be written as: Y + Z = 22.
In order to find the length of the smallest piece, we need another equation or data.
More data is required.
1. Two adjacent angles of a parallelogram are in the ratio of 1:3. What is the smaller
angle of the two?
(a) 30.
(b) 45.
(c) 90.
(d) 135.
(e) 180.
The sum of two adjacent angles in a parallelogram is 180. 180 divided by 4(1+3) is 45
and that is the size of the smallest among the angles.
2. Two adjacent angles of a parallelogram are in the ratio of 2:3. What is their
average size?
(a) 30.
(b) 40.
(c) 45.
(d) 90.
(e) 180.
The ratio doesn’t give us anything, two adjacent angles of a parallelogram always sum
up to 180 degrees. And 180/2 is always 90 degrees.
3. The angles of a triangle are in the ratio of 3: 2: 1. The largest angle in the triangle
is:
(a) 36.
(b) 45.
(c) 72.
(d) 90.
(e). 108.
The sum of all the angles is 180. Divide 180 by 6 (3+2+1) and we’ll get 36; this is the
size of the smallest angle. The largest angle is three times bigger, thus 36 x 3 = 108
degrees.
4. The perimeter of a circle is approximately 6.3 centimeters. The area of the same
circle is A. which of the following is true?
(a) 1 < A < 2.
(b) 2 < A < 3.
(c) 3 < A < 4.
(d) 4 < A < 5.
(e) A > 5.
The perimeter of a circle is 2 ⋅ π ⋅ R , 2 ⋅ π ⋅ R ≅ 6.3 R ≅ 1 cm.
A = π ⋅ R 2 , Therefore A is approximately pie (3.14).
8. John bought grocery products for 10 dollars using 55 coins. If John used quarters
and dimes, what is the difference between the numbers of dimes to the number of
quarters that he used?
5.
10.
15.
25.
30.
Define X as the number of dimes that John used. Just a reminder, dimes are 10 cents
coins.
The number of quarters that he used is (55 – X). We can write the following equation:
10X + 25(55 – X) = 1000. Notice that 1000 is the money he spent in cents.
Therefore (–15X = -375)
X = 25.
The number of dimes is 25 and the number of quarters is (55 – 25 = 30).
The difference between the amounts is 5 coins.
9. Rick deposited \$850 to his bank account using \$5 and \$15 dollar bills
only. If Rick came to the bank with 70 bills and left with 10 bills only,
how many \$15 dollar bills did he deposit?
5.
25.
50.
55.
65.
Rick came to the bank with 70 bills and left with 10 and therefore he
deposited 60 bills.
Define X as the number of \$5 dollar bills that he deposited and so (60 –
X) is the number of \$15 bills that he deposited.
We can write the following equation: 5X + 15(60 – X) = 850
-10X = 50
X = 5.
The number of \$15 dollar bills is (60 – 5) 55.
10. The average (arithmetic mean) of four numbers is equal to three times
the largest number. If the largest number is equal to 3, what is the sum of
the other three numbers?
28.
33.
35.
38.
42.
Let’s say the four numbers are: X, Y, Z and W.
The average of all four numbers is equal to 3 times the value of the
largest number:
(X+Y+Z+W)/4 = 3W. W is equal to 3 and therefore (X+Y+Z+3) = 3 x 9
X+Y+Z = 36-3 = 33.
11. What is the reciprocal of (AB)/(A + B)2 ?
(AB)/(A2 + 2AB + B2).
A/B + B/A + AB.
B/A + A/B + 2.
(A2 + B2)/AB.
A/B + 2AB.
The reciprocal of X is 1/X and therefore the reciprocal of (AB)/(A + B)2
is (A + B)2/AB.
Simplify the expression: (A + B)2/AB = (A2 + 2AB + B2)/AB = A/B +
B/A + 2.
12. Naomi drives to the beauty parlor in 60 minutes. On the way back,
her average speed is half the average speed as it was to the way to the
parlor. How much time will it take Naomi to travel two round trips to the
beauty parlor?
3 hours.
4 hours.
4 hours and 20 minutes.
5 hours and 50 minutes.
6 hours.
If the average speed from the beauty parlor is half of the average speed to
the parlor then the time back from the parlor is twice the time it takes her
to get to the parlor, thus 120 minutes.
The total round trip will take Naomi (60 + 120 = 180) minutes, which is 3
hours.
Two round trips will take her 6 hours.
13. It takes Tanya 50 minutes to drive to the country club. If the average
speed of the entire round trip to the club is 87.5% of the average speed on
the way to the club, how many minutes approximately will it take Tanya
to drive home from the country club?
42 minutes.
48 minutes.
52 minutes.
54 minutes.
66 minutes.
Define X as the average speed to the country club and Y as the average
speed on the way back.
The average speed of the entire round trip to the club is 87.5% of the
average speed on the way to the club: (X + Y)/2 = 0.875X
X+Y=
1.75X
Y = 0.75X.
The average speed on the way back is 75% of the speed to the club.
If the time it takes her to get to the country is T, then T/0.75 is the time it
will take to get back home.
50/0.75 = approximately 66 minutes.
14. It costs \$4 for the first ¼ hour to use the laundry machine at the
Laundromat. After the first ¼ hour it costs \$12 per hour. If a certain
customer uses the laundry machine for 3 hours and 25 minutes, how
much will it cost him?
\$25.
\$32.
\$36.
\$40.
\$42.
The customer uses the machine for 3 hours and 25 minutes. The first 15
minutes cost him \$4 and he has 3 hours and 10 minutes left, which is (12
x 3 = \$36) + 10 minutes.
10 minutes are 1/6 of an hour, which is (1/6 x 12 = \$2).
The total cost will be: 4 + 36 + 2 = \$42.
15. The mall charges 50 cents for the first hour of parking and \$3 for each
additional hour until the customer reaches 4 hours, after that the parking
fee is \$1 per hour. If a certain customer parked his in the mall for 7 hours
and 30 minutes, how much is he going to pay?
\$11.5.
\$12.
\$13.
\$14.5.
\$15.
The customer parked for 7 hours and 30 minutes. Divide the problem into
parts:
The first hour cost him 0.5 dollars. He has 6.5 hours left.
The next three hours cost him 3 dollars per hour, \$9 in total for that time.
He has (7.5 – 4 = 3.5 hours left) at a wage of \$1 per hour, it sums up to
Sum it all up: 0.5 + 9 + 3.5 = \$13.
16. If ( 0 < X < Y ), X is an odd number and Y is a prime number, which
of the following can be the value of X + Y ?
11.
13.
17.
10.
7.
Every prime number except 2 is an odd number. If Y = 2, than X must
have been 1 because X is smaller than Y but this answer doesn’t appear
among the answers and therefore Y is odd.
If X and Y are both odd numbers, their sum must be an even number. The
17. It takes Avery 3 hours to build a brick wall while Tom can do it in 2.5
hours. If the two start working together and after an hour Avery leaves,
how much time will it take Tom to complete the wall on his own?
25 minutes.
30 minutes.
40 minutes.
55 minutes.
1 hour and 20 minutes.
The output of Avery is 1/3 walls in one hour and the output of Tom is 2/5
walls in one hour.
The two worked together for one hour, their combined output is (1/3 +
2/5 = 11/15) wall and that’s the fraction of the wall that they completed
together.
Tom has (1 – 11/15 = 4/15) wall left, with his current output it will take
him ((4/15) / (2/5) = 2/3) hours, which is 40 minutes.
18. There are three foam generators in the factory, the first two can
generate 14 liters of foam in one hour and the third can generate 18 liters
in an hour. The three generators start working together at the same time
and after one hour and a half one of the first generators stops working and
two hours after that the third generator stops working and only one
generator is left. If 5 hours after they all started to work the last generator
stops working, how many liters of foam were generated?
120.
132.
146.
154.
166.
In the first hour and a half all the generators worked and produced (14 +
14 + 18) x (1.5) = (69) liters.
In the next two hours two generators worked and produced (14 + 18) x
(2) = (64) liters.
The rest of the time (5 – 1.5 – 2 = 1.5 hours) only one generator worked
and produced (14 x 1.5 = 21 liters of foam).
The total amount of foam that was created is (69 + 32 + 21 = 154 liters).
19. Mike, Jim and Bob are all professional fisherman. Mike can catch 21
fish in one hour, Jim can catch twice as much and Bob can catch 50%
more than Jim. If the three started to fish together and after 40 minutes
Mike and Bob left, how many fish did the three fishermen catch in one
hour?
64.
72.
86.
98.
112.
Mike can catch 21 fish in one hour, Jim can catch 42 fish in one hour and
Bob can catch
(1.5 x 42 = 63 fish) in one hour. 40 minutes is 2/3 of an hour.
After 2/3 hours they all caught (21 + 42 + 63) x (2/3) = 84 fish.
In the next 1/3 hour that’s left, Jim is left alone and so he can catch (1/3 x
42 = 14) fish.
The total number of fish that they caught is (84 + 14 = 98).
20. A certain church bell rings the bell twice at half past the hour and four
times at the hour plus an additional number of rings equal to what ever
time it is. How many rings will the clock make from 6:20 in the morning
to 10:10 in the morning?
32.
36.
42.
46.
50.
Let’s start from the top. The first two rings will be at 6:30 from there
there’ll be 2 rings ever half past the hour until ten, at 7:30, 8:30 and 9:30.
That will sum up to 8 rings total.
The bell will also ring at 7:00 8 rings, at 8:00 9 rings, at 9:00 10 rings and
at 10 11 rings.
The total number of rings is: 8 + 8 + 9 + 10 + 11 = 46 rings.
21. A 75-liter solution of cool-drink is made from 8% jasmine water. If 3
liters of jasmine and 12 liters of water were added to the solution, what
percent of the solution is jasmine?
10.3%.
11.5%.
10%.
12.2%.
12%.
8% of the solution is made of jasmine, 8% of 70 is 6 liters.
If 3 liters of jasmine and 12 liters of water are added, the amount of the
jasmine is 9 liters and the percent of the jasmine out of the entire solution
is ((9)/(75 + 15) = (9/90) = 10%.
22. A 340-liter solution of Kola is made from 88% water, 5%
oncentrated Kola and the rest is made from sugar. If 3.2 liters of sugar, 10
liter of water and 6.8 liters of concentrated Kola were added to the
solution, what percent of the solution is made from sugar?
6%.
7.5%.
9.2%.
10.5%.
11%.
(100% - 88% - 5% = 7%) of the solution is made from sugar, which is
(0.07 x 340 = 23.8 liters).
3.2 liters of sugar were added to the solution, so there are 27 liters of
sugar in the solution.
The total volume of the solution is: 340 + 3.2 + 10 + 6.8 = 360 liters.
(27 / 360 = 3/40), which is 7.5% percent.
1. If Peter spends 460 lirettas on three pairs of shoes, how much did the
least expansive shoes cost?
(1) The ratio between the most expansive shoes to the least expansive
shoes is 3 to 1.
(2) The ratio between the least expansive shoes to all the other ones is 1
to 5.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
answer the question, requiring more data pertaining to the problem.
We are to find what is the price of the least expansive among three shoes.
Statement (1) gives us the ratio between the most expansive shoes and the
least expansive one but that’s not sufficient because the median is not
unequivocal and it could be anything.
Statement (2) is sufficient. If the ratio between the least expansive to all
the shoes is 1 to 5 than we can calculate the price of the least expansive
shoes, (460/6).
Therefore statement (2) is sufficient by itself.
2. What is the average price of three different DVD’s?
(1) The first DVD costs 125 dollars.
(2) The sum of the prices of the second and the third DVD’s is 300
dollars.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
answer the question, requiring more data pertaining to the problem.
Statement (1) gives us information about the first DVD only and
therefore it’s not sufficient.
Statement (2) gives us information only about two of the DVD’s and
therefore this statement by itself is not sufficient either.
Combining both statements we have enough data to calculate the average.
If X, Y and Z are the prices of the three DVD’s, statement (1) tells us the
value of X and statement (2) the value of the other two so we can
calculate the average: (X+Y+Z)/3.
Therefore, the two statements combined are sufficient.
3. What percent of 20 is Y?
(1) 50 percent of Y is 5.
(2) Y percent of 200 is 20.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
answer the question, requiring more data pertaining to the problem.
The question, in other words, is asking what is the value of (Y/20 x 100).
So all we need to find out is if the value of Y is known. Statement (1)
gives us Y explicitly, its 10.
Statement (2) is also sufficient, it tells us that Y is also equal to 10.
Both statements are sufficient by themselves and the answer is D.
4. How much is Y percent of X?
(1) 400% of Y is 90.
(2) X percent of Y is 25.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
answer the question, requiring more data pertaining to the problem.
Statement (1) gives us data on Y only regardless to X and therefore it’s
insufficient.
Statement (2) tells us how much is X percent of Y. Make up numbers, X
= 25 and Y = 100.
X percent of Y is 25 but we also know how much is Y percent of X, 100
is 400% of 25 and therefore this statement is sufficient by itself.
5. Did it take Reese more than 24 minutes to bake the carrot cake?
(1) Reese spends more than 1260 seconds on the cake.
(2) Reese spends less than 1560 seconds on the cake.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
answer the question, requiring more data pertaining to the problem.
Define X as the time Reese spends on the cake.
Statement (1) tell us that X > 1260 seconds, which is 21 minutes and so
this statement is insufficient because X can be 22 or even 28 and we
cannot determine for sure that X is smaller or bigger than 24.
Statement (2) tells us that X < 26 and its not sufficient because X can be
either 25 or 21.
Combining both statements wouldn’t help (21 < X < 26) because X can
be bigger than 24 (25) or smaller (22).
6. Did Sammy drive more than 21 kilometers last night?
(1) Sammy drove more than 20,000 meters last night.
(2) Sammy drove less than 20,500 meters last night.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
answer the question, requiring more data pertaining to the problem.
Define Y as the number of kilometers that Sammy drove last night. We
are asked if Y > 21.
Statement (1) tells us that Y > 20 kilometers which is not sufficient by
itself because Y can be smaller or bigger than 21 kilometers.
Statement (2) tells us that the number of kilometers that Sammy drove is
less than 20.5 and therefore this statement is sufficient by itself.
8. One person won the lottery this week, what is the probability that it
was a woman over the age of 40?
(1) 55% of all the lottery participants are male.
(2) 60% of all the lottery participants are over the age of 40.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
answer the question, requiring more data pertaining to the problem.
We need to know what is the probability that one of the lottery
participants is a woman over the age of 40. Statement (1) tells us that
45% of the participants are woman and statement (2) tells us that 60% of
the participants are over the age of 40. Combine both statements, we still
9. If a kid is chosen randomly from his class, what is the probability that
he would have blue eyes?
(1) The class is in Denmark, where 95% of the population has blue
eyes.
(2) 5% of the class has brownish eyes.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
answer the question, requiring more data pertaining to the problem.
We need to know the ratio between the numbers of blue-eyed kids in the
class to the others.
Statement (1) gives us data about the average population but not
specifically on the class.
Statement (2) implies that 5% have brown eyes, but that doesn’t
necessarily mean that others have blue eyes, they could have green eyes
as well.
More sufficient data is required.
10. In a bulb factory there are different kinds of bulbs, what is the
probability that a bulb chosen randomly is a halogen?
(1) There are three times as many halogens than any other bulb in the
factory.
(2) The ratio between the halogen to all the other bulbs is 2 to 7.
(a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
(b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
(c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
(d)Either statement BY ITSELF is sufficient to answer the question.
(e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
answer the question, requiring more data pertaining to the problem.
We are asked what is the ratio between the halogen to the not.
Statement (1) tells us that halogen can be found 3 times more often in the
factory. The problem with this statement is that we don’t know 3 times of
what?
Statement (2) is sufficient, it gives us enough data to solve the question.
If the ratio is 2:7 then 2/9 is the probability of choosing a halogen at
random.
1. How many people are in the van?
(1) The average weight of each person is 75 Kg.
(2) The people and the van together weigh 5000 Kg.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
STATEMENT (1) alone is not sufficient. We still need the total weight of
the people: then we can divide by the average weight to obtain the
number of people. STATEMENT (2) tells us how much the people and
the van together weigh, but we don't know how much the people weigh.
So STATEMENTS (1) and (2) together are not sufficient.
More sufficient data is required.
2. How many kids are spinning on a carousel?
(1) The average weight of each kid is 25 Kg.
(2) The kids and the carousel together weigh 400 Kg.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
STATEMENT (1) alone is not sufficient. We still need the total weight of
the kids: then we can divide by the average weight to obtain the number
of kids. STATEMENT (2) tells us how much the kids and the carousel
together weigh, but we don't know how much the kids weigh. So
STATEMENTS (1) and (2) together are not sufficient.
More sufficient data is required.
3. If ( 0 < X < 30), what is the value of X?
(1) When X is divided by 6 the remainder is 0.
(2) When X is divided by 12 the remainder is 0.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) by itself is insufficient because the possibilities are many:
6, 12, 18 and 24.
Statement (2) by itself is insufficient because there are more than one
options: 12 and 24.
Even if we combine both statements, still we have two options and X is
not distinct.
4. If ( 0 < X < 70), what is the value of X?
(1) When X is divided by 6 the remainder is 0.
(2) When X is divided by 11 the remainder is 4.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) by itself is insufficient because the possibilities are many:
6, 12, 18, ….
Statement (2) by itself is insufficient because there are more than one
options: 15, 26, …
When we search among all the numbers from statement one that are
divisible by 6, we can see that only one of them will give a remainder of
4 when divided by 11 and that would be 48.
Therefore, the crosslink between the two statements is sufficient.
5. How much was the original price of a car, which was sold for \$25,000?
(1) The original price was more than half of the price it was sold.
(2) The car has appreciated in value by 35% from its original value.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) is insufficient since it doesn’t give us exact information
about the original price, more than half is not accurate enough. Statement
(2) is sufficient since it tells us that the original value is 65% of the price
it was sold.
6. If a full glass of water holds 0.236 liters, how many liters are there in
one pint?
(1) One pint is 1/8 of a gallon.
(2) One gallon is 3.78 liters.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
The data about the glass of water is irrelevant, all you're asked is how
many liters are there in a pint.
Statement (1) is insufficient because we are told the conversion from pint
to gallons only.
Statement (2) is insufficient because we are told the conversion from
gallons to liters only and we have no connection to pints.
Combine both statements so you know how many gallons is one pint and
how many liters are in 1/8 of a gallon. Both statements taken together are
sufficient.
7. One cubic centimeter is equal to 0.001 liters, is a volume of a
rectangular tank larger than 0.001 liters?
(1) The rectangular tank holds 0.3 teaspoons. There are 0.0049 liters in
one teaspoon.
(2) The dimensions of the tank are 0.5 x 0.6 x 4 centimeters.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) gives us the amount of teaspoons that the tank holds and it
gives us that conversion between teaspoons and liters. This statement is
sufficient.
Statement (2) is also sufficient since we are given the dimensions of the
tank, we can calculate the volume and compare it to the volume of one
cubic centimeter.
8. A television set cost \$65 in 1981, did it cost more than \$65
in 1983?
(1) In 1981, the average family had to work three weeks in order to save
up enough money to buy a television set.
(2) In 1983, the average family had to work two weeks in order to save up
enough money to buy a television set.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
STATEMENTS (1) and (2) together are insufficient. You need to know
whether the wages of the average family changed. 3 weeks of work in
1981 could be worth more or less than 2 weeks of work in 1983. More
sufficient data is required.
9. Last year, a bottle of milk cost \$1. Does it cost more than \$1 today?
(1) Last year, the average worker had to work 10 minutes to pay for a
bottle of milk.
(2) Today, the average worker had to work 8 minutes to pay for a bottle
of milk.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
STATEMENTS (1) and (2) together are insufficient. You need to know
whether the wages of the average worker changed. 10 minutes of work
last year could be worth more or less than 8 minutes today.
10. Is the line to the rollercoaster getting longer or shorter by the minute?
(1) Each rollercoaster can process 48 people in 4 minutes.
(2) Every two minutes, 20 people join the line.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
STATEMENT (1) tells us the rate at which people are getting out of the
line, but we need to know if anyone is joining the line to be able to
answer the question. STATEMENT (2) gives us this information.
Therefore the two together are sufficient to see that the net gain in people
on the line is 2 per minute; the line is getting shorter.
11. There is a traffic jam in the freeway, is the number of cars getting
bigger by the minute?
(1) 25 cars escape the traffic by exiting the freeway at the nearest exit
every minute.
(2) 40 new cars get stuck in the traffic jam every two minutes.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
STATEMENT (1) tells us the rate at which people are getting out of the
traffic jam, but we need to know if anyone is joining the line to be able to
answer the question. STATEMENT (2) gives us this information.
Therefore the two together are sufficient to see that the net gain in
number of cars on the line is 5 per minute; the number of cars is getting
smaller.
12. Did the value of the house grow this year?
(1) If the neighborhood becomes more crowded the value of the house
drops.
(2) This year the neighborhood has gotten less crowded yet a new railway
was built nearby.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) gives a casual link between the value of the house and the
neighborhood, yet no data on the neighborhood was given in this
statement. Statement (2) tells us that the neighborhood has gotten less
crowded and so the value of the house grew. Statement (2) also tells us
about a new railway that was built nearby but no link has been given to us
about the connection between the value of the house and the rails.
Both statements, taken together are sufficient.
13. Did the average cost of flying abroad decline this year?
(1) If the geo-political situation in the area is evolving, the average cost of
(2) The geo-political situation in the area is not evolving, yet more people
fly this year than the previous years.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) gives us a link between the cost of flying abroad and the
geo-political situation.
Statement (2) tells us the geo-political situation and thus we know the
changes in the average cost of flying abroad. The second piece of
information given in statement (2) is irrelevant, we are not told about a
connection between the number of people and the price of flying abroad.
Both statements, taken together, are sufficient.
14. Derrick is flying a small Cessna light plane for 2 hours. What is
Derrick's average flying speed in miles per hour?
(1) Derrick flew 250 miles.
(2) Derrick's top speed was 200 miles per hour and his low speed was 100
miles per hour.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
The average flying speed can be found by dividing the total distance by
the total time.
Statement (1) gives us the total distance and the total flying time is given
to us in the question and therefore this statement is sufficient.
Statement (2) is irrelevant, the changes in velocities during the time
interval don’t affect the average speed. And therefore this statement is
insufficient.
15. Fred drove from city A to city B. What is Fred's average speed?
(1) Fred drove from city A to city C during a 4 hour period and from city
C to city B during a 2-hour period.
(2) The distance between city A and city B is 600 miles.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) presents us with another city, which we don’t know her
relative location from city A or B. City C can be between city A and B or
it can be somewhere else.
Statement (2) tells us the distance between city A and B yet it doesn’t
give us any of the other distances. The average speed is calculated by
dividing the total distance (which is unknown) by the total travel time
(which is known).
More sufficient data is needed.
16. What is the sum of the digits of a two digits number?
(1) The sum of the digits is a number, which is divisible by 4.
(2) The two-digit number is a prime number.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) is insufficient since there are plenty of double-digit
numbers who fit this statement.
Statement (2) is insufficient since there are plenty of double-digit prime
numbers.
The combination of both statements is still not sufficient.
Take 31 and 71: they both fit statement (1) and (2) but each has a
different sum of digits.
17. How many liters of lime are needed in order to paint the entire wall of
a castle?
(1) The length of a wall is 45 yards.
