Gre math practice test 5

Test 5
GRE
MATH
Complete GRE MATH Preparation Material
http://studymaterialcollection.blogspot.com/2015/12/complete-gre-math-preparation-material.html

GRE Math Tests
94
Questions: 24
Time: 45 minutes
1. Column A Column B
ab2
a2
b
Multiple-choice Question – Select One or More Answer Choices]
2. Which of the following are divisible by both 2 and 3? [Note: If an integer is divisible by 3, then the
sum of its digits is also divisible by 3.]
(A) 1005
(B) 1296
(C) 1351
(D) 1406
(E) 1414
(F) 3456
3. Column A The digits of a two-digit number X
differ by 4.
Column B
The positive difference between
the squares of the digits of X
15
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Test 5—Questions
95
[Multiple-choice Question – Select One Answer Choice Only]
4. What is the remainder when 37
is divided by 8?
(A) 1
(B) 2
(C) 3
(D) 5
(E) 7
5. Column A 0 < x < 1 Column B
x2
x

GRE Math Tests
96
6. From the figure, which of the following must be true?
(I) x + y = 90
(II) x is 35 units greater than y
(III) x is 35 units less than y
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only
7. Which one of the following is true regarding the triangle shown in figure?
(A) x > y > z
(B) x < y < z
(C) x = y = z
(D) 2x = 3y/2 = z
(E) x/2 = 2y/3 = z
z°
x° y°1
1 1/2 2
C
B
A
l
m
35°
x° y°

Test 5—Questions
97
8. Column A |a| is the distance point a is from
the origin on the number line. x
0.
Column B
|x| + |–2| |x – 2|
9. Let u represent the sum of the integers from 1 through 20, and let v represent the sum of the integers
from 21 through 40. What is the value of v – u ?
(A) 21
(B) 39
(C) 200
(D) 320
(E) 400
10. Richard leaves to visit his friend who lives 200 miles down Interstate 10. One hour later his friend
Steve leaves to visit Richard via Interstate 10. If Richard drives at 60 mph and Steve drives at
40 mph, how many miles will Steve have driven when they cross paths?
(A) 56
(B) 58
(C) 60
(D) 65
(E) 80
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GRE Math Tests
98
11. How many different ways can 3 cubes be painted if each cube is painted one color and only the 3
colors red, blue, and green are available? (Order is not considered, for example, green, green, blue is
considered the same as green, blue, green.)
(A) 2
(B) 3
(C) 9
(D) 10
(E) 27
12. Column A Column B
2/3 – 3/4 3/4 – 4/5
13. If x is not equal to 1 and y =
1
x 1
, then which one of the following cannot be the value of y ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Test 5—Questions
99
14. Which one of the following numbers is the greatest positive integer x such that 3x
is a factor of 275
?
(A) 5
(B) 8
(C) 10
(D) 15
(E) 19
15. Column A The arithmetic mean (average) of
the numbers a and b is 17. The
geometric mean of the numbers a
and b is 8. The geometric mean of
two numbers is defined to be the
square root of their product.
Column B
a b
16. If x = 10a, y = 3b, z = 7c, and x : y : z = 10 : 3 : 7, then
7x + 2y + 5z
8a +b+ 3c
=
(A) 111/12
(B) 7/6
(C) 8/15
(D) 108/123
(E) 135/202

GRE Math Tests
100
17. A perfect square is a positive integer that is the result of squaring a positive integer. If N = 34
53
7,
then what is the biggest perfect square that is a factor of N ?
(A) 32
(B) 52
(C) 92
(D) (9 5)2
(E) (3 5 7)2
18. If 50% of x equals the sum of y and 20, then what is the value of x – 2y ?
(A) 20
(B) 40
(C) 60
(D) 80
(E) 100
19. Column A 1 Pound = 16 Ounces Column B
Weight of 16,000 ounces of
rice
Weight of 1,000 pounds of coal

Test 5—Questions
101
20. An old man distributed all the gold coins he had to his two sons into two different numbers such that
the difference between the squares of the two numbers is 36 times the difference between the two
numbers. How many coins did the old man have?
(A) 24
(B) 26
(C) 30
(D) 36
(E) 40
21. Column A In a jar, 60% of the marbles are
red and the rest are green.
Column B
40% of the red marbles in the
jar
60% of the green marbles in the
jar
22. A prize of $200 is given to anyone who solves a hacker puzzle independently. The probability that
Tom will win the prize is 0.6, and the probability that John will win the prize is 0.7. What is the
probability that both will win the prize?
(A) 0.35
(B) 0.36
(C) 0.42
(D) 0.58
(E) 0.88