(2) The width of the wall is 50% of the length.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Neither statement defined how many liters are needed in order to paint a
certain area of the wall and therefore we can’t convert the area into liters
of paint.
18. If one Pint is 1/8 of a gallon, how many pints are needed in order to
(1) The hogshead is a cylinder with a diameter of 25 inch.
(2) The height of the hogshead is 100 inch.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Neither statement gave us the conversion factor from inches cubed to
gallons or pints.
Therefore, we can only calculate the volume of the hogshead but we can’t
convert it into pints.
19. Rick and Nick are sitting in their cars waiting in line to be served at
the drive-in café. How many cars are in the line?
(1) There are 13 cars between Rick and Nick.
(2) There are 31 cars in front of Rick and 24 cars behind Nick.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Using both statements combined, we cannot determine who is in from
who, is Rick further in the line than Nick or the opposite. Because of this
we have two options for the number of cars.
The first (If Rick is in front): 31 + 13 + 24 = 68.
The second (If Nick is in front): 18 + 13 + 11 = 42.
More sufficient data is required.
20. Is A > B ?
(1) A2 > B2.
(2) B is positive.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Use both statements, find two sets of numbers that fit the statements yet
give a different result.
Take A = 2, B = 1
the numbers fit the statements and A > B.
Take A = -2, B =1
the numbers fit the statements and B > A.
Therefore more sufficient data is required.
21. Is the area that is blocked between the line Y = AX + B, the Y-axis
and the X-axis bigger than C?
(1) A = 2, B = -1.
(2) C = 10.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Using both statements, the line is Y = 2 – X and the area is 10.
Draw a rectangular axis system and draw a line.
You can see that the blocked area is a triangle with an area of (2 x 2 / 2 =
2) and therefore the area of the blocked area is smaller than 10.
Only if you use both statements together, you can answer the question.
22. In which of the following lines: Y1 = A1X + B1, Y2 = A2X + B2, is the
angle X bigger?
(1) X is the angle between the line and the X-axis.
(2) A1 = 2A2.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Both statements are sufficient.
Statement (1) defines the angle X of each line.
Statement (2) gives us the relevant data on each of the lines, the angle X
is determined by the coefficient of X, thus A1 and A2. The line with the
bigger coefficient is the one with the bigger angle X.
23. If A stamps can be bought with B dollars, how many stamps can be
bought with 10 dollars?
(1) B dollars are more than enough to buy 20 stamps.
(2) B = 5.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) is not accurate, it tells us that B dollars are more than
enough to buy 20 stamps, in other words, the number of stamps that can
be bought with B dollars is larger than 20.
Statement (2) completes statement (1) but still both statements are
insufficient together.
All we know from the statements is that more than 20 stamps can be
bought with 5 dollars.
More sufficient data is required.
24. What is the value of (X + Y)?
(1) Y is 20% more than X.
(2) X = 120.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) is insufficient by itself since we only know that Y = 1.2X
and the expression required is (X + Y), which becomes (X + 1.2X).
Using statement (2) combined with statement (1), we know that the value
of the expression becomes
(120 + 1.2 x 120 = 144) and therefore both statements, taken together, are
sufficient.
25. What is the value of (X + Y)/X?
(1) Y is 35% more than X.
(2) Y = 88.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
The expression can be written as: (X + Y)/X = 1 + Y/X and therefore we
need the ratio only.
Statement (1) is sufficient because it gives us the ratio between Y and X,
Y = 1.35X.
Statement (2) is insufficient by itself since it supplies no data on the
variable X.
26. How long will it take Jim to wax his car?
(1) It would take Jim and Mike 40 minutes to wax Jim’s car.
(2) It would take Mike 1 hour and 20 minutes to wax his car.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) gives us the output of Jim and Mike together on Jim’s car
only.
Statement (2) gives us Mike’s output on his car and not Jim’s car and
therefore we can’t conclude anything about the output of Mike on Jim’s
car. More sufficient data is required.
27. How much time will it take Gus to deliver 350 newspapers on his
bikes?
(1)Tom and Gus, working together each on his bike, can deliver 100
newspapers in one hour.
(2)Tom can deliver 25 newspapers in 30 minutes on Gus’s bike.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) gives us the output of Gus and Tom, together, in order to
answer the question we need to know the output of Tom on his bike
alone. Statement (2) gives us almost that, it gives us the output of Tom on
Gus’s bike and as far as we are concerned, his bike could be a lot faster or
slower and thus the output will change accordingly.
More sufficient data is required.
28. How much time will it take two different fire extinguishers to shut
down a level 3 fire?
(1) The first fire extinguisher can shut down a level 3 fire in 45 seconds.
(2) The second fire extinguisher can shut down a level 3 fire in one
minute and 20 seconds.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) gives you the output of the first fire extinguisher, which is
1/45 fires in a second.
Statement (2) gives you the output of the second fire extinguisher, which
is 1/110 fires in a second.
Combine the two statements, we know the total output of the two
extinguishers and so we can easily calculate the time it would take them
to shut down the fire.
29. What is the average number of questions that Laura can write in three
days if on the first day she wrote 20 questions?
(1) Every passing day the number of questions that Laura writes grows
by 20%.
(2) On the other two days, Laura wrote 53 questions.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Laura writes questions over a 3 day period.
In the first day Laura wrote 20 questions.
Statement (1) tells us that on the second day she wrote (1.2 x 20 = 24)
questions and on the third day she wrote (1.2 x 24). This statement is
sufficient in order to calculate the average.
Statement (2) is also sufficient, it gives us the sum of questions that Laura
wrote in the other 2 days and therefore the average can be easily
calculated.
Each statement is sufficient on its own.
30. How many pictures exactly did Sammy develop on Saturday?
(1) Sammy gave away 3 films for development.
(2) There are approximately 36 pictures in one film.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
The question asked explicitly for an exact number of pictures.
Statement (1) is not sufficient by itself because it gives us no data on the
pictures.
Statement (2) completes statement (1) but not accurately as the question
required and therefore more sufficient or accurate data is required.
31. How long exactly did it take Claudia to drive from the beach house to
her green house?
(1) Claudia drove at a constant speed of 55 miles per hour.
(2) The approximate distance between the beach house and the green
house is 200 miles.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
We are specifically asked about the exact time and not an approximation.
Statement (1) gives us the exact traveling speed of Claudia.
Statement (2) gives us the approximate distance between the two houses
and therefore we can only calculate the approximate duration and not the
exact one.
More accurate data is required.
1. Two giant identical poles have been planted in the ground. One of the
poles was planted dipper than the other pole. The shadow of pole A is 10
meters long and the shadow of pole B is 8 meters long. How tall is pole
B?
(1) Pole A is hoisted 14 meters in the air.
(2) Pole B is located 2 meters from pole A
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
The shadows are proportioned to their height.
Statement (1) gives us the height of pole A and so by using the proportion
we can calculate the height of pole B. The proportions are 10/14 = 8/Hb,
from that we know the height of pole B.
Statement (2) is insufficient because the distance between the poles is
irrelevant to the question and it doesn’t contribute anything.
2. A new taxi service charges money according to the weight of the
passengers and their luggage or according to the distance they wish to
travel. If the taxi service charges the highest among the two, according to
what will the Smith’s pay, weight or distance?
(1) The total weight of the Smith’s family including the luggage is 300
Kg.
(2) The total distance, which the Smith’s wish to travel is 100 Km.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
We are given two possibilities of charging fee, according to two different
criteria’s but both statements don’t provide us with the conversion factor
from weight to amount of money or from distance to amount of money.
More sufficient data is required.
3. Is the square root of A an integer?
(1) The last digit of A is 8
(2) A is divisible by 6
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
If you square each digit {0, 1, 2,..., 8, 9}, you will see that the only
possible last digits for a square are 0, 1, 4, 5, 6 and 9. Thus, if the last
digit of A is 8, A cannot be a square. So the square root of A is not an
integer. So statement (1) by itself is sufficient. Since 36 is divisible by 6
and its square is an integer, this statement is insufficient by itself.
4. How many Popsicles can Amy and Megan eat in 30 minutes?
(1)Megan eats twice as fast as Amy.
(2) Megan can eat 5 more Popsicles than Amy in 15 minutes.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) is insufficient by itself because it only gives us the ratio
between the two and we need a concrete number, which statement (2)
provides. Altogether we have two equations with two unknowns and the
solution is feasible by using both statements.
5. If there are 350 words in each page, how many pages can Susan type in
one hour?
(1) There is an average of 30 twenty-word lines in one page.
(2) Susan can type 15 twenty-word lines in two minutes.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
We are given the amount of words per one page, and we are asked how
many pages can be typed in one hour. In order to be able to answer this
question, we need to know the rate of Susan’s typing.
Statement (2) gives us just that, it tells us that Susan can type (15 x 20 =
300) words in two minutes and therefore this statement is sufficient by
itself.
Statement (1) is insufficient because it only gives us an average data of
20 words lines with out any correlation to Susan’s rate of typing.
6. A CNC machine produces metal parts through the machining process.
How many aluminum cubes can the machine produce in 40 minutes?
(1) The CNC machine produces 3 steel cubes in 20 minutes.
(2) The rate in which the machine produces steel cube is three times
slower than the rate of producing aluminum cubes due to the lower
density of aluminum relative to steel.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
We are to find the rate in which the CNC machine produces aluminum
cubes and then multiply it by the required time.
Statement (1) is insufficient because it gives us the rate of producing steel
cubes, which is a different type of metal. Statement (2) tells us that the
rate of production is 3 times quicker in aluminum and therefore using
both statements together, we can calculate the necessary rate.
7. Is the average of X consecutive numbers odd?
(1) The first number in the series is odd.
(2) The sum of the numbers is odd.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) is insufficient by itself, take X as 2: if the first number is
odd, the sum of the two numbers is odd. Take X as 3: if the first number
is odd, the sum of the three numbers is even.
Statement (2) tells us that the sum of the numbers is odd and therefore the
median must be odd.
If the median is odd the average of these numbers is also odd because that
means that there is an even amount of even numbers and an odd amount
of odd numbers.
This statement is sufficient by itself.
8. In a group of 200 foreign workers, how many workers can read and
write?
(1) 80 workers can write.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) is insufficient because it tells us nothing about workers who
Statement (2) is insufficient because it tells us nothing about workers who
can write.
Combine the two, still you don’t know if there is any overlapping
between the two groups and thus more sufficient data is required.
9. In a group of 350 academics, how many are either British or French?
(1) There are 100 British in the group.
(2) There are 180 French in the group.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
The question asks, how many are either British or French. In other words,
how many people are British or French. Statement (1) is insufficient
because it tells us nothing about the French. Statement (2) is insufficient
for the same reason, only that it doesn’t say anything about the British.
Combine the two statements, you know that (100 + 180 = 280) is the
number of people that are either British or French.
10. There are 45 people on the quay, what is the most and the least
number of people that are both sailors and erudite?
(1) The number of sailors is 28.
(2) The number of erudite people is 30.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Use statements (1) and (2) together. You can see that the number of
people in each of the groups together sums up to a number, which is more
than 45 and therefore there must be an overlapping between the groups.
The maximum number of people in both groups is the size of the smaller
group, thus 28 people.
The minimum number of people in both groups is (28 + 30 – 45 = 13).
Both statements, taken together, are sufficient.
11. 47 people are standing behind a large dais, what is least number of
people, behind the dais, that are blonde haired and over 6 feet tall?
(1)The number of blond people is 35.
(2) The number of people who are less than 6 feet tall is 25.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Use statements (1) and (2) together. You can see that the number of
people in each of the groups together sums up to a number, which is more
than 47 and therefore there must be an overlapping between the groups.
The maximum number of people in both groups is the size of the smaller
group, thus 25 people.
The minimum number of people in both groups is (25 + 35 – 47 = 13).
Both statements, taken together, are sufficient.
12. What is the value of X?
(1) 4X + 18 = 2X + 22.
(2) 8/X + 14 = 6/X + 16.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
We ought to find the value of the unknown X.
Statement (1) is sufficient because it presents us an equation where the
only unknown is X.
Statement (2) is also sufficient for the same reasons as the first one and
therefore each statement is sufficient by itself.
13. If X and Y are integers, what is the value of Y/X?
(1) X3 + X2 + 2X = 0.
(2) Y – 4 = 0.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
In order to find the value of Y/X, you need to use both statements to find
each of the variables.
Statement (1) gives only one possible root for X, the expression in
statement (1) can be written as
X(X2 + X + 2)
the only solution for X which is an integer is 0, the
other two are complex numbers.
Statement (2) tells us that Y = 4.
The value of Y/X is undefined, because the denominator is zero and
therefore both statements are not sufficient in order to determine the
value of Y/X.
14. If X and Y are integers, what is the value of XY?
(1) X3 – 3X2 – 2X – 8 = 0.
(2) 4 + 3Y = 2Y + 8.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) can be written as (X – 4)(X2 + X + 2) = 0.
The roots of this equation are one integer and two complex numbers,
which you should pay no attention to since you were told that X is an
integer.
Statement (2) is a simple equation, Y = 4.
The value of the expression XY is 16.
Both statements, taken together, are sufficient to answer the question.
15. What is the value of AB?
(1) A + 4 = 3A – 8.
(2) B2 – 12B + 36 = 0.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) is insufficient by itself. The value of A from this statement
is 6.
Statement (2) can be written as (B – 6)2 = 0 and therefore the value of B
is 6.
Combine both statements to calculate the value of the required expression
AB.
17. How many days will it take two windows cleaners to clean the entire
30 stories building?
(1) The first window cleaner can clean 15 windows in 10 minutes.
(2) The second window cleaner can clean twice as much as the first
cleaner in 15 minutes.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Both statements, taken together, are not sufficient. There is no data on
how many windows are in each story of the building and so we can’t
calculate the time it would take the two cleaners to finish the job.
More sufficient data is required.
18. How long will it take until Andy and Tim to finish swimming 1500
meters?
(1) Andy can swim at a constant rate of 100 meters in one minute.
(2) Tim can swim faster than Andy.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
There is a trick to this question, the time until both swimmers finish
swimming 1500 meters is the time it takes the slower swimmer among
the two to finish swimming.
Statement (1) tells us the rate of Andy, we can calculate the time it would
take him to finish 1500 meters. This statement is insufficient by itself
since we don’t know that Andy is slower than Tim.
Statement (2) completes statement (1) by telling us that Tim is faster and
so the time is determined by Andy’s time.
Both statements, taken together, are sufficient.
19. There are three sprinters on a racetrack. How much time will it take
all three to finish an 800 meters race?
(1) The first runner can run the fastest, he runs at a 110% of the
slowest runner.
(2) The slowest runner runs at a constant speed of 7 meters per
second.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
The trick to this question is to understand that the time it would take them
all to finish the track is the time it would take the slowest sprinter to
finish the track.
Statement (1) tells us that the first sprinter is the fastest among the three.
Statement (2) tells us that the third sprinter runs the slowest and therefore
they determine the time it would take all of them to finish the track. The
rate of the slowest sprinter is given and so this answer is solvable.
20. How many hours will it take ship A and ship B to transfer 50 cars
from one side of the river to the next?
(1) Ship A can transfer 5 cars in 10 minutes.
(2) Ship B can transfer twice as many cars in half the time.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
This is a simple rate problem. In order to find the time it will take both
ships to finish the job, we need to know the output of each ship.
Statement (1) gives us the output of ship A, which is 30 cars per hour.
Statement (2) gives us the output of ship A, which is 120 cars per hour.
Both statements, taken together, are sufficient.
21. Each of the 850 local villagers in Lucia owns either a Golden
Retriever or a Bernard. How many people own both?
(1) The number of villagers who own a Golden Retriever only is 300.
(2) The number of villagers who own a Bernard only is 280.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Each one of the villagers, according to the question, has to own at least
one of the two dogs.
Statement (1) is insufficient because it says nothing about the Bernard
owners.
Statement (2) is insufficient because it says nothing about the Golden
Retriever owners.
Combine the statements, all the information we need is present,(800 –
300 – 280) is equal to the number of people who own both races of dogs.
22. The town rules in Kid-Town require each house to have at least a
ping-pong table or a soccer-table. If there are 50 houses in Kid-Town,
how many houses carry both types of tables?
(1) The number of houses that have a ping-pong table only is 20.
(2) The number of houses that have a soccer table is 40.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Pay attention to the statements, (1) gives you the exact number of houses
who have a ping-pong table only as opposed to the other statement, which
tells you that 40 houses have a soccer-table.
There are two possible answers to the question: there are (40 + 20 – 50 =
10) houses with both tables in them or there are 20 houses (the size of the
small group).
The answer is not distinct and therefore more sufficient data is required.
23. An air-balloon is filled with air, how long will it take to fill it
assuming that its volume is 30 meters cubed?
(1) Every minute the pump fills 1 meter cubed and emits one fifth of
that amount.
(2) The net airflow to the balloon is 0.8-meter cube per minute.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
The volume is given in the question and so we ought to find the rate of
fulfillment.
Statement (1) tells us that every minute 1 meter cubed is going in and 0.2
is going out, making a total of 0.8 meter cube air going in per minute.
Statement (2) gives us right away the net flow of air going in to the
balloon.
Therefore, each statement is individually sufficient.
24. A waiter earns a basic amount of 500 pounds per month regardless of
her tips. How much did the waiter receive on tips during the month of
May?
(1) On May, the waiter earned a total amount of money that was 150%
of the basic.
(2) The waiter receives an average amount of 250 pounds on tips per
month.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
We need to find the amount of money that the waiter earned on tips.
Statement (1) tells us that the total amount money earned is 150% of the
basic, therefore the tips are 50% of 500 pounds, thus 250 pounds. This
statement is sufficient.
Statement (2) is insufficient since the amount of tips earned on may can
be over or under the average and so we can’t pinpoint the exact amount.
25. Willy the wale receives sugar cubes every time he does something
exquisite. How many sugar cubes did Willy get on yesterday’s show?
(1) Today, Willy performed the same show as yesterday.
(2) Today, Willy received 11 sugar cubes.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient
d)Either statement BY ITSELF is sufficient to answer the question
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
yesterday’s show, which is dependent on his performance.
Statement (1) tells us that Willy did the same show but it didn’t mention
how did he perform, did he do the same number of exquisite acts? This
statement is insufficient.
Statement (2) only completes statement (1) by telling us how many cubes
More sufficient data is required.
1. Last year, what was the average (arithmetic mean) number of cookies
consumed by a person in the Swanson family?
(1) Last year, the family consumed 45 boxes of cookies.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
We are required to find the number of cookies that each person
consumed.
Statement (1) and (2) tell us that (45 x 12) cookies were consumed by the
entire family last year.
This data is insufficient because we don’t know how many members are
in the Swanson family.
More sufficient data is required.
2. What was the maximum wind speed on January 1998?
(1) The average (arithmetic mean) of the maximum monthly wind
speed between March 1997 and January 1998 is 35 knots, which was
higher by 12 knots from the average (arithmetic mean) of the
maximum monthly wind speed between February 1997 and December
1997.
(2) The maximum wind speed on January 1998 was 8 knots higher
than the maximum wind speed on February 1997.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
The question doesn’t tell us much, go straight to the statements.
From statement (1) we can find the sum of the wind speeds between
March 1997 and January 1998, which is (11 x 35 = 385 knots). The
average of the other group of months is (11 x (35 – 12) = 253).
The difference between the two numbers is the sum of the maximum in
January 1998 and February 1997. Define J as the maximum on January
and F as the maximum on February, you can write the following
equation: J + F = (385 – 253 = 112). J is what we’re looking for.
Statement (2) can be written as J – F = 8.
We have two simple equations with two unknowns, both statements
together are sufficient.
3. Is A5 > A2?
(1) A is an integer.
(2) A is positive.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
There are 3 cases in which the inequality is not true: when A is negative,
when A is a fraction and when A is equal to 1. Statements (1) and (2)
cover up two of the cases above, yet A=1 fulfills both statements and still
the inequality will not be true.
More sufficient data is required.
5. X, Y and Z are three numbers. If Y = 5, what is their sum?
(1) X – Z = 10.
(2) Z – Y = 15.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
We need to find the value of X+Y+Z. Y is given to us so we need the
value of (X+Z).
Statement (1) is insufficient by itself since we are given the value of (X –
Z) and not (X + Z).
From statement (2) we can find the value of Z, and from there return to
statement (1) and find the value of X. Both statements, taken together, are
sufficient.
6. X, Y and Z are three positive integers. If Z = 2, what is their sum?
(1) X – Y = 5.
(2) 3Y + 15 = 3X.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
We need to find the value of X + Y since Z is already given to us.
Statement (1) is insufficient since we need the sum of X and Y.
Statement (2) can be written as: 3X – 3Y = 15
X – Y = 5, you can see
that both statements are the same and therefore more sufficient data is
required.
7. Bony and Clyde, each had to translate half of a new blockbuster movie.
If Bony finished her half after two hours and 20 minutes, how long will it
take Clyde to finish his half?
(1) Bony can translate 3 lines of speech in 1 minute, which is one and
a half faster than Clyde.
(2) The movie contains 840 lines of speech.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
We are told that each of the translators has to finish the same job (each
has to finish one half).
Statement (1) tells us that Bony can translate 1.5 faster than Clyde and
therefore it would take him 1.5 times more than Bony’s time. This
statement is sufficient, the data about the specific translation rate is
irrelevant and so is statement (2).
8. Jean and Jordy each had to wash half of a rectangular floor. If Jean
finished his part of the job after 45 minute, how long will it take Jordy to
finish his half?
(1) Jean can wash 10 meters square in 5 minutes, which is twice as
fast as Jordy.
(2) The area of the rectangular floor is 180 meters squared.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
We know how much time it took Jean to wash the floor and we need
Jordy’s time.
Statement (1) tells us that the rate of Jean is double than the rate of Jordy
and therefore it will take him twice as long to wash his half of the floor.
The numbers that describe the rate of Jean are irrelevant to the question
and so is statement (2) .
9. 35% of the students in Cambridge study engineering. How many
students in Cambridge study Aerospace engineering with honors?
(1) 18% of the total number of students in Cambridge, which is
10,000, study with honors.
(2) One fifth of the honor students study Aerospace engineering.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) tells us that (0.18 x 10,000 = 1,800) student’s study with
honors.
Statement (2) tells us that (0.2 x 1,800 = 360) is the number of students
that study Aerospace engineering with honors.
Both statements, taken together, are sufficient.
10. Is Y even?
(1) 2Y is even.
(2) Y2 is even.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) by itself is insufficient because every number that is
multiplied by 2 will result in an even number. Statement (2) is also
insufficient by itself since numbers like 6 fulfills this statement
although it’s not even. Combine the statements and Y must be an even
number.
15. If A is a prime number, what is the value of A?
(1) 0 < A < 10.
(2) (A – 2) is divisible by 3.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) narrows down A to be one of : 1, 2, 3, 5 and 7. This
statement is insufficient.
Statement (2) is also insufficient, there are a lot of numbers that fulfill
this statement, like 17, 23 and many more. Even after you combine both
statements, there are still two options: 5 and 2.
Both, when you subtract 2 you get a number that is divisible by 3.
16. If B is an odd number, what is the value of B?
(1) 20 < B < 30.
(2) (B – 1) is divisible by 3.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) narrows down B to the following numbers: 21, 23, 25, 27,
29. This statement is insufficient by itself. Statement (2) is also
insufficient by itself since more than one numbers fulfill the statement. If
you combine the two statements, you will narrow down B to be 25.
Therefore both statements, taken together, are sufficient.
19. As the new basketball season reopened, 40 players received either a
yellow ball or a red ball as a present. How many basketball players
received the yellow ball as a present?
(1) 16 players received the red ball.
(2) 50% more players received the yellow ball than players who
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) is sufficient since it gives us the number of players who
received red balls. The number of players who received yellow balls is
(40 – 16 = 24).
Statement (2) is sufficient, Define X as the number of players that
X + 3X/2 = 40
X = 16 and so (40 – 16 = 24) is the number of players
20. If the sum of six numbers is between 135 and 164, then the average
(arithmetic mean) of the six numbers could be which of the following?
15.4.
20.5.
25.5.
31.25.
32.5.
Define X as the sum of all the six numbers, 135 < X < 164.
The average of the six numbers, using the average formula, is X/6.
Divide both sided of the equation by 6: 135/6 < X/6 < 164/6
22.5 <
X/6 < 27.33.
The average has to be in that range and therefore the only answer could
be C.
21. Is X bigger or smaller than Y?
(1) X > Z.
(2) Y > Z.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Use both statements. Pick Z = 2.
X=3, Y=4 fulfill both statements and Y>X.
X=4, Y=3 fulfill both statements and Y<X.
Therefore more sufficient data is required.
24. A long rope was divided to three different parts. What is the length of
the smallest piece?
(1) The sum of the two smaller pieces is 14 inch.
(2) The sum of the two larger pieces is 22 inch.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Translate the statements into variables: Let X, Y and Z be the thee pieces
of the rope, X<Y<Z.
Statement (1) can be written as: X + Y = 14.
Statement (2) can be written as: Y + Z = 22.
In order to find the length of the smallest piece, we need another equation
or data. More data is required.
25. A hose was divided into 3 smaller and different in size hoses. What is
the difference between the length of the largest and the smallest hose?
(1) The sum of the two larger hoses is 45 feet.
(2) The sum of the two smaller hoses is 23.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Translate the statements into variables: Let X, Y and Z be the thee pieces
of the hose, X<Y<Z.
Statement (1) can be written as: Z + Y = 45.
Statement (2) can be written as: X + Y = 23.
In order to find the difference between Z and X, subtract the second
statement from the first to get:
Z – X = 22, and this is what we were looking for.
Both statements, taken together, are sufficient.
26. Fuel tanker A can fill the underground reservoir in 12 minutes. How
long will it take fuel tanker A and fuel tanker B to fill up the same
reservoir together?
(1) The reservoir contains 3000 liters of fuel.
(2) Fuel tanker A alone will require the same number of hours to fill
the same reservoir.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) is insufficient since the size of the reservoir is irrelevant.
Statement (2) is sufficient since it tells us that the second tanker has the
same output as the first one and so it will take them both half of the time
it took the first tanker alone.
27. If Z is an integer, is Z/3 an odd integer?
(1) Z/3 is an integer.
(2) Z/6 is an integer.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) is insufficient. If Z=6, Z/3 is even and if Z=9, Z/3 is odd.
Statement (2) is sufficient. If Z/6 = X (X is an integer according to the
statement) then Z/3 = 2X.
2X must be an even integer since it’s a multiplication of an integer by 2.
Therefore Z/3 is always even. Statement (2) is sufficient.
28. What is the ratio between A and B?
(1) A is the sum of X, Y and Z.
(2) B is the average (arithmetic mean) of X, Y and Z.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) tells us that A = X + Y + Z.
Statement (2) tells us that B = (X + Y + Z)/3.
Using both statements together: A/B is 3.
Both statements together are sufficient.
29. If X and Y are both integers different from zero, what is the value of
(X + Y)?
(1) X3 = Y3.
(2) Y = 10.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) gives us the relations between X and Y, they must be equal
because the power is odd.
Statement (2) gives us Y and by using the first statement, we know the
value of X also.
Both statements, taken together, are sufficient.
30. If X and Y are both integers different from zero, what is the value of
(X + 2Y)?
(1) X4 = Y4.
(2) X = 5.
a)Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer
the question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
Statement (1) tells us that X and Y are equal? No, they could have
different signs.
Statement (2) gives us X, which is not sufficient.
Both statements together are also insufficient since Y can be –5 or 5.
More sufficient data is required.
5. M.A.S (Most Affordable Speed) is defined as the speed where the fuel
consumption of a car is the lowest. The average family car consumes 3
liters of fuel per 36 kilometers at the M.A.S with only one passenger (the
driver). A pickup truck consumes twice as much as a family car does.
1
3
Assuming the fuel consumption of both cars rises by 3 % of the original
consumption for each additional passenger, how many km per litter
would a pickup truck do if the driver has three additional passengers?
(a) 10 km.
(b) 5.8 km.
(c) 6 km.
(d) 5.4 km.
(e) 4 km.
This is a question with a lot of dispensable text, it teaches us to focus on
relevant information only. A family car consumes 1 liter for 12 Km, a
pickup truck consumes twice as much, 1 liter for 6 Km. There are 3
additional passengers so the consumption rises by 10%,
6 Km × 0.9 = 5.4 Km .
6. A Super-Jet airplane is flying at an average speed of 500 miles per
hour. The average speed of a Turbo-Prop airplane is 15% lower than that
of a Super-Jet. How long will it take a Turbo-Prop airplane to fly 950
miles?
(a) 2 hours.
(b) 1 hour and 30 minutes.
(c) 3 hours and 15 minutes.
(d) 2 hours and 24 minutes.
(e) 4 hours.
The average speed of a Turbo-Prop airplane is 85% of that of a Super-Jet
airplane, meaning 425 mph. (15 % is 3/20 fraction, so 17/20 * 500 is
425).
Traveling at 425 mph, it would take the airplane 2 hours to travel 950
miles.
8. A windmill is taking advantage of strong air currents in order to
produce electrical energy. On a typical day the wind speed is around 20
mph and in that speed the windmill produces 800 kw/h (kilowatts per
hour). On a stormy day a windmill produces 20% more energy. How
much kw/h can three windmills produce in two hours on a stormy day?
(a) 2880.
(b) 4860.
(c) 5780.
(d) 5760.
(e) 6380.
On a stormy day, a windmill will produce 20% more energy. 20% of 800
is 160, so each windmill will give out 960 kw/h. Three windmills will
give 3 x 960 = 2880, which is answer (a), but we want two hours so the
answer is 2880 x 2 = 5760.
6. If 0.22z = 118.8, then z =
(a) 540
(b) 622
(c) 830
(d) 991
(e) 1000
Because the answer choices are so far apart, you can ballpark this
problem. Multiply both sides by 100 to eliminate the decimal points:
22z = 11880, divide both sides by 22.
z = 540.
7. If 0.45x = 101.25, then x =
(a) 180
(b) 225
(c) 328
(d) 444
(e) 448
Because the answer choices are so far apart, you can ballpark this
problem. Multiply both sides by 100 to eliminate the decimal points:
45x = 10125, divide both sides by 45.
x = 225.
8. If 0.01z = 9.99, then z =
(a) 99.9
(b) 999.9
(c) 999
(d) 9999.9
(e) 9999
Because the answer choices are so far apart, you can ballpark this
problem. Multiply both sides by 100 to eliminate the decimal points:
z = 999.
9. If 0.06x = 52.2, then x =
(a) 540
(b) 622
(c) 830
(d) 870
(e) 920
Because the answer choices are so far apart, you can ballpark this
problem. Multiply both sides by 100 to eliminate the decimal points:
6x = 5220, divide both sides by 6.
x = 870.
15. A taxi company costs \$2.75 for the first quarter-mile and 12.5 cents
for each additional quarter mile. What is the maximum distance you can
travel with \$6.50?
(a) 4 miles
(b) 5 3/4 miles
(c) 6 1/2 miles
(d) 7 3/4 miles
(e) 8 1/4 miles
Since you spend \$2.75 for the first quarter mile, you have \$3.75 (6.5 –
2.75) left to spend on ¼ mile intervals. \$3.75 divided by \$.125 equals 30
quarter miles. The 30 miles, plus the initial quarter mile, makes a total of
31quarter miles. 31 quarter-miles equal 7.75 miles.
16. A photo-mat shop charges \$0.55 for the first photo inside a film, and
40% cents less for each additional photo. How many photos can we
develop with \$52.58 if each film contains 36 photos?
a) 4 films and 12 photos
b) 5 films and 6 photos
c) 5 films and 14 photos
d) 6 films and 4 photos
e) 6 films and 28 photos
Each film costs: 0.55 + 35 x (60% of 0.55 \$ is 0.33) = 0.55 + 11.55 =
12.1\$ per film.
52.58\$ = 4 x 12.1\$ + 4.18\$.
4.18\$ - 0.55\$ = 3.63\$. 3.63\$/0.33\$ = 11 All together gives 4 films +
12 photos.
17. In a fuel station the service costs \$1.75 per car, every liter of fuel
costs 0.65\$. Assuming that a company owns 12 cars and that every fuel
tank contains 55 liters and they are all empty, how much money total will
it cost to fuel all cars?
(a) 320\$
(b) 380\$
(c) 420\$
(d) 450\$
(e) 480\$
The cost of fuel per car is: 1.75 + 0.65 x 55 = 37.5 \$.
There are 12 cars so the sum is 37.5 x 12 = 450 \$.
18. In a fuel station the service costs \$1.15 per car, every liter of fuel
costs 0.4\$. Assuming that you own 2 sports cars and 2 executive cars and
all fuel tanks are empty. How much will it cost to fuel all cars together if
a sports car tank is 32 liters and an executive car tank is 75% bigger?
(a) 37.5\$
(b) 75\$
(c) 87.5\$
(d) 94.5\$
(e) 98.4\$
The cost of fuel per a sports car is: 1.15 + 32 x 0.4 = 13.95 \$.
The cost of fuel per an executive car is: 1.15 + (32 x 1.75) x 0.4 = 1.15 +
56 x 0.4 = 23.55 \$.
The sum of the fuel price is: 13.95 x 2 + 23.55 x 2 = 75\$
19. In a fuel station the service costs \$2.05 per car, every liter of fuel
costs 0.6\$. Assuming that you fill up 3 mini-vans and 2 trucks, how much
money will the fuel cost to all the cars owners total, if a mini-van’s tank
is 65 liters and a truck’s tank is 120% bigger and they are all empty-?
(a) 122.6\$
(b) 128.9\$
(c) 243.7\$
(d) 298.85\$
(e) 312.12\$
The cost of fuel per mini-van is: 2.05 + 65 x 0.6 = 41.05\$.
The cost of fuel per an executive car is: 2.05 + (65 x 2.20) x 0.6 = 2.05 +
143 x 0.6 = 87.85 \$.
The sum of the fuel price is: 3 x 41.05 + 2 x 87.85 = 298.85\$
20. The average (arithmetic mean) of seven numbers is 12.2. If the sum of
four of these numbers is 42.8, what is the average of the other 3 numbers?
(a) 12.4
(b) 14.2
(c) 16.8
(d) 18.6
(e) 19.2
This is an average problem, so use the average formula. If the average of
7 numbers is 12.2, we can solve for their sum: 7 × 12.2 = 85.4. If four of
these numbers total 42.8, then by subtracting 42.8 from 85.4, we get the
sum of the other three numbers, 42.6. To find the average of these three
numbers, we divide their sum by their number: 42.6/3 = 14.2.
21. The average (arithmetic mean) of eight numbers is 44.1. If the sum of
half of these numbers is 158.4, what is the average of the other half?
(a) 12.8
(b) 24.2
(c) 48.6
(d) 72.1
(e) 96.8
This is an average problem, so use the average formula. If the average of
8 numbers is 44.1, we can solve for their sum: 8 × 44.1 = 352.8. If four of
these numbers total 158.4, then by subtracting 158.4 from 352.8, we get
the sum of the other four numbers, 194.4. To find the average of these 4
numbers, we divide their sum by their number: 194.4/4 = 48.6.
22. Eric, Nick and Archi make contributions to the Society Of Nature
Protection in the ratio of 5:3:2.5. If altogether they contribute 5145 Nis,
how much more money does Nick contribute than Archi?
(a) 128 Nis
(b) 212 Nis
(c) 234 Nis
(d) 245 Nis
(e) 288 Nis
Add the numbers in the ratio 5:3:2.5 = 10.5. Divide the 5145 by 10.5 and
you get the basic
Unit = 490 Nis. Nick contributes 0.5 more units than Archi, and since
each unit is 490, he contributed 245 Nis more.
24. Of 70 players on a football team, 37 are throwers. The rest of the team
is divided so one third are left-handed and the rest are right handed.
Assuming that all throwers are right handed, how many right-handed
players are there total?
(a) 54
(b) 59
(c) 63
(d) 71
(e) 92
70 – 37 are the rest. Meaning that 33/3 = 11 are left-handed. The overall
number of right handed: 37 + 22 = 59.
23. Irin, Ingrid and Nell bake chocolate chip cookies in the ratio of 9.18:
5.17: 2.05. If altogether they baked a batch of 148 cookies, what percent
of the cookies did Nell bake?
(a) 0.125%
(b) 1.25%
(c) 12.5%
(d) 125%
(e) 0.152%
Add the numbers in the ratio to get 9.18 + 5.17 +2.05 = 16.4.
You don’t need to relate to the number of cookies, it doesn’t contribute
anything.
The relative part of Nell is 2.05/16.4 = 0.125 = 12.5%
26. On a map, 1 inch represents 28 miles. How many inches would be
necessary to represent a distance of 383.6 miles?
(a) 5.2
(b) 7.4
(c) 13.7
(d) 21.2
(e) 28.7
This is a proportion problem. Dividing the requested amount of miles by
the reference amount would give us the answer in inches. 383.6 / 28 =
13.7 inches.
25. Of 15 players on a basketball team, one third are left handed. Out of
the right-handed players there are 80% over 2 meters high. Assuming that
out of the left handed players there are 40% players fewer than 2 meters.
How many players are there over 2 meters in height?
(a) 6
(b) 7
(c) 8
(d) 9
(e) 10
Start from the top. One third are left handed (5). Out of the right-handed
players there are 8 players over 2 meters. 40% of 5 are 2 players. All
together over 2 meters there are 10 players.
27. On a map, 1.5 inches represent 24 miles. How many miles
approximately is the distance if you measured 47 centimeters assuming
that 1-inch is 2.54 centimeters?
(a) 174.2
(b) 212
(c) 288.1
(d) 296
(e) 282.4
Dividing the number of centimeters in 2.54 gives you the number of
inches: 47/2.54 = 18.5 inches. 1.5 inches represent 24 miles, 1 inch
represents 16 miles.
(18.5 inches) x (16 miles) = 296 miles.
28. On a mechanical gear, 6 teeth represent a movement of 1.2 radians.
The gear is connected to a wheel that moves twice as much. If the wheel
moved 276 degrees how many teeth did the gear move assuming the each
(a) 11.5
(b) 14.2
(c) 28.3
(d) 34.7
(e) 41.3
The wheel moved 276 degrees the gear moved 138 degrees / 60 = 2.3
1.2 radians are 6 teeth one radian is 5 teeth’s. 2.3 x 5 = 11.5 teeth
movement.
29. A rotometer is a device that measures flow of liquid and gases. When
measuring liquid phase flows, 2.5 inches represent 60 liters per minute of
liquid. With gas measurements the rotometer moves 50% of the
movement he moves with the liquid phase. How many liters of gas
passed through the rotometer if it measured 4 inches?
(a) 176
(b) 192
(c) 202
(d) 218
(e) 284
The rotometer moves 50% of 2.5 meaning 1.25 inches is 60 liters of gas.
The rotometer moved 4 inches. 4/1.25 = 3.2 x 60 liters is 192 liters
measured.
30. Billy worked for three straight hours on his homework questions. If
he solved 132 questions in the third hour, which was twice as many as he
solved in the second hour, and three times as many questions as he solved
in the first hour, how many questions did he solve total?
(a) 242
(b) 312
(c) 424
(d) 525
(e) 622
132 / 2 are 66 questions in the second hour, 132 / 3 are 44 questions in the
third hour.
132 + 66 + 44 = 242 questions.
31. V is the volume of a cylinder; the radius of the cylinder is 3.4. The
height of the cylinder is 550% more than the radius. Which of the
following is true?
(a) 100 < V < 300
(b) 300 < V < 500
(c) 500 < V < 700
(d) 700 < V < 900
(e) 900 < V < 1100
You can start with the length. Length = 6.5 x 3.4 is approximately 22.
The Volume of the cylinder is the area of its face x its length.
Area of face = π ⋅ R 2 = π ⋅ (3.4) 2 ≅ 36 .
V is approximately 36 x 22 = 792 and the best answer is D.
1. Two trains are traveling on a collision course. If train A is traveling at
a speed of 350 mph and train B is traveling 28% slower, how much time
will it take the trains to collide if the initial distance between the two is
1505 miles?
(a) Two hours and 30 minutes.
(b) One hour and 10 minutes.
(c) Two hours and 25 minutes.
(d) Three hours and 15 minutes.
(e) Four hours and 20 minutes.
Train B is traveling at a speed of 0.72 x 350 = 252 mph.
The two trains are traveling in oposite directions. Thus, the distance
should be divided by the sum of their speeds = 252 + 350 = 602 mph.
1505 miles / 602 = 2.5 = two hours and 30 minutes.
2. Two cars are traveling towards each other. If car A is traveling at a
speed of 50 mph and car B is traveling 12% slower, how much time will
it take the cars to meet if the initial distance between the two is 705
miles?
(a) Six hours and 30 minutes.
(b) Seven hours and 30 minutes.
(c) Eight hours and 20 minutes.
(d) Nine hours and 15 minutes.
(e) Ten hours and 20 minutes.
Car B is traveling at a speed of 0.82 x 50 = 44 mph.
The cars are traveling in oposite directions. Thus, the distance should be
divided by the sum of their speeds = 50 + 44 = 94 mph.
705 miles / 94 = 7.5 = seven hours and 30 minutes.
4. Tom and Jerry are running on the same road towards each other. If
Tom is running at a speed of 2.5 meters per second and Jerry is running
36% slower, how much time will it take them to meet if the initial
distance between the two is 50 meters and Tom started running 20
seconds before Jerry did?
(a) 2 minute and 12 seconds.
(b) two minutes.
(c) 44 seconds.
(d) 20 seconds.
(e) 12 seconds.
Jerry is running at a speed of 0.64 x 2.5 = 1.6 meters per second.
Tom runs alone a distance of 2.5 x 20 = 50 meters. Tom caught Jerry offguard and Jerry didn’t even start running.
7. Rain is falling at a rate of 3 centimeters per hour all over Springefield.
Somewhere downtown in Springfield a group of pigeons is waiting for
the rain to stop. If the rain filled a round puddle the with a base area of
350 square centimeters and a depth of 13.5 centimeters, how long did the
pigeons wait for the rain to stop?
(a) 3 hours and 12 minutes.
(b) four hours and 30 minutes.
(c) four hours and 45 minutes.
(d) five hours and 10 minutes.
(e) five hours and 30 minutes.
The volume of the puddle is irrelevant since rain fell all over the city.
Thus, you should relate to the depth that fell only. 13.5 / 3 = 4.5 hours of
rain.
16. Ronald is now 4.5 years older than Andrew and third of that amount
older than Ingrid. If in 0.5 years, Ronald will be three times older than
Andrew, then in 2.25 years what would be Ingrid’s divided by Andrew’s
age multiplied by Ronald’s age?
(a) 8.125
(b) 12.5
(c) 12.875
(d) 14.875
(e) 15.225
Translate piece by piece into numbers. R (Ronald) = A (Andrew) + 4.5.
The second equation: R = I (Ingrid) + 1.5.
The third equation: R + 0.5 = 3(A + 0.5). We have three equations with
three variables.
Ronald is 6.25, Ingrid is 4.75 and Andrew is 1.75. In 2.25 years,
Ronald will be 8.5, Ingrid will be 7 and Andrew will be 4. The answer is:
7 / 4 x 8.5 = 14.875.
17. Richard is now 14.5 years older than Arthur and half of that amount
older than Sam. If in 2.75 years, Richard will be twice and a half older
than Arthur, then in 7 years what would be Arthur’s age approximately?
(a) 8
(b) 14
(c) 22
(d) 24
(e) 30
The best answer is B .
Translate piece by piece into numbers. R (Richard) = A (Arthur) + 14.5.
The second equation: R = S (Sam) +7.25.
The third equation: R +2.75 = 2.5(A + 2.75). We have three equations
with three variables.
Today Arthur’s age is approximately 6.9 (take 7). In 7 years he would
18. In 13.5 years Stacy will be as old as Carolyn is now. Thirty two years
ago Carolyn was two and a half the age of Stacy. How old will Stacy be a
(a) 36
(b) 47
(c) 51
(d) 64
(e) 71
Translate the data into numbers: s (Stacey) + 13.5 = c (Caroline) and
2.5(s-32) = c – 32.
From the equations Today Stacy is 41 years old, in a decade from now
she will be 51.
19. X years in the future, Zach will be Y years old. Z years in the future,
Zach will be how old?
(a) Z + X + Y
(b) Z + X – Y
(c) X – Y – Z
(d) Y – X + Z
(e) 2Z + X – y
Write the following equations: Zach + X = Y and Zach + Z =?
Put Zach from the first equation and place it in the second one: Y – X + Z
is the age in Z years.
21. An investment yields an interest payment of \$68 each week. If the
annual interest rate is 7.5%, what is the amount of the investment?
(a) \$28,600
(b) \$30,430
(c) \$34,330
(d) \$37,860
(e) \$43,520
Principal × percent interest = interest earned
Principle× (0.075)× 1/(12 x 4) = \$68.
Solve for the principal (68 × 12 x 4)/.075= \$43,520.
22. An investment yielded an interest payment of \$350 each month when
the annual interest rate was 9%, what is the amount of the investment that
should be invested with an annual interest rate of 8% if
We want to gain 15% more each month?
(a) \$60,375
(b) \$50,400
(c) \$41,300
(d) \$32,500
(e) \$25,100
The new monthly payment should be 1.15 x \$350 = \$402.5. Now we’ll
proceed with the formula:
Principal × percent interest = interest earned
Principle× (0.08)× 1/12 = \$402.5.
Solve for the principal (402.5 × 12)/.08= \$60,375.
23. An investment gained an interest payment of \$250 each month when
the annual interest rate was 8%, how much more should we invest
annually if we want to gain 12% more per month with a new annual
interest rate of 7.5% ?
(a) \$9,360
(b) \$9,100
(c) \$8,250
(d) \$7,300
(e) \$7,150
The new monthly payment should be 1.12 x \$250 = \$280. Now we’ll
proceed with the formula:
Principal × percent interest = interest earned
Solve for the principal for the 9% interest: (250 × 12)/.08= \$37,500, this
is what we invested before.
Solve for the principal for the 6.5% interest: (280 × 12)/.075= \$44,800,
this is what we should invest with the new interest. (44800 – 37500 =
\$7,300) is the amount to be added to the prime payment.
24. Mike earns \$14 per hour and Phil earns \$10.5 per hour.
Approximately how much less, as a percentage, does Phil earn than Mike
per hour?
(a) 25%
(b) 32.5%
(c) 37%
(d) 37.5%
(e) 40%
Mike earns (14 – 10.5) \$3.5 more than Phil, that is 3.5/14 = 0.25 = 25%.