GRE Math Tests
102
23. The buyer of a particular car must choose 2 of 3 optional colors and 3 of 4 optional luxury features.
In how many different ways can the buyer select the colors and luxury features?
(A) 3
(B) 6
(C) 9
(D) 12
(E) 20
24. In the two-digit number x, both the sum and the difference of its digits is 4. What is the value of x?
(A) 13
(B) 31
(C) 40
(D) 48
(E) 59
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Test 5—Solutions
103
Answers and Solutions Test 5:
Question Answer
1. D
2. B F
3. A
4. C
5. B
6. B
7. A
8. D
9. E
10. A
11. D
12. B
13. A
14. D
15. D
16. A
17. D
18. B
19. C
20. D
21. C
22. D
23. D
24. C
If you got 18/24 correct on this test, you are likely to get 750+ on the actual GRE by the time you
complete all the tests in the book.
1. If a = 0, both columns equal zero. If a = 1 and b = 2, the two columns are unequal. This is a double
case and the answer is (D).
2. A number divisible by 2 ends with one of the digits 0, 2, 4, 6, or 8.
If a number is divisible by 3, then the sum of its digits is also divisible by 3.
Hence, a number divisible by both 2 and 3 will follow both of the above rules.
Choices (A) and (C) do not end with an even digit. Hence, eliminate them.
The sum of digits of Choice (B) is 1 + 2 + 9 + 6 = 18, which is divisible by 3. Also, the last digit is 6.
Hence, choice (B) is correct.
This is also true with choice (F). The sum of digits of Choice (F) is 3 + 4 + 5 + 6 = 18, which is divisible by
3. Also, the last digit is 6. Hence, choice (F) is correct.
Next, the sum of the digits of choices (D) and (E) are 1 + 4 + 0 + 6 (= 11) and 1 + 4 + 1 + 4 (= 10),
respectively, and neither is divisible by 3. Hence, reject the two choices.
Hence, the answer is (B) and (F).

GRE Math Tests
104
3. Since the digits of the number differ by 4, the number can be any one of the numbers 15, 26, 37, 40, 48,
and 59 or any one of their reverse numbers, except 40. Hence, the positive difference between the digits
varies from 42
– 02
= 16 (for the number 40) to 92
– 52
= 56 (for the numbers 59 or 95). Hence, Column A is
a number between 16 and 56, inclusive, and therefore is always greater than 15, the value of Column B.
Hence, the answer is (A).
4. We know that 37
= 3 33
33
= 3 27 27 = 3(272
). The number immediately before 27 that is divisible
by 8 is 24. Hence, replace 27 with 24 + 3. Then we have
37
= 3(272
) = 3(24 + 3)2
= 3(242
+ 2 24 3 + 32
)
= 3 242
+ 3 2 24 3 + 3 9
Now,
37
8
=
3 242
+ 3 2 24 3+ 3 9
8
=
3 242
8
+
3 2 24 3
8
+
3 9
8
= Integer + Integer +
27
8
= Integer + Integer +
24 +3
8
= Integer + Integer + 3 +
3
8
Hence, the remainder is 3, and the answer is (C).
5. Since 0 < x < 1, we know that x is positive. Now, multiplying both sides of the inequality 0 < x < 1 by x
yields
0 < x2
< x
Hence, Column B is greater than Column A, and the answer is (B).
6.
Angle x is an exterior angle of the triangle and therefore equals the sum of the remote interior angles, 35
and y. That is, x = y + 35. This equation says that x is 35 units greater than y. So, (II) is true and (III) is
false. Now, if x is an obtuse angle (x > 90), then x + y is greater than 90. Hence, x + y need not equal 90. So,
(I) is not necessarily true. The answer is (B).
7. From the figure, we have the following inequality between the sides of the triangle: BC (= 2) > AB (= 1
1/2) > AC (= 1). In a triangle, the longer the side, the bigger the angle opposite it. Hence, we have the
following inequality between the angles of the triangle: A > C > B. Replacing the angles in the
inequality with their respective vertical angles in the figure yields x > y > z. Select Choice (A).
l
m
35°
x° y°