25. The original price of a car was \$25,200. Because the car owner
thought he could get more money for the car, he increased the price of the
car to 110% of its original price. After a week, the car had not sold, so
the owner then discounted the price by 10%, and the car was finally sold.
What price was the car sold for?
(a) \$25,200
(b) \$25,000
(c) \$24,948
(d) \$24,542
(e) \$23,658
Pay attention, when you raise a number by X% and than you reduce X%
you don’t get the original number again because the second time you took
X% off you reduced it from a larger number thus answer A is not the
correct one, let’s check:
25,200 x 1.1 = 27,720.
27,720 x 0.9 = 24,948 and not 25,200. The correct answer is C.
26. A frustrated greengrocer is trying to cell cucumbers at a price of \$1.5
per Kg. Unfortunately he has no success. The greengrocer gives a
discount of 18% on the original price but than the cucumbers are sold too
fast so he raises the price again by 10%. At that final price, how many
cucumbers can you buy for \$4.5 assuming that there are 12 cucumbers
per Kg and that only a whole number of Kgs are sold?
(a) 25
(b) 34
(c) 40
(d) 46
(e) 48
1.5 x 0.82 = \$1.23.
\$1.23 x 1.1 = 1.353\$
5 / 1.353 = 4 Kg and change = 48 cucumbers.
27. An air-conditioning unit costs \$470. On December there was a
discount for Christmas of 16%. Six months later, the holiday season was
over so the company raised the price of the air-conditioning by 16%.
How much will an air-conditioning unit cost in November?
(a) \$458
(b) \$470
(c) \$472
(d) \$484
(e) \$491
Pay attention, when you raise a number by X% and than you reduce X%
you don’t get the original number again because the second time you took
X% off you reduced it from a larger number thus answer A is not the
correct one, let’s check:
470 x 0.84 = 394.8.
394.8 x 1.16 = approximately \$458.
A simpler way to solve this problem is by knowing that the price would
be lower than the original price because we increased and decreased the
same amount of percentage.
28. In a rectangular coordinate system, what is the area of a triangle
whose vertices have the coordinates (4, 0), (6, 3), and (6, -3)?
(a) 7.5
(b) 7
(c) 6.5
(d) 6
(e) 5.5
First draw the x and y-axes, then plot the points and connect them. The
length of the base is 6 units [from (6, 3) to (6, -3)] and the height is 2
units [from (6, 0) to (4, 0)].
Area of a triangle = (base × height) / 2, so (6 × 2)/2 is 6.
30. In a rectangular coordinate system, what is the area of a rectangle
whose vertices have the coordinates (-4, 1), (1, 1), (1, -3) and (-4, -3)?
(a) 16
(b) 20
(c) 24
(d) 25
(e) 30
First draw the x and y axes, then plot the points and connect them, right
away you can see that the base is 5 units and the height is 4 units. The
area of the rectangle is 20.
31. In a rectangular coordinate system, what is the area of a rhombus
whose vertices have the coordinates (0, 3.5), (8, 0), (0, -3.5), (-8, 0)?
(a) 56
(b) 88
(c) 112
(d) 116
(e) 120
First draw the x and y axes, then plot the points and connect them.
The area of a rhombus is simply the product of its diagonals divided by 2.
The area is = 16 x 7 = 112/2=56.
32. In a rectangular coordinate system, what is the square root of the area
of a trapezoid whose vertices have the coordinates (2, -2), (2, 3), (20, 2),
(20, -2)?
(a) 7.5
(b) 9
(c) 10.22
(d) 12.25
(e) 14
First draw the x and y axes, then plot the points and connect them.
The area of a trapezoid is (base1 + base2) x (height) / 2.
Base1 = 5, base2 = 4, height = 18 thus the area is 9 x 9 = 81.
The answer to the question is the square root of 81, meaning 9.
33. How much interest will \$2,400 earn at an annual rate of 8% in one
year if the interest is compounded every 4 months?
(a) \$141
(b) \$150
(c) \$197
(d) \$234
(e) \$312
Here, it is enough to calculate the simple interest of 8% per year.
\$2,400 x 8/100 x 1 = \$192. since we are calculating as a compounded
rate, the interest should be a bit higher, or C as the best answer.
34. A GMAT class has a ratio of girls to boys of 1.5 to 3. If the class has
a total of 36 students, how many more girls are there than boys?
(a) 8
(b) 10
(c) 12
(d) 15
(e) 18
This is a standard ratio problem. 36 / (1.5 + 3) = 8.
The number of boys is 8 x 3 = 24.
The number of girls is 8 x 1.5 = 12.
The difference between the numbers is 12.
34. A Math-club class has a ratio of girls to boys of 1.5 to 4.5. Out of all
the boys 16.66% are left-handed, how many left-handed boys are there in
the class assuming that there are 24 students all together.
(a) 8
(b) 6
(c) 5
(d) 4
(e) 3
This is a standard ratio problem. 24 / (1.5 + 4.5) = 4.
The number of boys is 4 x 4.5 = 18.
The number of girls is 4 x 1.5 = 6.
16.66% out of 18 are 3 boys.
1. Is a>b?
(1) a2 > b2
(2) a+d > b+d
Explanation:
Plug in numbers. The first statement will work when a=3 and b=2, for
example and then a2=9 and b2=4. However, it will not work when a=(-2) and
b=(-3), then a2=4, and b2=9.
The second statement is sufficient, it is possible to subtract d from both sides
of the inequality and get: a+d-d>b+d-d, or a>b.
2. At a certain library, there are fiction and non-fiction books only. How
many of the books are non-fiction books?
(1) There are 13,200 books at the library.
(2) 35% of the books at the library are non-fiction books.
Explanation:
The first statement alone does not give the information needed to know how
many of the 13,200 books are fiction and how many are non-fiction. The
second statement does not tell us 35% of what number are non-fiction books.
The two statements together, give us enough information to calculate 35% of
13,200 and find the number of non-fiction books.
3. Is the integer X even?
(1) X is divisible by 7.
(2) X is divisible by 11.
Explanation:
The fact that X is divisible by 7 does not help figure out whether it is odd or
even, both even and odd numbers could be divisible by 7. The same applies
for 11; both even and odd numbers could be divisible by 11. Both statements
taken together do not shed a new light on the matter, there could be even and
odd numbers that are divisible by 7 and 11.
4. Is (a+b)2 + (a+b)3 even?
(1) a and b are positive.
(2) a>b.
Explanation:
Each statement alone is sufficient since there are only three possibilities:
(1) a and b are even.
(2) a and b are both odd.
(3) One is odd and the other is even.
Any of the options give us an even result, thus, the expression is always even.
5. Is the product xy divisible by 22?
(1) x is divisible by 4.
(2) y is divisible by 11.
Explanation:
The prime factors of 22 are 2 and 11. Hence, if x is divisible by 4, it is
divisible by 2 and if y is divisible by 11, surely xy is divisible by 22.
6. If A, B and C are integers. Is AB a factor of C?
(1) A is a factor of B.
(2) B is a factor of C.
Explanation:
Plug in numbers: A=4, B=8, C=16. A is a factor of B and B is a factor of C,
However, AB=32 is not a factor of C=16.
7. What is the value of (a+b)?
(1) a2-b2=133.
(2) a-b=7.
Explanation:
Since a2-b2=(a+b)(a-b), 133=(a+b)7, and (a+b)=19.
Both statement are needed to solve the question.
8. What is the value of x+z?
(1) x+y=11
(2) z+y=13
Explanation:
Each statement alone leaves out one of the terms x or z, so we cannot find
their sum using any statement alone. Moreover, even combining both
statements does not help:
x + y = 11 z + y = 13
y = 11 − x y = 13 − z
11 − x = 13 − z
z − x = 13 − 11
It is only possible to find z-x.
9. What was the total amount John earned on his two investments?
(1) John received an annual interest of 5% on one investment and 13%
on the other.
(2) John invested a total of \$15,000 on both investments.
Explanation:
Knowing the interest alone is not enough to calculate the profit.
Knowing the total amount invested is not enough to calculate the profit,
unless we have the interest rate.
Since we have no knowledge of the amount invested in each investment, there
is no way to know how much was earned.
10. What percent of the employees In X Company are managerial
employees?
(1) 30% of the employees are technical.
(2) There are exactly 45 clerical employees in the X Company.
Explanation:
The fact that 30% are technical or that 45 are clerical, does not tell anything
about the rest of the workers. In this question, 3 types of workers appear,
technical, clerical and managerial. However, there is no mention of whether
other types of workers exist also. Since we cannot assume there are no other
types, we cannot answer the question.
11. What is the average of a sequence of integers?
(1) There are 15 integers in the sequence.
(2) The sum of the integers in the sequence is 1275.
Explanation:
An average of a set of integers is calculated as the sum of the integers divided
by the number of integers in the sequence. Statement 1, gives the number of
integers, which is not enough by itself. Statement 2, gives the sum of the
integers in the sequence, which is not enough by itself. Using the data from
both statements is enough to find the average.
12. What is the sum of the two smallest integers in a set of different
positive integers?
(1) There are 4 integers in the set.
(2) The average of the integers in the set is 3.
Explanation:
The only two possible sets that have 4 different positive integers and an
average of 3 are: [1, 2, 3, 6] and [1, 2, 4, 5]. The sum of the two smallest
integers in both sets is always 3.
13. If cookies are put in a jar and the jars of cookies are packed in a carton
box, how many cookies does one carton box contain?
(1) Every carton box is filed to half its volume.
(2) Twenty cookies are put in each jar, and 12 jars are put in each
carton box.
Explanation:
The first statement gives no information of the number of cookies. The
volume of the carton box or the percent of its volume filled, do not help in
finding any number. The second statement gives all the information needed
in order to find the number of cookies packed in each carton box. The
number of cookies per jar, and the number of jars per box are sufficient to
calculate real numbers.
14. The total volume of a swimming pool, when filled to capacity, is 2,652
gallons. How long will it take for the pool to fill up?
(1) Water is being purred into the empty pool at the rate of 120 gallons
per minute.
(2) It takes 5 hours to empty the pool when it is half full.
Explanation:
It is sufficient to know the rate of water being purred and that the pool
was empty to calculate the time needed for the pool to fill up:
2,652
= 22.1min.
120
Therefore, statement 1 is sufficient. Statement 2 does not give information
regarding the rate of filling the pool, thus, it is not sufficient.
1. X is an even number, which of the following is odd?
(a) X2.
(b) (X +1) 2
(c) (X+2) 2
(d) X3 + X
(e) 2X2
The easiest way is to try out a number, lets say X=2.
You can see that B is 9, and that is always an odd number.
2. X is an integer, which of the following must be even?
(a) X (X +2) + 2
(b) X (X +1) +1
(c) X (X + 1)
(d) X2
(e) X3 +1
You don’t know whether or not X is even. In answer C you have a
multiplication of two consecutive numbers so one of them must be
even and an even number multiplied by an odd number is always
even.
3. x, y, z, and w are integers. The expression x-y-z is even and
the
Expression y-z-w is odd. If x is even what must be true?
(a) y-z must be odd.
(b) w must be even.
(c) w must be odd.
(d) z must be even.
(e) Z must be odd
The first expression is even and the second is odd, the differences
between the two expressions is x instead of w. (remember, there is no
difference in odd/even numbers if the number is positive or negative so yz is like z-y). Therefore if x is even w must be odd.
X is an even number and Y is a positive odd number. Which of the
following expressions mustn’t be even?
(a) (XY) Y
(b) X3Y3
(c) X3
(d) XY
(e) Y2
The fastest way to solve this problem is by trying out some numbers.
Lets say: X = 2, Y = 1.
According to answer e: 1 x 1 = 1 and that must be an odd number.
X is a prime number bigger than 10. Also, Y = X+X3+X5+X7 .
What is definitely true about Y?
(a) Y is a prime number.
(b) Y is odd.
(c) Y is even.
(d) Y can be equally divided by 3.
(e) Y can be equally divided by 7.
Because X is a prime number bigger than 10, he must be odd. Ignoring the powers of
X in the expression of Y, you’ll see that Y is a sum of 4 odd numbers therefore it must
be even.
P is divisible by 4.Q is divisible by 3. Which of the following is definitely
odd?
Q(P+1)
2P+3Q
PQ2
P2Q3
None of the above.
P must be even but Q is either even or odd (3,6,…). None of the following answers
are definitely odd although some can be.
A and B are integers. If 2A-B = B-A, than which of the following is true?
B<A
A<B
A and B are even.
A is even and B is either even or odd.
B is even.
2A-B = B-A
3A = 2B. Therefore 3A must be an even number and since 3 is odd, A
must be even and B can be either even or odd.
8. A is even and B is odd. Which of the following expressions can’t be
an integer?
(a)
(b)
(c)
(d)
(e)
A +1
B +1
A +1
B
B− 1
A2
A
64
A4
A⋅ B
In this question we are looking for an expression: odd/even, which can’t
be an integer. In answer A: The numerator is odd and the denominator is
even therefore it can’t be an integer.
1. If X is a positive integer, does X have six distinct positive factors?
(1) X is the product of four different positive primes.
(2) X = 18.
You can find out how many factors a number has if you know what that
number is, or something about its prime factorization. Look for this as
you move on to the statements.
Statement (1) is sufficient. If X is the product of four different prime
numbers it will have six different factors; 1, the four prime numbers, and
itself.
Statement (2) is also sufficient. 18 has 6 different factors: 1, 2, 3, 6, 9 and
18.
2. If Y is a positive integer, does Y have four distinct positive factors?
(1) Y = 8.
(2) Y is a multiplication of two different odd numbers.
Statement (1) is sufficient since 8 has the following factors: 1, 2, 4 and 8.
Statement (2) is not sufficient. For example, take 1 and 3, the product is
3, which has only two factors. But if you take 3 and 5, the product is 15
and we have 1,3,5 and 15 as factors of y, and we have four factors.
2. What is the value of (X + Y)?
(1) 3X + 8 = 14 – 3Y.
(2) (X + Y)2 = 4.
From statement (1) we can find that 3X + 3Y = 6
(X + Y) = 2. (1) is
sufficient.
Statement (2) is insufficient, (X + Y) can be 2 or -2.
What is the value of (X2 + Y2)?
(1) 4X2 – 7 = 17 – 4Y2.
(2) (X + Y)2 = 6 + 2XY.
Statement (1) can be written as 4X2 + 4Y2 = 24
X2 + Y2 = 6. (1) is
sufficient.
X2 + Y2 =
Statement (2) can be simplified: X2 + 2XY + Y2 = 6 + 2XY
6. This statement is also sufficient.
If X and Y are integers and X + Y < 0, can X be greater than Y?
(1) X < -2.
(2) Y > -4.
From each of the statements by itself we cannot determine if X can be
greater than Y. Using both statements, we know the values each of the
variables can get:
X: -3, -4, -5, …
Y: -3, -2, -1, …
We can see that X, at the most, can be equal to Y but it cannot be greater.
Both statements, taken together, are sufficient.
If X + Y = 17, is X < 0?
(1) X < 17.
(2) Y < 17.
Statement (1) is insufficient. X can be 16 and then Y = 1 or X could be -2
and then Y would be 19.
Statement (2) is sufficient, if Y is smaller than 17 than X must be
negative in order to balance the expression back to 17.
X, Y and Z are three positive prime integers. What is the value of Y?
(1) The product XYZ is divisible by 4.
(2) X is an odd number.
The question alone provides little information. Notice that it does not say
that X, Y and Z are different.
Statement (1) tells us that XYZ is divisible by 4 and therefore two of the
prime numbers are 2, meaning that only one of the prime numbers are
odd.
Statement (2) completes statement (1) by adding that X is odd and
therefore Y and Z must be equal to 2. Both statements, taken together, are
sufficient.
If X3 = Y, is Y a fraction?
(1) X2 is a fraction.
(2) X > Y.
Statement (1) is sufficient. If X2 is a fraction, X must also be a fraction.
Meaning that X3 and Y will also be fractions.
Statement (2) is sufficient. If X3 is a fraction then X must be greater than
X3, which is also equal to Y. We get that from X > Y, so X3 and Y are
fractions.
Is 12x + 2 = 10 + 3x?
(1) 5x is smaller than or equal to 12
(2) 2x is greater than or equal to 4
We can simplify the given question ' is 9x = 8? ' or ‘ is x = 8/9 ‘ ?
Statement (1) tells you that 5x is smaller than or equal to 12, 5x < 12,
which essentially means, x < 12/5. If x < 12/5, it is unknown whether x
does or does not equal 8/9. In other words, statement (1) may or may not
be sufficient.
Statement (2) tells you that 2x is greater than or equal to 4. This means
that x could never be less than two, therefore could never be equal to 8/9
and this statement is sufficient.
Is 22Y = 10 + 7Y?
(1) -5Y is greater than or equal to -10.
(2) -6Y is smaller than or equal to -6.
We can simplify the given question ' is 15Y = 10? ' or ‘ is Y = 2/3 ‘ ?
Statement (1) tells you that -5Y is greater than or equal to -10, -5Y > -10,
which essentially means,
Y < 2. If Y < 2, it is unknown whether Y does or does not equal to 2/3.
Statement (1) is insufficient.
Statement 2 tells you that -6Y is smaller than or equal to -6 or that Y > 1.
This means that Y could never be less than one, therefore could never be
equal to 2/3. This statement is sufficient.
What is the value of (A + 3B/7)?
(1) 5600A + 2400B = 12,000.
(2) 50B - 50 + 250A = 9700 – 4200A – 1900B.
Simplify the (1) statement by dividing both sides by 5600: A + 3B/7 =
15/7. This statement is sufficient.
Simplify the (2) statement by adding similar items to get: 4550A + 1950B
= 9750. Divide both sides by 4550 to get: A + 3B/7 = 15/7 this statement
is also sufficient.
What is the value of (X + 2Y/5)?
(1) 292X – 675 + 80Y = 100 – 18X – 44Y.
(2) 300X + 80Y – 830 = 82.5 – 85X – 66Y.
Simplify statement (1). Add similar items, 310X + 124Y = 775 ----> X +
2Y/5 = 2.5. This statement is sufficient.
Do the same to statement (2) and you’ll see that 385X + 146Y = 912.5,
when divided by 385: X + 146Y/385 = 2.5. This statement is insufficient.
What is the par value of the stock Y?
(1) The purchase price of stock Y was 120 Australian dollars.
(2) Stock Y increases in value by 8.5% each year.
We are asked to specify the exact value of Y today.
Statement (1) is insufficient since it gives us the purchase value of the
stock. However, no purchase date is provided.
Statement (2) is also insufficient since it doesn’t specify how many years
have passed since the stock was bought. Both statements together are also
insufficient since they don’t mention the time that has passed since the
stock was bought.
A turtle is crossing a field, how many meters total did he pass?
(1) The average speed of the turtle is 2 meters per minute.
(2) Had the turtle walked 1 meter per minute faster than his average speed
it would have finished the same path 40 minutes earlier.
Statement (1) gives us the average speed of the turtle; this statement is
insufficient since the time is not given.
Statement (2) is also insufficient by itself since we don’t know what the
average speed is.
Both statements combined are sufficient since we can calculate the
distance
2*T=D and 3(T-40)=D, solve to get D=240 meters.
What is the distance that Cynthia has to travel in order to get from the
university to the dorms?
(1) Cynthia can walk half the distance in 7 minutes when walking at her
fastest possible rate.
(2) Cynthia is walking at an average speed of 1 mile in 12 minutes.
Statement (1) is insufficient since it only gives the time period of
reaching the half point at the maximum speed.
Statement (2) is also insufficient by itself since we are only told the
average speed.
Both statements, taken together, are insufficient since the average speed,
mentioned in (2) might be different than Cynthia’s fastest possible rate.
The net value of a certain stock increased at a constant rate during the
ten-year period between 1990 and 2000. What was the value of the stock
in the year 1998?
(1) In 1991, the value of the stock was 130 U.S dollars.
(2) In 1992, the value of the stock was 149.5 U.S dollars.
We are told that the stock increases its value by a constant rate and
therefore we need to find some kind of pattern in order to know the value
every year.
Statements (1) and (2) taken together are sufficient since they give us the
percent increase of the stock from 1991 to 1992, which is 15%. The value
of the stock in 1998 can be easily calculated, add 15% every year until
1998.
A spaceship in orbit rotates around the planet Pluto. How many full
rotations can the spaceship complete in 20 hours.
(1) The radius of the rotation is 21,000 miles.
(2) The spaceship travels at 35 miles per second.
In order to calculate the time it would take to complete one rotation, you
need the radius and the velocity. Statement (1) provides the radius of
rotation and (2) provides the velocity.
The distance the spaceship has to travel in order to complete one rotation
is 2 ⋅ π ⋅ R = 2 ⋅ π ⋅ (21,000) and the speed is 35 miles per second. Divide the
first by the second and you’ll get the time it would take the spaceship to
complete one rotation.
What is the total number of students that will finish their first degree this
year at the Willhunt University?
(1) The number of male students that will finish their first degree ten less
than three-fifths of number of female students.
(2) The number of male students is 360.
Let m = males and f = females.
From Statement (1) you know that m = 3/5f - 10, but you don't know the
actual values of m and f. From Statement (2) you know m, which can be
fed into the equation derived from statement (1).
These two equations with two variables are sufficient and therefore both
statements, taken together, are sufficient.
Is X > Y?
(1) 12X = 4C.
(2) C = 3Y4.
Since we need to compare between X and Y, look at both statements
together.
(1) and (2) state that: 12X = 4C = 12Y4
X = Y4.
Take Y=-1, X=1: X is bigger than Y.
Take X=1 and Y=1: X is not bigger than Y.
We can see that the answer is not distinct and therefore more data is
required in order to solve the question.
If X is a prime number, is Y even.
(1) X = Y + 1.
(2) X = 5.
Statement (1) is insufficient by itself since X can be even (2) or odd (3).
Statement (2) is insufficient since we don’t know the relation between X
and Y.
Both statements together are sufficient since we know that X is odd and
that Y is an odd number plus one, meaning an even number.
Is Y/X odd?
(1) Y is a prime number.
(2) X is a prime number.
Each statement alone is insufficient since it tells us nothing of the other
variable.
(1) and (2) together are trick. If you take X=Y=5 or any other prime
number the result of Y/X will be odd but if you take X=5 and Y=7 for
instance the result wouldn’t even be an integer.
The answer cannot be determined and more data is required.
If A and B are two different integers, is A/B even?
(1) A is prime.
(2) B is prime.
Each statement alone is insufficient since each statement refers to one
variable only.
(1) and (2) together are sufficient since dividing any two different primes
will never result in an integer. A prime is a number that has exactly two
natural divisors, 1, and itself and is therefore not divisible by any other
integer.
How many mini sports cars does little Timmy own?
(1) Timmy has 10 red sports cars.
(2) The number of blue sports cars is 50% larger than the number of red
sports cars.
Statement (1) gives us the number of red cars and (2) the number of blue
cars. We are not told that there are only two colors of sports cars and
therefore more data is required.
How many keys are found on an average keyboard?
(1) There are 20 number keys on an average keyboard.
(2) 20% of the keys on the keyboard are number keys.
Statement (1) gives us the number of number keys only and thus is
insufficient.
Statement (2) alone is insufficient because it relates to the number stated
in statement (1).
Both statements together are sufficient since we are told that 20 buttons is
20% of the total and therefore there are 100 buttons on the average
keyboard.
What is the amount of interest paid on an X dollars loan over a 6 months
period?
(1) X = 12,000.
(2) The interest rate is 5%.
Statement (1) gives out the amount of money loaned.
Statement (2) gives the interest with out mentioning during what time is
the interest 5%.
Both statements are insufficient since we cannot determine what the
interest on \$12,000 is during a 6 months period. The interest could be
monthly or annually or anything else.
Merline made a \$360,000 mortgage on a house. How much interest total
will she pay?
(1) The simple interest rate is 8.5% annually.
(2) It will take Merline 12 years to return the loan on the house.
Statement (1) implies that the annual interest is 8.5%. it is not sufficient
by itself since we do not have the time period of the return.
Statement (2) implies that 12 years have passed until Merline paid the
loan but we do not have the interest rate.
During the 12 years, we can calculate the interest every year until we
reach 12 years. Both statements together are sufficient.
A simple interest can be calculated using the following formula:
Principle (money loaned or invested)*rate (percent) *time = interest paid
Nicola bought a new cellular phone. How much did it cost him?
(1) Every month Nicola pays 45 franks to the cellular company, which
includes the calls and the cellular device itself.
(2) Every month, Nicola spends twice on calls than on the cellular device.
Statement (1) gives the total cost of the device and the calls, define C as
calls and D as device. You can write the following equation: C + D = 45.
(1) is insufficient.
Statement (2) is also insufficient by itself, it implies that C = 2D.
(1) and (2) together are sufficient to find D and C but that is not enough.
D is the money spent on the device per month while we want the total
price of the device.
If we knew how many months he paid we would have enough data.
How much did Nancy pay for her new air-conditioner?
(1) Nancy paid 12 payments, one every month.
(2) On April, Nancy paid \$130.
Statement (1) tells us that Nancy paid one statement every month.
Statement (2) gives us a specific payment on a specific month.
Each statement by itself is insufficient and (1) and (2) together are also
insufficient since it is not mentioned that the all the payments are equal.
What is the perimeter of circle O?
(1) The circle inscribes a square.
(2) The perimeter of the square is 10.
If a circle encircles a square then the diagonal of the square is the
diameter of the circle, which is sufficient to find the perimeter.
Statement (1) tells us about the square that is blocked with out any further
data.
Statement (2) gives us the perimeter, which is equal to 4 times the side of
the square.
If we know the side of the square, we know its diagonal.
Both statements, taken together, are sufficient.
A map is divided into blocks. Each block is 1 inch long and 1 inch wide.
Every inch on the map represents 20 miles. What is the maximum actual
total distance that the Flanders family traveled?
(1) The Flanders were on three different adjacent blocks on the map.
(2) The Flanders traveled in a straight line through the blocks.
Statement (1) is insufficient since we don’t know how many times the
family was in each block and how their trip looked like.
Statement (2) is also insufficient. It is not known through how many
blocks the Flanders drove.
Both statements together are sufficient since the longest distance traveled
through three blocks is the diagonal starting at the bottom corner of the
lowest block and ending at the top opposite corner of the top block. The
length of this line can be calculated using the pythagorean thorem.
A paint shop sells spray cans at a flat charge of 50 cents per can. If a
customet bought 10 cans and the owner decided to give that customer a
special discount on the last two cans, what was the price of the two
discounted cans?
(1) The customer paid four dollars and twenty cents total for the ten cans.
(2) The customer bought the ten cans for an average price of 42 cents per
can.
Statement (1) tells us that 10 cans cost 4.2 dollars instead of 5 dollars and
therefore the last two cans were sold for 20 cents. This statement is
sufficient.
Statement (2) also tells us that the entire amount of cans cost 4.2 and
therefore this statement is also sufficient.
The line Y = X/2 is drawn on a rectangular axis system. If the line is
rotated, on which quadrant will he be found?
(1) The rotation is done counter clockwise.
(2) The line is rotated 270 degrees.
Draw an axis system and draw a line from the coordinate (0, 0) in the first
From (1) we can learn that the rotation is done counter clockwise which
is insufficient.
From (2) we can learn that the rotation is 270 degrees, but in which way?
Join the statements, we know that a 270 degrees turn counter clockwise
was made and therefore the line is now in the fourth quadrant.
The line Y = 3X is drawn on a rectangular axis system. If the line is
rotated, on which quadrant will it be found?
(1) The rotation is done clockwise.
(2) The line is rotated 180 degrees.
Draw an axis system and draw a line from the coordinate (0, 0) in the first
From (1) we can learn that the rotation is done clockwise which is
insufficient.
From (2) we can learn that the rotation is 180 degrees, but in which way?
It doesn’t matter since the rotation is half a circle, which in both ways
will be parallel to the original position.
This statement is sufficient and the line will be in the third quadrant.
Travis parked at a certain parking lot that charges more for the first hour
of parking than any other hour. If it cost Travis 5.5 dollars, how much
time total did he park in the parking lot?
(1) Parking charges are \$1.5 for the first hour and \$1 for any additional
hour of parking or a part of it.
(2) If the charges for the first hour were \$2 then Travis would have paid
\$6 for parking.
Statement (1) gives the data on the charges but it says that every fraction
of an hour is rounded up and therefore, even if Travis was 4 and 10
minutes, he still has to pay for 5 hours.
We cannot determine the exact time that Travis parked.
Statement (2) is insufficient since it doesn’t change the accuracy problem
introduced in (1).
More data is required.
12 liters of a certain water-based color contain A liters of water and B
liters of color. How many liters of water are in the water-based color?
(1) A2 + 16 = 8A.
(2) B2 – 2B = 48.
Statement (1) can be written as: (A – 4)2 ---> A = 4 and B = 12 – 4 = 8.
Sufficient.
Statement (2) can be written as: (B + 6)(B – 8) ---> B = -6 or B = 8 but B
must be a positive, so B = 8 ---> A = 12 – 8 = 4.
Either statement by itself is sufficient.
Does the product of XYZW = 16?
(1) Y = 1.
(2) X = 4Y and ZW = 4Y2
From (1) we know the value of Y only, which is 1.
From (2) and (1) we know the value of all the other parameters, X = 4
and ZW = 4.
Therefore, (1) and (2) together are sufficient.
How many of the girls in a group of 200 children have an average score
of 80 in their final exams?
(1) 45% of the children have an average score of 80 in their final exams.
(2) 50% of the children in the group are girls.
From (1) by itself we can only learn that 90 kids have good grades. This
statement is insufficient.
From (2) we can learn that there are 100 girls in the group.
Combining the statements doesn’t help much since we know nothing
about the overlapping of (1) and (2) and more data is required.
From January 1948 to March 1981, the value of an antique house in the
downtown area tripled. What was its value back in January 1948?
(1) From March 1981 to September 2001, the value of the house doubled.
(2) The value of the house in September 2001 was \$300,000.
Use both statements and solve the problem backwards.
In 2001 the value was \$300,000, which is double the value in 1981.
In 1981, the value of the house was \$150,000, which is triple the value in
1948.
In 1948, the value of the house was \$50,000.
Both statements, taken together, are sufficient.
If A = 2B, is A4 > B4?
(1) A2 = 4B2.
(2) 2A + B < A/2 + B.
Statement (1) is insufficient. Take A=0 and B=0, (1) is correct yet A4 is
not bigger than B4.
Take different numbers, A=6 and B=12 A4 is larger than B4.
Statement (2) is sufficient. The only possible way that A will not be
larger than B is if they are both zero. (2) claims that A < 0 and therefore
A cannot be zero and this statement is sufficient, A4 is bigger than B4.
The Ponds basketball team played 45 games this season, how many did
they win?
(1) The Ponds won their first 25 games in this season.
(2) The team won none of their last 5 games and of the rest they won
75%.
Statement (1) only refers to the first 25 games, all the others are unknown
and therefore this statement is insufficient by itself.
Statement (2) is sufficient. The rest of the games after reducing the 5 last
ones that were lost is 40 games. 75% of the 40 games give 30 games that
the team won.
Is X negative?
(1) X + 12 < 0.
(2) 12X > 14X.
Statement (1) is sufficient. Subtract 12 from both sides to get X < -12 and
therefore X is negative.
Statement (2) is also sufficient. In order for the left side of the inequality
to be bigger than the right side, X must be negative.
Is A an odd integer?
(1) AB + AC is an even integer.
(2) B = 15 – C.
Statement (1) can be written as A(B + C), which means that A or (B+C)
must be even.
Statement (2) tells you that (B + C) is odd and therefore, using statement
(1) A must be even.
Both statements, taken together, are sufficient.
Is Y even?
(1) WX + YW + ZW is an odd number.
(2) X = 8 – Z.
From (1) we can learn that W(X + Y + Z) is odd and therefore W is odd
and also (X+Y+Z).
From (2) we can learn that (X+Z) is even.
Use both statements together: (X + Y + Z) is odd yet (X + Z) is even and
therefore Y must be odd.
In the past few years, the import of cars to Italy increased significantly. In
the years 1994 to 2001 the number of imported cars increased by 12%
each year. How many cars were imported to Italy in the year 2000?
(1) In 1997, 12,500 cars were imported to Italy.
(2) During the years 1997 to 1999, 42,180 cars total were shipped to
Italy.
From the question we know that the number of cars grew by 12% each
year.
From (1) we can learn that if in 1997 12,500 cars were shipped then in
1998 (1.12 x 12.500) cars were shipped and so on until 2000. This
statement is sufficient.
From (2) we can learn that during a certain period 42,180 cars were
shipped, if you define X as the number of cars that were shipped in 1997
then: 42,180 = X + 1.2X + 1.2(1.2X) and so we can find X and anything
else we wish to find. Either statement alone is sufficient.
Mike spends 50% of his time studying and 20% of the rest of his time
going to the Jym. Jim spends 60% of his time in the Jym and 85% of the
rest of his time studying. How many hours more does Mike spend on
studying than Jim?
(1) Jim spends 22 hours a week on working out.
(2) Mike spends 18 hours a week on studying for the finals.
(1) Is insufficient since it gives us information regarding Jim only.
(2) Is insufficient since it gives us information regarding Mike only.
In order to compare the number of hours Mike and Jim spend studying,
we need a real data regarding the number of hours each spends on any
activity. Therefore, we need to use both statements.
5 numbers are randomly chosen. If their average is 20, how many of the
numbers are larger than 15?
(1) One of the numbers is 15.
(2) The average of three of the numbers is 15.
Use both statements.
Lets look at two cases, where each one will give out different results.
One of the numbers is 15 and three more are 15, 15 and 15 (with an
average of 15).
The sum of all the numbers should be (20 x 5 = 100).
The sum of the numbers we picked up is (15 x 4 = 60) and therefore the
last number should be 40 which is greater than 15. The answer in this
case is 1.
Take another case: One of the numbers is 15 and three more are 14, 15
and 16 (with an average of 15). In this case there will be 2 numbers over
15. More data is required.
A certain basketball player receives a \$700 bonus for every dunk that he
performs and a \$1,000 bonus for every game that the team wins. Last
month, the player earned \$9,100 on bonuses only, how many dunks did
he preform last month?
(1) Last month the player preformed at least 3 dunks.
(2) The number of dunks, last month, was two and a half times smaller
than the number of wins.
Statement (1) is insufficient since it gives two possible options: 13 dunks
or 3 dunks and 7 wins.
Statement (2) is sufficient since it eliminates all the answers but one: 7
wins and 3 dunks.
Bob earns twice as much as Bobby and Bobby earns two thirds as much
as Buddy does. How much did Bob earn?
(1) All three earn a total of 18,000 dollars.
(2) Buddy earns \$6,000.
Pay attention, the question presents two equations with three unknowns
and therefore one more equation is required in order to solve the question.
Statement (1) is sufficient since it presents another equation.
Statement (2) is also sufficient since it tells us one of the unknowns.
Either statement alone is sufficient.
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What is the value of the integer X?
(1) X2 = 81.
(2) X is a square of a prime number.
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Statement (1) is insufficient since X can be either 3 or –3.
Statement (2) is insufficient since X can be 22, 32, …
Both statements together are sufficient since (2) eliminates the negative
option of (1).
X is 3.
What is the value of P?
(1) P is even.
(2) P is a square of a prime number?
Statement (1) alone is insufficient since there is infinite number of
possibilities.
Statement (2) alone is insufficient since there are lots of prime numbers.
The combination of the statements is sufficient since the only even square
of a prime number is can be 22, which is 4. Both statements, taken
together, are sufficient.
If AB = 40, what is the value of AB(A + 2B)?
(1) A – B = -18.
(2) A2B = 80.
Statement (1) alone is insufficient since we need the value of A + 2B.
A=2 and
Statement (2) is sufficient. A2B = A(AB) = A(40) = 80
B=20.
Now, we know that A + 2B = 42 and we can calculate the required
expression.
If X3Y = 24, what is the value of (X3Y3 – X2Y2)?
(1) X2Y2 = 36.
(2) X3Y2 = 72.
Statement (1) is insufficient since the expression given is only one of two
needed.
Statement (2) by itself is sufficient since X3Y2 = (X3Y)Y = 24Y = 72
Y = 3.
If Y=3 then X3=8 and so X=2.
The expression required is a combination of X and Y and therefore it is
calculable.
There are two bus routes that go through the chapel and the cemetery. For
each route, the bus company may use one of two types of buses, a
standard bus or a duplex bus. How many passengers total can travel in the
duplex bus?
(1) When both buses are used for the routes, as many as 90 passengers
can travel from the chapel to the cemetery.
(2) On Saturday, the busiest day of the week, two duplex buses are
used for each of the routes and as many as 120 passengers can travel
from the chapel to the cemetery.
Define S as the number of passengers that can travel in the standard bus
and D as the number of passengers in the duplex bus.
Statement (1) can be written as S + D = 90. Insufficient.
Statement (2) can be written as 2D = 120.
Combine the statements, you got all the data you need to solve the
question and therefore they are sufficient together.
X equals to Y% of what number?
(1) X = 3Y.
(2) 6Y+2X = 56X/14.
From (1) we have X and Y and therefore we can find A easily, A = 300.
(2) is identical to (1), simplify it and see that it can be written as X = 3Y.
Either statement by itself is sufficient.
Which expression is larger 1/(5 - X) or X/5?
(1) X < 8.
(2) X > -8.
The easiest way to solve such a problem is to plug in numbers.
Use both statements to see that they are both insufficient even together.
Take X=0: 1/(5 – X) = 1/5 and X/5 = 0, in this case the first expression is
larger.
Take X=7: 1/(5 – X) = -1/2 and X/5 = 7/5, in this case the second
expression is larger.
We can see that the answer is dependent on which numbers we choose
and more data is required in order to determine the answer.
X is a two-digit number. If the ratio between the units digit and the tens
digit is 1 to 2, what is the value of X?
(1) The sum of the digits multiplied by the tens digit is 54.
(2) The product of the digits divided by 2 is 9.
There are a limited number of possibilities: 21, 42, 63 and 84.
According to statement (1), the only number that is compatible is 63.
According to statement (2), the only number that is compatible is also 63.
Either statement alone is sufficient.
Loren bought a roll of cloth and sold it for a 5% profit. If Loren ‘s profit
was \$45.5 total on the cloth, how much did it cost her to buy the cloth?
(a) \$455.
(b) \$525.5.
(c) \$675.
(d) \$810.5.
(e) \$864.5.
5% of the total price of the cloths is 45.5 dollars, multiply this number to
get the entire 100% of the total selling price: (45.5 x 20 = \$910). Now
subtract the profit \$45.5 to get the cost: 910-45.5=\$864.5.
The bus that travels from St. Louis to Chicago and back comes to St.
Louis every 4 hours while the bus that travels from St. Louis to
Springfield and back comes to St. Louis every 6 hours. If at 12:00
o’clock both bused arrived at St. Louis, at what time will they meet in
St. Louis again?
16:00.
18:00.
20:00.
22:00.
24:00.
In order for both buses to meet again, we ought to look for the
smallest common factor of 4 and 6, thus 12 hours. After 12 hours one
of the buses will complete 3 roundtrips while the other will complete
only 2, they will meet at (12:00 + 12 = 24:00).
Monica planned her birthday party. She prepared 5 muffins for each of
her guests and kept aside two additional muffins in case someone will
want extra. After the party, it turned out that one of the guests didn’t
come but every one of the guests that did come ate six muffins and 3
muffins remained. How many guests did Monica plan on?
3.
4.
5.
6.
7.
X is the number of guests that were suppose to show up at the party,
and so Monica prepared 5X + 2 muffins. (X – 1) is the number of
guests that did come to the party and the total number of muffins is
6(X – 1) + 3. The number of muffins that Monica prepared is equal to
the total number of muffins and so we can compare the following
expressions: 5X + 2 = 6(X – 1) +3 ---->X = 5.
If 1/3 < A < 2, which of the following expressions can have the largest
value?
A2/3.
A.
A/2+1/3.
A + 1/3.
A3-4.
Try the answers for the largest and smallest values A can get.
Take A = 1.9, answer E will be the largest and so this is the expression
that can be the largest among the other expressions.
In an isosceles triangle the sum of the sides is 2 inches longer than the
base. What is the ratio between the length of the side and the length of
the base?
1.5
1.
1.75.
2.
Not enough information.
Try taking the base of the triangle as 1 inch. Both of the sides are 3
inches.
And the portion required is (1.5 / 1) = 1.5.
Try taking different numbers, take 10 inches as the length of the base.
The length of both sides is (10 + 2 = 12 inches).
The portion required is (6 / 10).
We can see that a randomly pick of two numbers will give two
Tim and Élan are 90 miles away from one another. They are starting
to move towards each other simultaneously, Tim at a speed of 10 Mph
and Élan at a speed of 5 Mph. If every hour they multiply their speeds,
what is the distance that Tim will pass until he meets Élan?
30 miles.
35 miles.
45 miles.
60 miles.
65 miles.
Tim is traveling at twice the speed of Élan, and so will be after they
multiply their speeds.
In other words, their speeds will always be at a 2:1 ratio no matter
what and therefore the ratio between the roads that they’ll pass will
also be 2:1 or 60 miles to 30 miles. Tim will go through 60 miles.
The distance between West-Town to East-Town is 15 kilometers. Two
birds start flying simultaneously towards one another, the first leaving
from West-Town at a speed of 4 kilometers per minute and the second
bird, leaving from East-Town, at a speed of 1 kilometers per minute.
What will be the distance, in kilometers, between the meeting point
and West-Town?
3.
7.
10.
12.
15.
The ratio between their speeds is 4:1 and so will be the ratio between
the distances that they’ll pass.
The first bird will pass a distance of 12 Km and the second bird will
pass only 3Km.
The meeting point will be 12 Km from West-Town.
A car traveled from San Diego to San Francisco at an average speed of
48 miles per hour. If the journey back took twice as long, what was
the average speed of the trip?
24.
32.
36.
42.
44.
Average speed can be found by dividing the total distance by the total
time. If the journey back took twice as long then the car traveled at
half the speed, thus 24-mph. Take the time the car traveled from San
Diego to San Francisco as T, and the time it took to get back as 2T.
The total distance is 48T+24*2T= 96T, now divide it by the total time,
3T to get 32 mph.
What kind of flowers does a florist have more of, lilacs or roses?
(1) The number of roses he has is less than 6 times the number of
lilacs that he has.
(2) One fifth of the number of Lilacs is less than the number of roses
that he has.
Define R as the number of roses and L as the number of lilacs.
From the first statement we can write the following inequality: R <
6L.
Both L = 1 and R = 2, L = 2 and R = 1 fit the inequality and therefore
this statement is insufficient.
From statement (2) we can write the following inequality: L/5 < R.
Both L = 1 and R = 2, L = 2 and R = 1 fit the inequality and therefore
this statement is insufficient.
Both statements combined are also insufficient, use the same numbers
to prove it and therefore more data is required.
If “Alfa” is defined as 2 percent of 5 degrees, how many “Alfa’s” are
there in two circles?
0.72.
7.2.
72.
720.
7200.
“Alfa” is defined as (0.02 x 5 degrees = 0.1 degrees).
In a circle there are 360 degrees, in two circles there are 720.
(720 degrees / 0.1 = 7200 “Alfa’s”) in two complete circles.
What percent is A of B?
(1) B is 32 more than the square root of A.
(2) A is more than 12 less than B/2.
In this question you have two unknowns, A and B, and thus you need
two equations.
Statement (1) is not enough to find A and B.
Statement (2) can be written as: A > B/2 – 12.
The second statement is an inequality and not an equation and another
equation is needed to solve the question.
What percent is X of Y?
(1) Y is bigger than 2X by 54.
(2) X is smaller than 3Y by 72.
Statement (1) can be written as: Y = 2X + 54.
Statement (2) can be written as: X = 3Y – 72.
Combining both statements, we have two different equations
containing X and Y and so we can solve and find the value of X and Y
and calculate what is X percent of Y.
A store bought Q windows at \$150 per window and W shelves at \$75
per shelve. What is the total price of the windows and the shelves?
(1) The Q windows cost \$600.
(2) Q + W/2 = 12.
Statement (1) is all about the windows and therefore it’s not sufficient
by itself.
Statement (2) tells us that Q + W/2 = 12, this is sufficient because the
total price of windows and shelves is 150Q + 75W.
Multiply the data in statement (2) be 150 to get: 150Q + 75W =
18,000.
Statement (2) is sufficient by itself.
What is the smallest possible common multiple of three integers, all
larger than 26?
27.
54.
846.
19,656.
21,924.
All of the integers have to be greater than 26, thus 27 and up. The
question didn’t mention that they have to be different and so they can
all be equal to 27.
The smallest common multiple of 27 is 27 itself.
What is the smallest possible common multiple of two integers, both
bigger than 260?
261.
262.
524.
12,542.
18,244.
All of the integers have to be greater than 260, thus 261 and up. The
question didn’t mention that they have to be different from one
another and so they can all be equal to 261.
The smallest common multiple of 261 is 261.
A customer asks the clerk for a paintbrush and a bucket of whitewash
for a total price of B dollars. If the paintbrush costs 200 cents less than
twice the price of a bucket of whitewash, what is the price of half a
bucket of whitewash in dollars?
(B + 200)/2.
(B + 2)/6.
(2B + 200)/3.
(B + 2)/3.
(B + 2)/6.
Define W as the price of the bucket of whitewash and P as the price of
a paintbrush.
You can write the following equations: P + W = B and P = 2W – 2.
Notice that 200 cents are translated into 2 dollars.
From the equations we can conclude that W = (B + 2)/3.
Therefore the price of half a bucket is (B + 2)/6.
If 34 boxes of cucumbers in brine cost A dollars, how much will B
boxes cost in cents?
AB/34.
17AB/50.
34/AB.
50AB/17.
100AB/17.
If 34 boxes cost A dollars then one box costs A/34 dollars or 100A/34
cents = 50A/17.
B boxes will cost B x 50A/17 = 50AB/17.
If X boxes of pineapple juice cost 342 cents, how much will Y boxes
cost in dollars?
342Y/X.
X/(342Y).
XY/(342).
3.42X/Y.
3.42Y/X.
If X boxes cost 342 cents (or 3.42 dollars), then each box costs
3.42/X.
Y boxes will cost 3.42Y/X.
Jeremy bought 2Q steaks for W dollars. Jerome buys R steaks for a
50% discount, how much will the steaks cost him in cents?
50RW/Q.
50QR/W.
25RQ/W.
25RW/Q.
RW/(4Q).
Jeremy paid W dollars (or 100W cents) for 2Q steaks and therefore
each steak cost him 50W/Q.
Jerome has a 50% discount, so every steak costs him 25W/Q.
Jerome wants to buy R steaks; it will cost him 25RW/Q.
An electrical appliances store sold this month 400% more than the
average of all the other months in the year. The sales total for this
month was approximately what percent of total sales for this year?
14%.
21%.
31%.
37%.
43%.
Plug in 1\$ as the sales for each of the other 11 months.
On that special month, the store sold 400% more (400% of 1 is 4), or
\$5. The sales of this month relative to the sales of the entire year are
5/(11 + 5). 5/16, which is a bit less than 1/3 (33%).
What is the units’ digit of (9)7(17)3(3)3 ?
3.
5.
7.
8.
9.
(17)3(3)3 = (17 x 3)3 = (51)3 = 51 x 51 x 51 ---> the units’ digit is just
1 x 1 x 1 = 1.
We are left with the units’ digit of 97.
(9)7 = 9 x 81 x 81 x 81 ---> the units’ digit is 9.
And therefore the units’ digit of the entire expression is 9.
A certain factory produces buttons and buckles at a uniform weight. If
the total weight of 2 buttons and 2 buckles is one third of 11 buckles
and 3 buttons, then the weight of 3 buttons and 2 buckles is how many
times that of 5 buckles and 6 buttons?
7/15.
4/9.
6/11.
5/9.
8/15.
Define B as the weight of a button and K as the weight of a buckle.
The total weight of (2B + 2K) is equal to
1
(11K + 3B) ---> 3B = 5K
3
B = 5K/3.
The question requires the ratio between (3B + 2K) and (5K + 6B).
The first factor is equal to (5K + 2K = 7K).
The second factor is equal to (5K + 2(5K)) = (15K).
The ratio between them is 7:15 and therefore the first factor is 7/15 of
the second one.
In a recent survey, Q people were asked whether they eat after 22:00
O’clock. 25 percent of the people answered positively and 40 percent
of the rest were asked, at what time do they get up in the morning.
Which of the following expressions represents the number of people
who do not eat after 22:00 and were not asked about the time they get
up in the morning?
3Q/20.
Q/10.
5Q/9.
3Q/10.
9Q/20.
75Q/100 is the number of people who eat after 22:00. 60% of that
number is the number of people who weren’t asked upon, when do
they get up at the morning.
60 x (75Q/100) / 100 = 45Q/100 = 9Q/20.
In a recent tender, X people participated. 35% of the X people, who
made an offer won the specific tender they participated in. 70% of the
rest, were disappointed from the result of the tender. Which of the
following expressions represents the number of people who weren’t
disappointed although they didn’t win the tender?
39X/200.
25X/50.
19.5X/200.
35X/250.
90X/200.
65X/100 is the number of people who didn’t win the tender. 30% of
that number is the number of people who weren’t deeply disappointed
about the fact that they didn’t win.
30 x (65X/100) /100 = 19.5/100 = 39/200.
A and B are integers. The expression (A+1)(B+1) is even.
What can be said about A and B?
They are both even numbers.
At least one of them is even.
At least one of them is odd.
They are both odd.
Nothing can be said surly on A and B.
Because the given expression is even, at least one of the phrases in
one of the parenthesis must be even therefore either A or B must be
odd.
In orange county one fifth of the people are gathering mushrooms and
one seventh of the people are collecting apples. What can be the
number of people in Orange County?
60.
42.
85.
140.
252.
The answer must be a number that is divisible by both seven and four.
The only possible number is 140.
In Tukitu village, one forth of the people are raising flowers, one ninth
are growing wheat and one eleventh are growing vegetables.
What could be the number of people in the village?
792.
540.
198.
132.
346
The answer must be a number that is divisible by 4, 9 and 11 together.
The only possible answer is A.
Danny can divide his herd into 5 equal parts and also to 6 equal parts,
but not to 9 equal parts. What could be the number of cows Danny has
in his herd?
155.
336.
180.
120.
456
The number of cows is divisible by 5 and 6 but not by 9. Meaning it must
end with a 5 or a 0 and be divisible by 3 (the sum of its digits is divisible
by 3). That leaves answers C and A only. However, 180 is also divisible
by 9 and is ruled out.
The number of bunnies in Peter’s yard increases by 4 times every
week. How many weeks will it take for the number of bunnies to be
divisible by 8 assuming that he started out with 3 bunnies?
1.
2.
3.
4.
Never.
In the beginning he has 3, one week later he has12 and two weeks
later he has 48 and that can be equally divided by 8.
If X = 23 x 52 x 7, then the expression X/8 is not divisible by:
1.
2.
5.
7.
25.
From the given expression we learn that X/8 = 25 x 7, which is not
divisible by 2.
The remainder when dividing the expression (X + Y) by 5 is 4.
The remainder of X divided by 10 is 2. What is the remainder of Y
divided by 5?
1.
2.
3.
4.
5.
Plug in numbers that fit the conditions. (X+Y) = 19 and so 19/5 gives
a remainder of 4. X = 12 and so 12/10 gives a remainder of 2.
Therefore Y is 7. 7/5 = 1. The remainder is 2.
Q is a prime number larger than 10. What is the smallest positive
number (except 1) that 3Q can be equally divided by?
3Q.
Q
3
Q+3
2Q
3Q is a prime number so it can be divide equally by 3Q, by 1 and by
the components 3 and Q. The smallest number is 3.
A is a prime number (A>2). If C = A3 than in how many different
integers is C divisible?
5
6
3
4
7
Factorize C: C = A x A x A. Therefore C can be equally divided into
1, A, A2, and A3 =C 4 numbers all together.
A can be divided by 11 with no remainder. Which of the following
expressions could be divided by 11 leaving a remainder of 1?
A-20.
A-12.
A-9.
A-10.
A-13.
A/11=X, so A=11X. Now change A in answer D with 11X to get:
11X-10. Plug in numbers in X to find out if it could leave a remainder
of 1 when divided by 11.
Let’s try X=2 and A = 22. In expression D: 22-10=12.
12/11 = 1 with a remainder of 1. try plugging the same number (2) to
each of the expressions to find the right answer.
Eggs are sold in packages of six or eleven only. If Doris bought 70
eggs exactly, what could be the number of large packs Doris bought?
6.
2.
3.
5.
4.
If she bought 2 large packs (22 eggs) than she has 48 eggs left. 48
eggs can be equally divided into eight boxes of 6.
A, B, C are three consecutive positive integers (A>B>C).
What is the value of the expression 2A +B +3C?
6A+7.
5A+1.
5A-1.
6A-5.
6A-7.
A different way is by expressing B and C according to A.
C = A –2, B = A – 1. So the expression becomes: 2A +(A-1) +3(A-2)
= 6A – 7.
Q, R, S, and T are four consecutive positive numbers. Which of the
following expressions must be odd?
QR + ST.
Q + R + S + T.
Q2 + S3.
Q2 + R2.
Q2 + 2R
You can ignore the powers since odds and evens remain as they were
under powers. Q and R are consecutive numbers and therefore one of
them is even and the other one is odd. The result of the sum of an even
number and an odd number must be an odd number.
X, Y and Z are consecutive numbers (X>Y>Z). X +2Y +3Z = 5Y + 4.
What is Z?
5.
6.
3.
4.
2.
X + 2Y +3Z = 5Y +4 ---> X + 3Z = 3Y +4
Lets try the first answer: Z = 5, so Y = 6 and X = 6.
Lets check the equivalence: 7 + 15 = 22 = 18 + 4.
The sum of 3 consecutive numbers is definitely:
Positive.
Divisible by 2.
Divisible by 3.
Divisible by 4.
Divisible by 5.
This is a rule: The sum of 3 consecutive integers is divisible by 3. For
example take 3 + 4 + 5 = 12.
352 - 342 =?
35 – 34.
35 + 34.
352.
2 x 35 x 34.
34.
352 - 342 = (35 – 34)(35 +34) = 1(35 + 34). B is the answer.
Which of the following expressions is independent to variable X?
(a) (4X – 3)/(X – 3).
(b) X – (1 + 2X)/2.
(c) 4X – 1 – 4(1 – X ).
(d) (X + 2)2 – X2.
(e) X/(4X) + 4X/4.
Simplify all the expressions to see where X disappears.
Answer B: X – (1+2X)/2 = X – ½ - X = -1/2 and this answer as you can see is
not dependent on the variable X.
If 4XZ + YW = 3 and XW + YZ = 6, what is the value of the expression
(2X + Y)(2Z + W) ?
9.
18.
16.
12.
15.
(2X + Y)(2Z + W) = 4XZ + 2XW + 2ZY + WY.
Now, place the given data to get: 3 + 2 x 6 = 15.
If (A + 2)2 = (A + 5)2, what could be the value of A?
1.5.
-2.5.
-3.5.
2.5
3.
Plug in the answers to backsolve this question. Input A = -3.5 and you’ll get
(-1.5)2 = (1.5)2.
If A2 (B + C) = 20, (A, B and C are all integers bigger than 1), what is the
value of the expression (B + C – A)?
1.
2.
3.
4.
9.
Because they are all integers bigger than 1, the only multiplication can be 4
x 5 = 20. Because A is an integer, A2 can’t be 5 therefore its 4 (A = 2). A =
2 B + C = 5. B + C – A = 5 – 2 = 3.
If (A-B-C+D = 13) and (A+B-C-D = 5), what is the value of (B-D)2?
16.
64.
8.
4.
12.
Subtract equation 2 from equation 1 and you’ll get: -2B +2D = 8 ---> (B-D)
2
= 16.
If (A+B) = 4, (B+C) = 9 and (C+D) = 3, what is the value of (A+D)?
16.
2.
7.
-2.
8.
Take equation 1 + equation 3 – equation 2 and you’ll get 4 + 3 – 9 = -2.
If X2 + Y2 = A +3, XY = 7 and (X + Y) 2 = 25, what is the value of A?
11.
6.
8.
5.
4.
A + 3 = X2+Y2 = (X + Y)2 – 2XY = 25 – 7 x 2 = 11
A = 8.
If 4-X < (2-5X)/3, which of the following is correct?
X < -5.
X > -5.
X > 5.
-5 < X < 0.
0 < X < 5.
Multiply both sides by 3: 12 – 3X < 2 – 5X
X < -5.
If Y < X and XM < YM, what must be true?
M < X.
M < Y.
X < 0.
M < 0.
Y < 0.
If Y < X, multiply both sides by M and see that M must be negative since it
changed the sign of the inequality. Remember that multiplying both sides of
an inequality by a negative number reverses the direction of the inequality
sign.
If X = (Y/Z), (-1 < Z < 0) and 4 < Y, which of the following is correct?
X > 4.
0 < X <4.
-4 < X < 0.
X < -4.
X < -20.
Plug in numbers that fit the conditions of the question, Y=5; Z = -1/2
= 5/(-1/2) = -10.
X
If (B+A < B – A < A – B), which of the following is correct?
A<B<0.
B<A<0.
B<0<A.
0<B<A.
B>A>0.
Divide the equation in to two: (B+A < B – A) and (B – A < A – B).
From the first one: A<0. From the second one: A>B. Therefore B is the
A is an integer. Which of the following expressions must be even?
A (A+2) – 1.
A(A – 1) +1.
(A+1)(A+2).
(A – 1)(A +3).
A2 – 1.
Answer C is a multiplication of two consecutive numbers, therefore one of
them must be even, and an even number multiplied by a different number is
an even number.
N is a prime number bigger than 5. Which of the following expressions
must be even?
(N+2)2.
N2+2.
N(N+2).
(N+1)(N+2).
(N – 2)2.
Answer D is a multiplication of two consecutive numbers, therefore one of
them must be even, and an even number multiplied by a different number is
an even number.
X and Y are integers. If (4X + 3Y = 3Y – X), which of the following
is true?
X is even.
X is odd.
Y is even.
Y is odd.
None of the above.
4X + Y = 3Y – X ---> 3X = 2Y. The right side of the equation is definitely
even. Therefore X must also be even since it is multiplied by an odd
number (3).
To a prime number bigger than 2, 1 is added, the result is multiplied by
three. What can be the result of these actions?
51.
54.
43.
41.
53.
A prime number bigger than 2 must be odd, adding one to that number
gives you an even number, multiply that number by 3 and again the result is
even. The only even answer is B.
A seven-digit phone number is divisible by 3. After adding the two digits of
the area code the number is not divisible by 3, which of the following
cannot be the area code?
06.
02.
07.
04.
05.
Adding one number that is divisible by 3 to another, the result will still be a
number that is divisible by 3; the only number among the answers that is
divisible by 3 is 06.
In a Greek restaurant there is a custom to break plates during celebrations.
If after 8 celebrations there were only 6 plates left, which of the following
could be the original number of plates before the celebrations?
30.
32.
34.
36.
40.
24 is divisible by 8 (celebrations).
If a, b, c and d are consecutive integers (a<b<c<d).
What is the value of (d+c) – (b+a)?
1.
2.
3.
4.
0.
Plug in some numbers. Try a=1, b=2, c=3 and d=4.
(d+c) – (b+a) = 7 – 3 = 4. It will work with any number since the difference
between any two consecutive integers is always 1.
A, B and C are consecutive integers. If X = (A x B x C)/2, what can be said
X is even.
X is divisible by 3.
X is divisible by 5.
X is positive.
X is a fraction.
A product of three consecutive integers is divisible by 6, therefore when
divided by 2 it is still divisible by 3.
If A, B and C are consecutive integers (A<B<C) and 6A – 4B = A, what is
the value of C?
6.
5.
4.
3.
2.
B is the number following A, thus 6A – 4(A+1) = A ---> A = 4 ---> C = 6.
A cup can hold one third of the amount a bowl can hold. A pot can hold six
times more than a cup. How many pots can be filled with a liquid that takes
up 6 bowls?
1.
2.
3.5.
4.
5.5.
Try some numbers. One cup can hold 1 liter one bowl can hold 3 liters, a
put can hold 9 liters.
6 bowls hold 18 liters and that can fill up two pots.
Michael, Steve and Tyler shared a box of cookies. Michael ate 1/8 of the
cookies, Steve ate one half and Tyler ate 150 more cookies than Michael. If
no cookies remain in the box, how many cookies were in the box?
1200.
600.
800.
400.
550.
The fastest way is to try the answers, take 600. Michael ate (600/8) 75
cookies, Steve ate 300 and Tyler ate 225. Adding the numbers and you’ll
The price of a pasta box in the neighborhood grocery store is \$7; the price
of a pasta box in the market is \$5. A bus ticket to the market costs \$4.70
(one way). What is the minimum number of pasta boxes that must be
bought so the trip would be economically worthwhile?
5.
6.
3.
4.
7.
The price difference between the two places is (7 – 5) = \$2 per box. The bus
ride back and Forth is \$9.4. If he buys at least 5 boxes the trip to the market
will be worthwhile.
Following an increase in prices, the price of a candy box was 10 pounds and
the price of a can of soda was 6 pounds. If the price of a candy box was
raised by 25%, and the price of a can of soda was raised by 50%. What was
the price of a box of candy plus a can of soda before prices were raised?
11.
12.
13.
14.
14.5.
For the candy box, 10 pounds is 125%; therefore the original price was 8
pounds.
The same thing goes with the soda, 6 pounds is 150%, and therefore the
original price was 4.
The price of both products together was 12p before the rise.
In a chocolate store, all chocolates are either vanilla or cocoa flavored only.
10% of the chocolates are cocoa flavored, 90% of the rest are squashed.
What percentage of the chocolates are both vanilla flavored and not
squashed?
1%
5%
9%
10%
2%
Pick a number of chocolates; it is best to take 100 as an example.
10 are cocoa, 90% of the rest (0.9 x 90 = 81) are squashed.
That means that only 9 are both vanilla flavored and not squashed.
Z is 120% of Y. X is smaller than Z by 80%.
What percentage is X of Y?
96%
24%
50%
40%
45%
Pick numbers: Y=100 ---> Z=120 ---> X = 0.2 x 120 = 24.
X/Y = 24%
A baker sold all of his donuts for a total amount of 216 pounds. If on each
donut the baker made a profit of 8% of the cost, how much did it cost the
baker to make all the donuts?
210.
200.
190.
180.
170.
If on each donut he made a profit of 8% than he made the same profit on all
of his donuts.
That means that 216 pounds are 108% of the cost, and 100% is 200 pounds.
The average age of Eric and George is 10 years smaller than the average
age of Martha and Bella. If Martha is six years older than Eric, how much
older is Bella from George?
2.
8.
10.
12.
14.
Write the equation: (Eric + George)/2 – 10 = (Bella + Martha)/2
Eric+George+20=Bella+Martha,
The girls are 20 years older than the boys, if one is older only by 6 than the
other one has to be older by 14.
The grade point average of the entire class is 90. If the average of one third
of the class is 96, what is the average of the rest?
92.
88.
89.
86.
87.
Choose a representative number of students, 3 for example.
Use the average formula: (96 + 2X)/3 = 90 ---> 2X = 174 --->X = 87.
In a workshop there are 4 kinds of beds, 3 kinds of closets, 2 kinds of
shelves and 7 kinds of chairs. In how many ways can a person decorate his
room if he wants to buy in the workshop one shelf, one bed and one of the
following: a chair or a closet?
168.
16.
80.
48.
56.
You must multiply your options to every item. (2 shelves) x (4 beds) x (3
closets + 7 chairs) = 80 possibilities.
In how many combinations can we choose 2 students out of 10 if each
student is needed to fill a different roll in the student’s council?
110.
45.
55.
90.
100.
For the first roll there are 10 free students, for the second roll there are only
9 left. Therefore we have (10 x 9) 90 combinations total.
Three people are to be seated on a bench. How many different sitting
arrangements are possible if Erik must sit next to Joe?
2.
4.
6.
8.
10.
Treat the two who must sit together as one person. You have two possible
sitting arrangements. Then remember that the two that sit together can
switch places. So you have two times two arrangements and a total of four.
How many 3-digit numbers satisfy the following conditions: The first digit
is different from zero and the other digits are all different from each other?
648.
504.
576.
810.
672.
For the first digit you have 9 options (from 1 to 9 with out 0), for the second
number you have 9 options as well (0 to 9 minus the first digit that was
already used) and for the third digit you have 8 options left.
So the number of possibilities is 9 x 9 x 8 = 648.
Barbara has 8 shirts and 9 pants. How many clothing combinations does
Barbara have, if she doesn’t wear 2 specific shirts with 3 specific pants?
41.
66.
36.
70.
56.
There are (8 x 9) 72 possibilities of shirts + pants. (2 x 3) 6 Of the
combinations are not allowed. Therefore, only (72 – 6) 66 combinations are
possible.
A credit card number has 6 digits (between 1 to 9). The first two digits are
12 in that order, the third digit is bigger than 6, the forth is divisible by 3
and the fifth digit is 3 times the sixth. How many different credit card
numbers exist?
27.
36.
72.
112.
422.
First digit is 1, the second is 2, the third can be (7,8,9), the forth can be
(0,3,6,9), the fifth and the sixth are dependent with one another. The fifth
one is 3 times bigger than the sixth one, therefore there are only 3 options
there: (1,3), (2,6), (3,9).
All together there are: 1 x 1 x 3 x 4 x 3 = 36 options.
In jar A there are 3 white balls and 2 green ones, in jar B there is one white
ball and three green ones. A jar is randomly picked, what is the probability
of picking up a white ball out of jar A?
2/5.
3/5.
3/10.
3/4
2/3.
The probability of picking the first jar is ½, the probability of picking up a
white ball out of jar A
Is 3/(3+2) = 3/5. The probability of both events is 1/2 x 3/5 = 3/10.
Out of a box that contains 4 black and 6 white mice, three are randomly
chosen. What is the probability that all three will be black?
8/125.
1/30.
2/5.
1/720.
3/10.
The probability for the first one to be black is: 4/(4+6) = 2/5.
The probability for the second one to be black is: 3/(3+6) = 1/3.
The probability for the third one to be black is: 2/(2+6) = 1/4.
The probability for all three events is (2/5) x (1/3) x (1/4) = 1/30.
The probability of pulling a black ball out of a glass jar is 1/X. The
probability of pulling a black ball out of a glass jar and breaking the jar is
1/Y. What is the probability of breaking the jar?
1/(XY).
X/Y.
Y/X.
1/(X+Y).
1/(X-Y).
Let Z be the probability of breaking the jar, therefore the probability of both
events happening is Z x (1/X) = (1/Y). Z = X/Y.
An ant walks an average of 500 meters in 12 minutes. A beetle walks 15%
less distance at the same time on the average. Assuming the beetle walks at
her regular rate, what is its speed in km/h?
2.215.
2.5.
2.775.
3.2.
3.5.
A beetle moves (0.85 x 500 = 425 meters) in 12 minutes
that is equal to
0.425 Km in 1/5 of an hour. The speed in the right units is 0.425 x 5 = 2.125
Km/h.
A car was driving at 60 Km/h for 20 minutes, and then at 90Km/h for
another 40 minutes. What was its average speed?
80.
75.
70.
65.
54.
The average speed is equal to: (Total distance)/(Total time) = (60 x 1/3 + 90
x 2/3)/1 = 80 Km/h.
The distance from Steve’s house to work is 30 Km. On the way back Steve
drives twice as fast as he did on the way to work. Altogether, Steve is
spending 6 hours a day on the roads. What is Steve’s speed on the way back
from work?
10.
20.
5.
14.
15.
Steve’s speed on the way back is double the speed to work therefore it takes
him half the time to return home ---> Divide 6 hours by 1:2 ratio, the time it
takes him to get home is 2 hours and the time driving to work is 4 hours. 30
Km in 2 hours is 15 Km per hour.
The running speed of a horse is three times faster than the jogging of a
donkey.
If a horse is running for 4 hours and a donkey is jogging for 3 hours, what is
the horse’s speed (in Km/h) if the sum of their distances is 45 Km?
15.
9.
3.
12.
8.
X is the donkey’s speed and 3X is the horse’s speed. The total amount of
distance is Equal to 45 and to 3X x 4 + X x 3 = 15X ---> X = 3, the speed of
a horse is 9.
Two ants are moving from their farms towards each other. Ant A is moving
at a speed of 9 Cm per hour and ant B is moving at a speed of 6 Cm per
hour. If the farms are 60 Cm away from each other, what will be the
distance ant A travels until meeting ant B?
18.
24.
36.
42.
48.
The ants are moving towards each other, therefore adding their speeds will
represent the speed that they are moving towards one another. 60 Cm / (9 +
6) = 4 hours.
The distance that ant A will travel in 4 hours is 4 x 9 = 36 Cm.
At 12:10 o’clock Bill leaves point A towards point B at a speed of 30 Km/h.
At 13:40 o’clock Richi leaves from the same point towards point B also, but
at a speed of 60 Km/h. At what time will Richi passes over Bill?
14:40.
15:10.
16:10.
15:40.
15:30.
At 13:40, Bill has already moved (30 x 1.5) 45 Km.
Their relative speed is (60 – 30) 30 Km/h; therefore it would take (45 / 30)
1.5 hours to close the gap between the two. 13:40 + 90 minutes are 15:10.
From the starting point in a boat race, one opponent started to sail north at a
speed of 1.6 Km/h, the other opponent started to sail west at a speed of 1.2
Km/h. What is the distance in Km between the two competitors after 5
hours?
10.
12.
12.5.
14.
15.4.
One is going north and the other is going west, therefore the distance should
be calculated using the Pythagorean theorem. The one made a distance of
1.6 x 5 = 8 Km, The second one did 1.2 x 5 = 6 Km.
The distance between them is the square root of (64 + 36) = 10 Km.
James can eat 25 marshmallows is 20 minutes. Dylan can eat 25 in one
hour.
In how much time will the two eat 150 marshmallows?
40 minutes.
1 hour and 30 minutes.
1 hour.
1 hour and 40 minutes.
2 hours and 15 minutes.
Calculate each of their output in one hour: James can eat 75 and Dylan can
eat only 25.
Together they can eat 100 marshmallows in 1 hour. It would take them 1.5
hours to eat 150 marshmallows.
A wolf eats 5.5 Kg of meat in 2 days. A baby tiger eats 3 Kg of meat in 4
days.
How much meat can the two combined eat in three days?
10.
8.
9.
7.
6.
A wolf eats 2.25 Kg a day; a baby tiger eats 0.75 Kg a day. Together they
eat 3 Kg of meat every day, in three days they will eat 9 Kg of meat.
Two grandfathers can nit a sweater in 6 days. Two grandfathers and one
grandmother can nit a sweater in 3 days. How many days will it take the
grandmother to nit a sweater all by her self?
4.5.
5.
5.5.
6.
6.5.
Two grandfathers and a grandmother can nit a sweater in 3 days, therefore
they can nit 2 sweaters in 6 days. Because two grandfathers can nit 1 in 6
days then the other sweater is done by the grandmother, she can nit 1
sweater in 6 days.
20 beavers, working together in a constant pace, can build a dam in 3 hours.
How many hours will it take 12 beavers that work at the same pace, to build
the same dam?
5.
2.
8.
4.
6.
20 beavers worked 3 hours; therefore 60 hours of work were needed to
build the dam. 60 hours / 12 beavers = 5 hours of work to complete the
same dam.
B and A are consecutive numbers. If A + B + X =15 than what is true
It can be equally divided by 5.
It can be equally divided by 3.
Positive.
Even.
Odd.
A + B is an odd number; in order for the sum to be equal to 15 X must
be an even number.
A and B are numbers between 0 and 9. When multiplying 56 by
another number the result is 1AB. which of the following can
represent A?
8.
2.
6.
4.
5.
We have two choices: 56 x 2 = 112 or 56 x 3 = 168.
Therefore A could be 1 or 6. The answer is A = 6.
Q and R are numbers between 0 and 9. When multiplying 71 by
another double-digit number the result is 7PQ. Which of the following
could represent Q?
1.
8.
3.
5.
4.
We have two choices: 71 x 10 = 710 or 71 x 11 = 781.
Therefore Q can be 0 or 1. The answer is A.
A and B are numbers between 1 and 9. What is ABAB/AB?
(AB is a two-digit number and ABAB is a 4-digit number).
11
BAB
101
AB
100.
Plug in numbers: A = 1, B = 2 AB = 12,
ABAB = 1212. 1212/12 = 101.
V, W, X, Y and Z are numbers between 0 and 9.
If XYZ / 15 = WV and V-W=X (WV is a 2 digit number, XYZ is a 3
digit number), which of the following numbers can represent XYZ?
321
215
633
570
414
We see that XYZ should be equally dividable by 15, or by 5 and 3.
The only number that fits those conditions is 570.
A, B and C are different numbers, each between 0 and 9.
If B = C+2, what is BCA – CBA? (CBA and BCA are both 3 digit
numbers).
170
180
173
198
146
Lets put in some numbers. C = 1, so B = 1 + 2 = 3. There is no
information about A so we’ll leave it as A.
BCA – CBA = 31A – 13A = 180. B is the answer.
Danny, Doris and Dolly flipped a coin 5 times and each time the coin
landed on “heads”. Dolly bet that on the sixth time the coin will land
on “tails”, what is the probability that she’s right?
1.
½.
¾.
¼.
1/3.
The probability of the coin is independent on its previous outcomes
and therefore the probability for “head” or “tail” is always ½.
In a deck of cards there are 52 cards numbered from 1 to 13. There are
4 cards of each number in the deck. If you insert 12 more cards with
the number 10 on them and you shuffle the deck really good, what is
the probability to pull out a card with a number 10 on it?
1/4.
4/17.
5/29.
4/13.
1/3.
The total number of cards in the new deck is 12 +52 = 64.
There are (4 + 12 = 16) cards with the number 10.
The probability of drawing a 10 numbered card is 16/64 = 1/4.
There are 18 balls in a jar. You take out 3 blue balls without putting
them back inside, and now the probability of pulling out a blue ball is
1/5. How many blue balls were there in the beginning?
9.
8.
7.
12.
6.
After taking out 3 balls there are 15 left. 15/5 = 3 blue balls is the
number of left after we took out 3 therefore there were 6 in the
beginning.
In a box there are A green balls, 3A + 6 red balls and 2 yellow ones.
If there are no other colors, what is the probability of taking out a
green or a yellow ball?
1/5.
1/2.
1/3.
1/4.
2/3.
The number of green and yellow balls in the box is A+2.
The total number of balls is 4A +8.
The probability of taking out a green or a yellow ball is
(A+2)/(4A+8)=1/4.
The probability of Sam passing the exam is 1/4. The probability of
Sam passing the exam and Michael passing the driving test is 1/6.
What is the probability of Michael passing his driving test?
1/24.
1/2.
1/3.
2/3.
2/5
Indicate A as the probability of Michael passing the driving test.
The probability of Sam passing the test is 1/4, the probability of both
events happening together is 1/6 so: 1/4 x A = 1/6 therefore A = 2/3.
In a blue jar there are red, white and green balls. The probability of
drawing a red ball is 1/5. The probability of drawing a red ball,
returning it, and then drawing a white ball is 1/10. What is the
probability of drawing a white ball?
1/5.
½.
1/3.
3/10.
¼.
Indicate A as the probability of drawing a white ball from the jar.
The probability of drawing a red ball is 1/5.
The probability of drawing both events is 1/10 so, 1/5 x A = 1/10.
Therefore A = ½.
Out of a classroom of 6 boys and 4 girls the teacher picks a president
for the student board, a vice president and a secretary. What is the
probability that only girls will be elected?
8/125.
2/5.
1/30.
1/720.
13/48.
The basic principle of this question is that one person can’t be elected
to more than one part, therefore when picking a person for a job the
“inventory” of remaining people is growing smaller.
The probability of picking a girl for the first job is 4/10 = 2/5.
The probability of picking a girl for the second job is (4-1)/(10-1) =
3/9.
The probability of picking a girl for the third job is (3-1)/(9-1) = 1/4.
The probability of all three events happening is: 2/5 x 3/9 x ¼ = 1/30.
Two dice are rolled. What is the probability the sum will be greater
than 10?
1/9.
1/12.
5/36.
1/6.
1/5.
When rolling two dice, there are 36 possible pairs of results (6 x 6).
A sum greater than 10 can only be achieved with the following
combinations: (6,6), (5,6), (6,5).
Therefore the probability is 3/36 = 1/12.
The probability of having a girl is identical to the probability of
having a boy. In a family with three children, what is the probability
that all the children are of the same gender?
1/8.
1/6.
1/3.
1/5.
¼.
The gender of the first-born is insignificant since we want all children
to be of the same gender no matter if they are all boys or girls.
The probability for the second child to be of the same gender as the
first is: ½. The same probability goes for the third child. Therefore the
answer is ½ x ½ = ¼.
On one side of a coin there is the number 0 and on the other side the
number 1. What is the probability that the sum of three coin tosses
will be 2?
1/8.
½.
1/5.
3/8.
1/3.
The coin is tossed three times therefore there are 8 possible outcomes
(2 x 2 x 2). We are interested only in the three following outcomes:
(0,1,1), (1,0,1), (1,1,0).
The probability requested is 3/8.
The average length of 6 snakes is 80 cm. If the average length of one
third of the snakes is 70 cm, what is the average of the other snakes?
75.
85.
90.
100.
94.
The length of all six snakes is 6 x 80 = 480 cm.
Third of the snakes are by average 70 cm long therefore their sum is
140.
The sum of the length for the remainder of the snakes is 480 – 140 =
340.
340 / 4 snakes is 85 cm.
There are ten players in the basketball team. If the average height of
the players is 170 cm, what will be the new average height if a 192 cm
player will join the team?
181.
172.2.
172.
168.
184.
The new player is (192 – 170 = 22 cm) above average. Dividing the
extra height among 11 players is 2 cm per player, thus the new
average height is (170 + 2 = 172 cm).
The grade point average of one third of the classroom is 69; the grade
point average of the rest is 60. What is the grade point average of the
whole class?
61.
63.
65.
67.
It depends how many people are there in the class.
Because the number of people in the classroom is unknown, take 3.
One person has 69 and the rest have 60. Their sum is 189.
189/3 students is 63.
In the Hillside summer camp there are 50 children. 90% of the
children are boys and the rest are girls. The camp administrator
decided to make the number of girls only 5% of the total number of
children in the camp. How many more boys must she bring to make
that happen?
50.
45.
30.
40.
25.
The total number of girls in the camp is (0.1 x 50 = 5). In order for 5
girls to be 5% there has to be a total number of 100 kids in the camp
therefore the camp’s administrator should bring 50 more boys.
Kelly used to get a 30% discount on movie tickets. When the price of
a movie ticket increased by 50%, she still got the same dollar amount
of discount. What is the percent of discount Kelly got of the new
Ticket price?
10%
20%
25%
35%
38%
The price of the ticket is unknown. Take 100 as an example.
30% discount of 100 is \$30, that amount remained the same after the
price of a ticket grew by 50%.
The new price of a ticket is \$150, so 30/150 is 20%.
A and B are numbers between 1 and 9. If A = 4B than by what number
is the two digit number BA not divisible?
1
2
7
14
3
You have two choices: 1) 4 = 4 x 1 2) 8 = 4 x 2.
In the first choice, (BA = 14) can be divided by all the numbers except
3. In the second choice, (BA = 28) can be divided by all the numbers
except 3. Therefore, E is the answer.
If X percent of 2.5X are 3X, then X could be?
90.
120.
150.
170.
180.
X percent is X/100.
X percent of 2.5X is (X/100)(2.5X) = (X/100)(5X/2) = 3X
both sides by X.
(X/40) = 3 ---> X = 120.
divide
In a flower shop, there are 5 different types of flowers. Two of the
flowers are blue, two are red and one is yellow. In how many different
combinations of colors can a 3-flower garland be made?
4.
20.
3.
5.
6.
We want to make a 3-flower garlands, each should have three colors
of flowers in it.
There are two different types of blue and two different types of red.
The options are (2 blue) x (2 red) x (1 yellow) = 4 options.
In an engineering company there are 3 engineers and an expert who
earn 725,000 pounds together annually. The senior engineer earns
60% of the total income. If the rest of the money is divided in a way
that the two remaining engineers earn together three times more than
the expert, how much money (in thousands) does the expert earn in a
year?
45.
55.
60.
75.
90.
The senior engineer earns (60% of \$725,000 = \$435,000).
The rest of the money (735,000 – 435,000 = \$300,000) is divided in a
3:1 ratio in favor of the engineers.
And therefore the expert earns (1/4 x \$300,000 = \$75,000).
Gina and Tina’s average grade is 23 points higher than Tina’s grade.
What is the difference between their grades?
23.
46.
16.
11.5.
15.
Write the equation, T + 23 = (G + T)/2 2T + 46 = G + T
= G G – T = 46.
T + 46
If Z is a positive integer and (192)5 is a multiple of 8Z, what is the largest possible
value of Z?
5.
7.
8.
10.
12.
Factorize (192)5 and see what can be the largest value of Z.
192 = 64 x 3 = 8 x 8 x 3 ---> (192)5 = (8 x 8 x 3)5 = 35 x 810.
The largest possible value of 8Z which is a factor of (192)5 is the largest possible
value of Z of which 8Z is a factor of 810. Z = 10.
If X is a positive integer and (405)4 is a multiple of 3X, what is the
largest possible value of X?
5.
12.
16.
20.
26.
Find the factors of (405)4 and see what the largest value of X can be.
405 = 81 x 5 = 9 x 9 x 5 = 3 x 3 x 3 x 3 x 5
(405)4 = (3 x 3 x 3 x 3
x 5)4 = 316 x 54.
The largest possible value of 3X that is still a factor of (405)4 is the
largest possible value of X and that is 316. X = 16.
The flying acrobatic team is made up of 120 airplanes. The team
wants to form a rectangular formation with X planes in a row and Y
planes in a column. If the number of airplanes in a row is no less than
4 and no more than 30, how many different combinations of
rectangular shapes are possible?
4.
5.
6.
8.
10.
Use the factors of 120 are: 1x120, 2x60, 3x40, 10x12, 4x30, 5x24,
6x20 and 8x15.
We are looking for combinations of (row x column) that are all
between 8 and 30.
The possibilities are: 8x15, 15x8, 4x30, 30x4, 5x24, 24x5, 10x12 and
12x10. 8 possibilities total.
What is the perimeter of a rectangle having an area of 60?
(1) The length and width of the rectangle are even integers smaller
than 25.
(2) The length of the rectangle is larger than three times the width.
The question tells you that the area is 60, the area of a rectangle is
equal to length x width, in order to find the perimeter, one should find
the value of the length and the value of the width.
Statement (1) tells us that both the length and the width are even
integers and therefore we know their exact values because 60 can be
factorized to: 1x60, 2x30, 3x20, 4x15, 5x12 and 6x10.
The only even integers smaller than 25 are (6x10) and we know the
perimeter.
The perimeter of a rectangle is 136, what is the area of the rectangle?
The length is more than twice the width.
The length and width are both prime numbers larger than 30.
Define L as the length and W as the width.
2L + 2W = 136 ---> L + W = 68. We need one more equation in order
to find the area.
Statement (1) does not give you exact values and therefore it’s not
sufficient.
Statement (2) tells us that both L and W are prime numbers larger than
30, using the equation in the question; the only option to get 68 is with
31 and 37. This statement is sufficient; the area is 31 x 37.
Which of the following is the greatest possible common divisor of two
different positive integers, both smaller than 124?
123.
122.
63.
62.
61.
A divisor is actually a factor of a number.
Check each of the answers individually:
124 is divisor of itself, but obviously he has no other divisors smaller
than 124.
63 is a divisor of itself and 126, which is bigger than 124.
62 is on the limit, we were asked for divisors less than 124.
This is the right answer. 61 is a divisor of 61 and 122.
X is a positive integer, is X even?
(1) 9X2 is divisible by 4.
(2) 3X + 2 is divisible by 8.
We are to find if X is an even number.
Statement (1) tells us that 9X2 is an even number (it’s divisible by 4)
and therefore X2 must be an even number. An odd number squared is
also odd and therefore X must be even, this statement is sufficient.
Statement (2) tells us that 3X + 2 is an even number (it’s divisible by
8).
If you subtract 2 from (3X+2) the result will still be even. If 3X is
even than X must be even. This statement is also sufficient to answer
the question.
M is a positive integer, is M odd?
2M3 + 2M is divisible by 8.
M + 10 is divisible by 10.
We are being asked if M is an odd number.
Statement (1) tells us that 2M3 + 2M is divisible by 8 and so M4 + M
is divisible by 4 and is even. We have two choices: M3 and M are
either odd or even. This statement is insufficient.
Statement (2) is sufficient, if M + 10 is divisible by 10 then M + 10 is
an even number. This statement is sufficient and the answer is D.
How many of the positive divisors of 120 are also multiples of 4 not
including 120?
3.
4.
5.
7.
8.
Write down all the factors of 120: 60, 40, 30, 24, 20, 15, 12, 10, 8, 6,
5, 4, 3, 2, 1.
Among these numbers only the following are multiples of 4: 4, 8, 12,
20, 24, 40 and 60.
What is the sum of squares of the first two positive odd integers if the
sum of squares of the first two positive even integers is X?
X.
X/3.
X/2.
X/6.
3X/4.
The squares of the first two odd numbers are: 12 + 32 = 10.
The squares of the first two even numbers are: 22 + 42 = 20.
A carousel spins at a rate of ½ a round per second. If a point on its
circumference is located 0.5 meters from the center of rotation, how
many times approximately will this point reach its starting point in
two minutes?
25.
30.
60.
120.
180.
If the carousel spins half a turn in one second then it spins once every
two seconds. In two minutes there are 120 seconds and therefore the
point will reach its starting point (120 / 2 = 60) times, no matter how
far it is from the center of the circle.
There are 7 players in a bowling team with an average weight of 85
Kg. If two new players join the team, one weighs 110 Kg and the
second weighs 60 Kg, what will be the new average weight?
75 Kg.
80 Kg.
85 Kg.
90 Kg.
92 Kg.
The trick to this question is to notice that the average weight of the
two new players is exactly 85 Kg and so when they join the team, the
average weight stays the same.
If 10% of the employees of the state fare are police officers, what is the
number of employees who are not police officers?
(1) 5% of the police officers empoyed in the fare are woman.
(2) 45% of the employees at the state fare are woman.
In order to know how many people aren’t officers you need a fix value, in
other words you need to translate percentage into real values.
Statement (1) doesn’t give any real numbers and therefore it’s not
sufficient.
Statement (2) is the same as one in that matter, if we wanted to know the
percentage of the woman officers, the statements would have been
satisfying. More data is required.
If 90% of the people in Rich-Town read the Rich-Town magazin how
many people read the Rich-Town news bulletin?
(1) There are one thousand residents in Rich-Town.
the Rich-Town news bulletin.
No statement here tells us that all people read any magazin at all,
moreover no staement gives any data regarding the News bulletin readers
that do not read the Ric-Town magazin. More data is needed to solve this
question.
If Y = 2X – 10, what is the value of Z?
(1) Z = Y + 2X.
(2) X = Z – Y.
The question presents to us one equation with two unknowns and asks
We need two more different equations.
Statement (1) and statement (2) are both independent and so combining
them is sufficient.
If Q = W + 22, what is the value of (Q + W)?
(1) Q = -86 – W.
(2) W = 2Q + 10.
We are given an equation with two unknowns and we are asked upon the
sum (Q+W).
Statement (1) is sufficient by itself with out even using the data in the
question, Q + W = -86.
Statement (2) is also sufficient by itself, use the equation given in the
question along with statement (2) to solve.
Each of the statements is sufficient by itself.
Is X greater than 1?
(1) X > X2.
(2) –X < -X2.
Statement (1) tells you that X is greater than X2, that is true only if X is
between zero and one and so the answer to the question is no.
Statement two is identical to one, multiply both sides by (-1) and don’t
forget to change the sign of the inequality.
Either statement is sufficient by itself.
Is 0 < Y < 1?
(1) 1/Y is positive.
(2) Y > Y2.
From statement (1) Y can be 2 or ½ and therefore this statement is
insufficient.
Statement (2) tells us that Y > Y2. that is true only if Y is between zero
and one. So, Statement (2) is sufficient.
Is the triangle ABC isosceles?
(1) Angle A is equal to the sum of angles B and C.
(2) Side AB is different from CB.
Draw a triangle.
From statement (1) we can conclude that angle A is 90 degrees because it
is equal to half of the degrees in the triangle, this statement is not
sufficient since the triangle can still be a non isosceles right triangle.
Statement (2) is also insufficient; all it tells us is that two sides are
different from one another. More data is required.
Is the triangle ABC equilateral?
(1) Angle A is half the sum of angles B and C.
(2) AC is equal to AB.
Let angle A be X. The sum of the angles in triangle is equal to 180
degrees.
From statement (1) we can write: X + 2X = 180 ---> X = 60.
The sum of the two other angles is equal to 120, but they’re not
necessarily equal and therefore this statement is not sufficient alone.
Statement (2) alone is insufficient, unless the question asked if ABC is an
isosceles triangle.
Combine both statements; if one angle is equal to 60 and the triangle is
isosceles then two of the angles must be equal to 60, the third must also
be 60 and the triangle must be equilateral.
What is the sum of two angles in a triangle?
(1) One of the angles is equal to the sum of the other two.
(2) One of the sides is equal to the other.
Both statements together help us know that the triangle is isosceles.
We are asked upon two arbitrary angles and so we cannot determine what
is their sum. More data is required.
The number of employees at the justice department tripled between 1994
and 1996. How many employees were in the department in 1994?
(1) The number of employees in 1996 is bigger by 12 than the number
in 1994.
(2) In 1995 the number of employees was 12.
Define X as the number of employees in the year 1994 and Y as the
number of employees in 1996.
From the question we can write that Y = 3X.
Statement (1) tells us that Y = X + 12 and so we have two equations with
two unknowns. This statement is sufficient by itself.
Statement (2) adds an unnecessary parameter to the equation and
therefore its insufficient.
The water level in lake Victoria increased over the last week. By what
percent did the level of water increase?
(1) There were 45 million liters of water in the lake last week.
(2) Water is constantly being pured into the lake at a rate of 2 million
litters a day from a nearby stream.
We are to find the increase of the water in percent terms.
Statement (1) gives us the exact amount of water that was in the lake last
week.
Statement (2) holds no data regarding water leaving the lake or other
water being pured in.
What is the average (arithmetic mean) of X and Y?
(1) XY = 12.
(2) 2X = 26 – 2Y.
Use the average formula: the average of X and Y is (X+Y)/2.
Statement (1) gives us the value of XY and not (X+Y) and is therefore
insufficient.
Statement (2) can be simplified to: 2X + 2Y = 26 ---> (X+Y) = 13. This
statement is sufficient by itself.
What is the value of XY?
(1) X2 – 2X + 1 = 0.
(2) Y2 + 8Y + 16 = 0.
Statement (1) can also be written as (X – 1)2 and therefore X = 1.
Statement (2) can also be written as (Y + 4)2 and therefore Y = -4.
Using both statements together, we know the value of X and Y and so we
can calculate XY.
What is the value of (X + Y)?
(1) X2 –4X + 4 = 0.
(2) Y2 + 6Y + 8 = 0.
Statement (1) can also be written as (X – 2)2 and therefore X = 2.
Statement (2) can also be written as (Y + 4)(Y +2) and therefore the value
of Y is not distinct.
(X + Y) can have two different values, and therefore more data is
required.
At the Rocket propulsion center, hybrid missiles are being tested 4 times
a day and scramjet missiles are being tested 6 times a day. If at a certain
day 186 tests were held, what is the difference between the numbers of
hybrid missiles to scramjet missiles that were tested?
(1) There are 37 missiles total in the Rocket propulsion center.
(2) There are 19 scramjet missiles in the Rocket propulsion center.
Define H as the number of hybrid missiles and S as the number of
scramjet missiles.
The question lets us set up the equation: 4H + 6S = 186.
We need one more equation in order to answer the required question.
Statement (1) can be written as H + S = 37.
Statement (2) can be written as S = 19.
Either statement by itself is sufficient because it presents us with a second
equation.
If red buckets can be filled with 0.5 liters of sand and blue buckets can be
filled with 0.8 liters of sand, how many buckets were filled?
(1) The buckets were filled with 8 liters of sand total.
(2) Three more red buckets than blue were filled with sand.
Define R as the number of red buckets and B as the number of blue ones.
From the question and statement (1) we can write the following equation:
0.5R + 0.8B = 8. This equation has three different solutions: (0 red and 10
blue), (0 blue and 16 red), and (5 blue and 8 red). Since it is possible to
have no red or no blue buckets, all three solutions apply and there is not
enough information.
From statement (2) we can write the following equation: R = B + 3.
Combining both statements will result in two equations with two
unknowns, so we can solve the question. And get the third solution only.
In order to pass the Bar exam, one should answer 65% of the answers
correctly. Did Kevin pass the Bar exam?
(1) Kevin made a mistake on less than 40% of the questions.
From statement (1) we can conclude that the number of correct answers is
more than 60%. This statement isn’t sufficient on its own because we
don’t know if Kevin answered more or less than 65%. Statement (2) is
sufficient, it tells us that two third (66.7%) of the questions were correct
and therefore Kevin passed the test.
On an IQ exam, each correct answer grants the examinee with 3 points
but every wrong answer deducts 1 points. How much did Ernst get on the
IQ exam?
(1) There are 50 questions in the IQ exam.
(2) The ratio between the right answers and the wrong answers that
Ernst answered is 9 to 1.
Define C as the number of correct answers and W as the number of
Statement (1) tells us that C + W = 50.
Statement (2) tells us that (50 / (9+1) = 5) is the number of wrong
answers and so 45 is the number of right ones. Using the numbers
presented in the question we can answer the question of how much Ernst
got. Thus, Statement (1) and (2) combined are sufficient.
Is A a multiple of B?
(1) B is a multiple of A.
(2) 2A is a multiple of B.
Rephrase the question: is A/B an integer?
Statement (1) can be written as: B/A is an integer. Take A = 5, b = 10:
B/A is an integer but A/B isn’t.
Take A=B=4, A/B = B/A and they are both integers and therefore this
statement is insufficient.
Statement (2) can be written as 2A/B is an integer. Take A=10, B=20:
2A/B is an integer but A/B isn’t.
Take A=B=1: 2A/B is an integer and also A/B is an integer and therefore
this statement is also insufficient.
If A and B are two different two-digit numbers, is (A + B)/2 an integer?
(1) AB is an odd number.
(2) (B – A) is an even number.
We are required to find if the sum of A and B is an even number.
Statement (1) tells us that AB is odd. The multiplications of two odd
numbers only will give an odd number and so A and B are both odd
numbers and therefore their sum is an even number. This statement is
sufficient by itself.
Statement (2) is also sufficient, if the difference between two numbers is
even then the numbers can either be both odds or both even. In any of the
cases, their sum is even.
Tom divided his cards between Tim and Din so each one received an equal odd
amount of cards. The number of cards that Tim received multiplied by the number of
cards that Din received is a number larger than 49 and smaller than 121. How many
cards did Tom have in the first place?
16.
22.
18.
14.
32.
Answers A and E are disqualified immediately because those are even
numbers that cannot be divided into two odd numbers. 22 is 11 + 11
but
11 x 11 is 121. The same goes with 14=7 x 7=49. Therefore, the
answer is 18. 18 = 9 + 9. 9 x 9 = 81.
The number of hats that Sarah owns is 5 times bigger than the number
Nicky owns. If Sarah gives 6 hats to Nicky, they will have the same
number of hats. How many hats does Sharon own now?
5.
2.
3.
4.
6.
Let X be the number of hats that Nicky owns, Sarah owns 5X.
5X – 6 = X + 6 X = 3.
Jason washes 3 plates after each and every diner. After how many
dinners will Jason wash the 26th plate?
10.
9.
8.
7.
6.
Only after he finishes each dinner he washes three plates. So after his
8’Th dinner he’ll finish the 24’Th plate and only after the 9’Th dinner
he’ll wash plates number 25, 26 and 27.
Arnold and Danny are two twin brothers that are celebrating their
birthday. The product of their ages today is smaller by 9 from the
product of their ages a year from today. What is their age today?
7.
2.
9.
4.
5.
Back solve using the answers. Take the age 4.
4 x 4 = 16. 16 + 9 = 25. And in one year they’ll be 5 so 5 x 5 = 25.
Brian got in his latest test a grade 3 times higher that he anticipated. In
spite of that, he decided to appeal. After the appeal he got 30 points
lower than the original grade but still the grade was 30 points higher
than his anticipation. What was the grade Brian anticipated?
15.
20.
30.
25.
35.
The grade Brian anticipated is x. The grade that he really got was 3x.
After the appeal he got 3x – 30 that was still x +30. Therefore x = 30.
Steve’s monthly income is bigger by \$4,000 than John’s salary.
If they both earned (together) \$144,000 in one year, what is John’s
monthly income?
\$1,000.
\$2,000.
\$3,000.
\$4,000.
\$5,000.
Steve every month: x + 4,000.
Steve and John together every month: x + x + 4,000 = 2x +4,000.
Steve and John in a year: 12(2x + 4,000) = 144,000 x = 4,000.
The value of a stock is X dollars. On Sunday the stock’s value grew
by half of its value, but on Monday its value dropped to a third of its
new value. What is the stock’s value at the end of Monday?
X.
X/2.
2X/3.
X/3.
X/4.
Lets say that the original value of the stock was 100.
After Sunday its value was 150, after Monday its value was 50 thus
one half of its original value.
Heather has 35 stamps in her stamp collection. The stamps are divided
into three groups: old, new and foreign. The number of stamps in the
old group is one half of the number of stamps in the new group and
one seventh of the total amount of stamps. How many foreign stamps
does Heather have?
21.
20.
15.
7.
18.
If the number of old stamp is 1/7 of the total amount then there are 5
old stamps. Five old stamps are 1/2 of the new ones so there are 10
new stamps, therefore there are (35 – 10 – 5 = 20) foreign stamps.
The number of pizza slices that are sold in Joey’s Pizza are only in
quantities of 5 and 8 slices per customer. How many slices cannot be
bought?
42.
33.
22.
20.
38.
42 = 2 x 5 + 8 x 4.
33 = 5 x 5 + 8.
20 = 4 x 5.
38 = 5 x 6 +8.
The only a number which cannot be bought is 22.
In the beginning of the season, the owner of a football team bought T
players for the price of 4R dollars each. At the end of the season the
owner sold the players in a total profit of X. How many dollars did the
owner get for all the players?
X – 4TR.
4X + 4TR.
4TR + X.
4(TR – X).
4TR – X.
The owner bought T player that cost him altogether 4TR.
He had a profit of X so he sold them for 4TR + X.
Elizabeth is interested in dividing the rooms in the house among the
family. Unfortunately, they do not divide equally. It turns out that in
order for the rooms to be dividable, Elizabeth has to build two more
rooms and kick out one of the family members. Which of the
following can describe the number of the initial rooms in the house
and the initial number of family members (in the order Rooms,
family)?
20; 8.
30; 9.
15; 6.
10; 5.
22; 10.
First we can eliminate answers where the number of rooms can be
divided by the number of people, such as answer D.
We are looking for an answer that if you subtract 1 from the right and
add 2 to the left, the left number will be divisible by the right number.
The only answer that fits this description is B. (30 +2)/(8) = 4.
In a hair cut competition, the number of blonde girls is three times
bigger than the number of brown haired girls, and the number of
brown haired girls is twice the number of red haired girls. Assuming
that there is no other hair color, what is the proportion of the brown
haired girls?
2/3.
4/9.
1/3.
5/9.
2/9.
Pick a number for the red haired girls: 1.
The number of brown haired girls is twice as much, thus 2.
The number of blondes is three times as much, thus 6.
The total number of girls is 9 and the brown ones are 2 out of 9.
In store A there are 10 pairs of pants for every 40 store B has. The
price ratio between the pants in store B and the pants in store A is 3:4.
If all the pants were sold in both places until the stock ran out, what is
the ratio between the total amount store A earned to the total amount
store B earned?
3:16.
2:3.
1:3.
3:4.
2:5.
Plug in numbers. Pants in store A cost \$3 and in Store B \$4.
In Store A they sold 10 pairs so they earned \$40, in store B they
earned (40 x 3) \$120. The ratio between the money earned is 1 to 3.
In a barrel of juice there is 30 liters; in a barrel of beer there are 80
liters. If the price ratio between a barrel of juice to a barrel of beer is
3:4, what is the price ratio between one liter of juice and one liter of
beer?
3:2.
2:1.
3:1.
4:3.
3:4.
Pick numbers: a barrel of beer costs \$40 and a barrel of juice costs \$30
according to the given ratio. One liter of beer will cost \$0.5 and one
liter of juice will cost \$1. Therefore the price ratio is 2:1.
Chris has 100 gold bars and 60 bronze bars. After replacing 10 gold
bars with bronze bars, he had an equal amount of gold and bronze
bars. The price ratio between a gold and a bronze bar is?
2:3.
1:2.
1:3.
3:4.
2:5.
After replacing 10 gold bars into bronze ones he had 90 of each thus
10 gold bars are worth 30 bronze ones, and this is their price ratio.
Car A travels at three times the average speed of car B. Car A started
to travel at 12:00 o’clock, car B started to travel at 16:00 o’clock.
What is the speed of car B (in Km/h) if the total distance that both cars
traveled until 18:00 was 1000 Km?
10.
25.
30.
38.
50.
The speed of car B is X; the speed of car A is 3X.
Car A traveled 3X x 6 hours = 18X Km.
Car B traveled X x 2 hours = 2X Km.
1000 = 20X ---> X = 50 Km/h.
A bird is flying from an oak tree to a pine tree in a speed of 6 Km/h.
On her way back, she flew at a speed of 4 Km/h and therefore the trip
lasted 4 hours more. What is the distance between the trees (In Km)?
12.
24.
36.
48.
52.
The distance to the pine tree is 6 x X, where X represents the time of
the trip. The distance back to the oak tree is 4(X+4), assuming the trip
back is equal in length.
Therefore 6X = 4(X+4) ---> X = 8. The length of the trip is 8 x 6 = 48
Km.
Liz drove from point A to point B at 40 Km/h. On her way back she
drove at 50 Km/h and therefore her way back lasted one hour less.
What is the distance (in Km) between A and B?
150.
200.
600.
450.
500.
The length of the road from A to B is 40 x X, where X is the time.
The road back is: 50(X – 1). The road is the same length so
40X = 50(X – 1) ---> X = 5 hours. The distance is 200 Km.
The average speed of a Zebra is 4 times faster than that of a horse.
The Zebra started to run at 5:00 o’clock and the horse started two
hours later. What is the Zebra’s speed if the total distance that the two
animals traveled until 10:00 o’clock is 46 Km?
11.5.
12.
16.
8.
6.5.
Let X be the horse’s speed. The following equation came from the
data: 4X x 5 + X x 3 = 46 ---> X = 2 Km/h. The zebra’s speed is 8
Km/h.
Two dung beetles start to run simultaneously towards each other when
they’re 150 feet apart. The first beetle runs at a speed of 20 feet per
minute and the second beetle runs at a speed of 10 feet per minute. At
the moment they start running towards each other a fly leaves the first
beetle and flies towards the second one at a speed of 70 feet per
minute, when he gets there he turns around and starts to fly towards
the first beetle and so on. What is the total distance in feet, that the fly
will travel until the beetles meet?
250.
200.
300.
350.
400.
The distance between the two beetles is 150 feet; they are running
together at a speed of (10 + 20) 30 feet per minute, therefore it will
take them 5 minutes to meet. The fly flies at a speed of 70 feet per
minute so he travels a total distance of 5 x 70 = 350 feet.
If X and Y are both two-digit numbers, is XY an even number?
(1) The sum of X and Y gives an even number.
(2) The value of Y is three times the value of X.
Statement (1) isn’t sufficient, X and Y can be both odd or both even, but
their multiplication can be either one. Statement (2) tells us that Y = 3X,
X and Y can both be even or odd from this statement and therefore this
statement is also insufficient.
Both of the statements imply the same thing and so combining them will
not help. More data is required.
Arthur and Bartholomew live in the same multi-story apartment building.
How many stories does the building have?
(1) There are 5 stories between the apartment of Arthur and
Bartholomew.
(2) There are 8 stories above Arthur’s apartment and 8 stories below
Bartholomew’s apartment.
Define A as the Arthur’s floor and B as Bartholomew’s floor.
Using both statements, we don’t know whether A is over B or the
opposite and therefore we cannot determine the number of stories in the
building.
If A is above B then: The number of stories is (5+8+8 = 21).
If B is above A then: The number of stories is (5+3+3 = 11).
Comp and Calc are two companies that are located on different floors in a
skyscraper. How many floors does the skyscraper have?
(1) There are 24 floors between Comp’s floor and Calc’s floor.
(2) There are 32 floors above Comp’s floor and 12 floors bellow
Calc’s floor.
Using both statements we know that Comp’s floor is the higher one
among the two because there are only 12 floors beneath Calc’s floor and
therefore it must be the lower one.
The number of floors in the building is (12 + 24 + 32 = 68).
The combination of the two statements is sufficient.
What is the distance between Greentown to Bluetown?
(1) The distance between Greentown and Redtown is 20 miles.
(2) The distance between Bluetown and Redtown is 5 miles.
Draw a guiding chart with three points on it.
Statement (1) tells us nothing about Bluetown and so it’s insufficient.
Statement (2) tells us nothing about Greentown and so it’s insufficient on
its own.
Even after you combine both statements, you don’t know if Redtown is
between both cities or not.
The distance from Greentown to Bluetown can be 25 or 15 miles. More
data is required.
How long does it take to drive from the factory to the warehouse?
(1) It takes 15 minutes to drive from the factory to the harbor.
(2) It takes 25 minutes to drive from the warehouse to the harbor.
From both statements we cant conclude where is the location of the
harbor relative to the other two places, in other words, the harbor could
be between the factory and the warehouse or it could be beyond the two.
The distance between the factory and the warehouse can be between 10
and 40 miles.
Jasmine is the oldest member of the “Brain Storm club”. If next year, the
age of Sam will be two thirds that of Jasmine, what is Sam’s age today?
(1) Rick is Sam’s twin brother.
(2) Three years ago, Rick’s age was half of that of Jasmine.
Define J as the age of Jasmine and S as the age of Sam.
The question tells us that in a year from now: (S + 1) = 2/3 x (Y + 1).
Statement (1) presents a new parameter, Rick’s age, which is equal to
Sam’s age.
Statement (2) tells us that: (X – 3) = ½ x (Y – 3).
Using both statements, we have two equations with two unknowns.
X is a prime number. Is Y odd?
(1) X is divisible by 7.
(2) YX is an even number.
Statement (1) BY ITSELF is sufficient to answer the question, but
statement (2) by itself is not.
Statement (2) BY ITSELF is sufficient to answer the question, but
statement (1) by itself is not.
Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the
question, even though NEITHER statement BY ITSELF is sufficient.
Either statement BY ITSELF is sufficient to answer the question.
Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
answer the question, requiring more data pertaining to the problem.
From statement one we conclude that X=7 because it’s the only prime
number that can be divided by 7. From statement two we conclude that if
X was a odd number Y must be an even number, therefore both
statements are required in order to solve the question.
In a jar there are balls in different colors: blue, red, green and yellow.
The probability of drawing a blue ball is 1/8.
The probability of drawing a red ball is 1/5.
The probability of drawing a green ball is 1/10.
If a jar cannot contain more than 50 balls, how many yellow balls are
in the Jar?
23.
20.
24.
17.
25.
If 1/8 is the probability of drawing a blue ball then there are 40/8 = 5
blue balls in the jar. And with the same principle there are 8 red balls
and 4 green ones. 40 – 5 – 8 – 4 = 23 balls (yellow is the only color
left).
A Barman’s train rails across an open track at 250 kilometers per hour. A regular
passenger train travels at 68% of the Barman’s train speed. If the two trains start
moving from the same station at the same time, how much time longer will it take
the passenger train than the Barman’s to travel 850 kilometers?
2 hours and 24 minutes.
1 hour and 24 minutes.
2 hours and 36 minutes.
1 hour and 36 minutes.
5 hours.
Let’s find the time it takes each train to travel 850 km.
The fast train: 850 / 250 = 3.4 hours.
The slow train: 0.68 x 250 = 170 km/h. So the time is 850 / 170 = 5 hours. The
difference between the travel times is 1.6 hours or 1 hour and 36 minutes.
A Hawk can glide for 4 consecutive hours without resting using thermals only. An
eagle can glide 14.5% longer without resting using the same thermals. If an eagle
makes 3 stops during a certain glide, each stop after gliding its maximum possible
time, how many hours long was the glide not including the resting time?
18.32.
13.74.
15.66.
9.16.
16.
14.5% of 4 are 0.58. Thus, an eagle can fly for 4.58 consecutive hours without
resting. Since the eagle rested 3 times, he could glide four periods 4 x 4.58 hours=
18.32 hours.
Ruth s age is two-thirds of Chris’s age. How old is Chris?
Two years ago Ruth was half the age Chris is today.
Four and a half years from now Ruth will be seven eights of Chris’s age.
Statement (1) BY ITSELF is sufficient to answer the question, but statement (2)
by itself is not.
Statement (2) BY ITSELF is sufficient to answer the question, but statement (1)
by itself is not.
Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
Either statement BY ITSELF is sufficient to answer the question.
Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From the question we know that R (Ruth) = 2 ⋅ C (Chris). That’s one equation
3
with two variables; we need one more equation to solve the problem. Both
statements are suitable equations and thus the answer is (d).
Bob is older than his brother, Jimmy. How old is Jimmy?
Two years ago Jimmy was one-third of bob’s age today.
In six years from today Bob will be three times Jimmy’s age today.
Build an equation from each statement both equations are identical. Since we need
two different equations to find two unknowns, we cannot solve this question.
If X is divisible by 4, is Y odd?
y = x + 3.
x = 4.
From the question one can conclude that x is even. From statement one: an even
number + odd number is an odd number. Thus, y must be odd. Statement two
doesn’t mention y at all, and is therefore insufficient
If W is divisible by 7, is Z even or odd?
Z = W+1.
W = 7.
a)Statement (1) BY ITSELF is sufficient to answer the question, but statement (2)
by itself is not.
b)Statement (2) BY ITSELF is sufficient to answer the question, but statement (1)
by itself is not.
c)Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the
question, even though NEITHER statement BY ITSELF is sufficient.
d)Either statement BY ITSELF is sufficient to answer the question.
e)Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the
question, requiring more data pertaining to the problem.
From the question we don’t know if W is even or odd (W = 7,14,21,28…) so
statement one also is not helpful. From statement two we get no connection to Z
so it’s not sufficient either.
If x = y+3+4.5x and y = x+18, what is the value of x/y ?
1/4.
5/6.
-6/5.
-7/20.
1.5.
Solve two equations with two variables. Put y in the first equation.
We’ll get x = (x+18)+3+4.5x, the value of x is -14/3 and y = 40/3. x/y = -7/20.
If X2=Y and Y=4.5X+2.5, which of the following could be the value of y?
25.
-0.5.
5.
10.
15.5.
Y=4.5X+2.5, x = 5 or -0.5. Y can be 25 (4.5*5 + 2.5).
If -3x+4y=28 and 3x-2y=8, what is the product of x and y?
428.
622.
464.
264.
642.
We can notice that adding both equations gives: 2y = 36 thus y = 18.
From one of the equations we can receive: x = 44/3, so the answer is 18x 44/3 =
6x44= 264.
If X+Y = 2X-2Z, X-2Y = 4Z and X+Y+Z = 21, what is the value of Y/Z?
-2.
-4.5.
3.
-1.7.
2.5.
There are three equations with three variables, the solution is:
X = 0, Y = 42 and Z = -21 so Y/Z is -2.
If X/Y = 3X and Y = 4-X, what can be the value of 2X?
Seven and two thirds.
Three and one third.
Seven and one third.
Two and one third.
Three and two thirds.
With both equations we’ll get the following equation: x (3x-11) = 0. So, X is
either 0 or 11/3 we want the value of 2X, meaning (c).
What is 0.05 * 0.05 in terms of percents?
25%
2.5%
0.25%
0.025%
.0.025%
Since 0.05 and 0.05 each have 2 decimal places, their product must have 4 (2 + 2)
decimal places. Because 5 × 5 is 25, you need to add 2 zeros to get the correct
number of decimal places, so the product of 0.05 and 0.05 is 0.0025. To change a
decimal to a percentage you multiply by 100 by moving the decimal point 2 places
to the right, so 0.0025 is 0.25%.
What is 0.04 x 0.03 x 0.2 in terms of percent?
24%
2.4%
0.24%
0.024%
0.0024%
The number we’re supposed to see in the answers is 4 x 3 x 2 = 24. It fits all
The real question is the decimal of the answer; the basic rule is to add the
decimals (2+2+1=5).
We want the answer in percent terms so we need to take two decimals down,
overall- 3.
The answer is 24/1000 (3 decimals) = 0.024%
What is 0.01 x 5 x 0.03 in terms of percent?
15%
1.5%
0.15%
0.015%
0.0015%
Since 0.01 and 0.03 each have 2 decimal places, their product must have 4 (2 + 2)
decimal places. Because 1 x 3 x 5 is 15, you need to add 2 zeros to get the correct
number of decimal places, so the product of 0.01, 5 and 0.03 is 0.0015. To change
a decimal to a percentage you multiply by 100 and move the decimal point 2
places to the right, so 0.0015 is 0.15%.
A computer factory produces 4200 computers per month at a constant rate, how
many computers are built every 30 minutes assuming that there are 28 days in one
month?
2.25.
3.125.
4.5.
5.225.
6.25.
4200/28 is 150 computers per day, 150/24 = 25/4 computers per hour = 6.25.
Every thirty minutes half of that number is made, 3.125.
A pizza house sells 30 pizzas on a Friday night. On a weekday it sells 11% less.
How many pizzas will the pizza house sell in a 28 days month assuming that
Saturday is a “no business” day?
320.
654.
235.
600.
540.
On a weekday the pizza house sells 89% of 30 = 26.7 pizzas.
In a month there are 4 Fridays ---> 4 x 30 = 120 pizzas.
There are 4 x 5 weekdays (there is no Saturday) ---> 20 x 26.7 = 534. All together
654 pizzas.
Bart is working as a paper delivery boy in Springfield. He delivers 620
newspapers every day except on Saturdays and Sundays. If Bart earns 4.2 cents
for every second newspaper he delivers, how much money can he earn in a month
with 28 days?
378.2 \$
376.5 \$
287.8 \$
260.4 \$
96.9 \$
(310 newspapers a day) x (5 days) x (4 weeks) x (4.2 cents)=26040 cents= 260.4\$.
A young and energetic cobbler fixes 3650 pairs of shoes every year, while an old
yet experienced cobbler fixes 20% less than the young cobbler. In a shoe factory
there are two old and one young cobblers working together. How many shoes can
the factory fix every day assuming that there are 365 days a year?
26.
34.
36.
44.
48.
An old cobbler fixes 0.8 x 3650 = 2920 shoes per year. In the factory there are two
old and one young cobbler ---> 3650 + 2920 + 2920 = 9490 shoes per year.
9490 shoes divided by 365 days is 26 shoes per day.
A carpenter makes 3 bunk beds every day. A military school needs to organize a
place to sleep for 143 soldiers. If there are 5 carpenters working on the job, how
many whole days in advance should they receive the order and start working in
order to finish the right number of beds assuming that each bunk-bed is used by
two soldiers?
3.
4.
5.
6.
7.