Test 5—Solutions
105
8. Suppose x = –2. Then Column A equals |x| + |–2| = |–2| + |–2| = 2 + 2 = 4, and Column B equals |x – 2| =
|–2 – 2| = |–4| = 4. Here, Column A equals Column B.
Now, suppose x = 2. Then Column A equals |x| + |–2| = |2| + |–2| = 2 + 2 = 4, and Column B equals |x – 2| =
|2 – 2| = |0| = 0. Here, Column A is greater than Column B.
This is a double case, and therefore the answer is (D).
9. Forming the series for u and v yields
u = 1 + 2 + ... + 19 + 20
v = 21 + 22 + ... + 39 + 40
Subtracting the series for u from the series for v yields
v u = 20 + 20 + + 20 + 20
20 times
= 20 20 = 400
The answer is (E).
10. Let t be time that Steve has been driving. Then t + 1 is time that Richard has been driving. Now, the
distance traveled by Steve is D = rt = 40t, and Richard's distance is 60(t + 1). At the moment they cross
paths, they will have traveled a combined distance of 200 miles. Hence,
40t + 60(t + 1) = 200
40t + 60t + 60 = 200
100t + 60 = 200
100t = 140
t = 1.4
Therefore, Steve will have traveled D = rt = 40(1.4) = 56 miles. The answer is (A).
11. Clearly, there are more than 3 color combinations possible. This eliminates (A) and (B). We can also
eliminate (C) and (E) because they are both multiples of 3, and that would be too ordinary, too easy, to be
the answer. Hence, by process of elimination, the answer is (D).
Let’s also solve this problem directly. The following list displays all 27 (= 3 3 3) color combinations
possible (without restriction):
If order is not considered, then there are 10 distinct color combinations in this list. You should count them.
RRR BBB GGG
RRB BBR GGR
RRG BBG GGB
RBR BRB GRG
RBB BRR GRR
RBG BRG GRB
RGR BGB GBG
RGB BGR GBR
RGG BGG GBB

GRE Math Tests
106
12. The least common multiple of the denominators 3, 4, and 5 of all the fractions is 60. Multiplying both
columns by 60 to clear the fractions yields
40 – 45 45 – 48
Subtracting the numbers yields
–5 –3
Since –5 < –3, Column B is greater than Column A and the answer is (B).
13. Since the numerator of the fraction
1
x 1
does not contain a variable, it can never equal 0. Hence, the
fraction can never equal 0. The answer is (A).
14. 27
5
= 3
3
( )
5
= 3
15
. Hence, x = 15 and the answer is (D).
15. The arithmetic mean of the numbers a and b is 17. Hence,
a +b
2
=17, or a + b = 34.
The geometric mean of the numbers a and b is 8. Hence, ab = 8, or ab = 82
= 64.
Solving the equation a + b = 34 for a yields a = 34 – b. Substituting this into the equation ab = 64 yields
(34 – b)b = 64
34b – b2
= 64
34b – b2
– 64 = 0
b2
– 34b + 64 = 0
(b – 32)(b – 2) = 0
b – 32 = 0 or b – 2 = 0
b = 32 or b = 2
Now, if b = 32, then a = 34 – b = 34 – 32 = 2. In this case, Column B is larger.
Now, if b = 2, then a = 34 – b = 34 – 2 = 32. In this case, Column A is larger.
Hence, we have a double case, and the answer is (D).

Test 5—Solutions
107
16. We are given the equations x = 10a, y = 3b, z = 7c and the proportion x : y : z = 10 : 3 : 7.
Substituting the first three equations into the last equation (ratio equation) yields 10a : 3b : 7c = 10 : 3 : 7.
Forming the resultant ratio yields 10a/10 = 3b/3 = 7c/7. Simplifying the equation yields a = b = c. Hence,
both a and b equal c. Substituting the result in the given equations x = 10a, y = 3b, z = 7c yields x = 10a =
10c, y = 3b = 3c, and z = 7c. Now, we have
7x + 2y + 5z
8a +b+ 3c
=
7 10c + 2 3c + 5 7c
8c +c + 3c
= because x = 10c, y = 3c, z = 7c, and a = b = c
111c
12c
=
111
12
The answer is (A).
17. Every positive integer can be uniquely factored into powers of primes. When the integer is squared, all
powers of these primes are doubled. Hence, a perfect square has only even powers of the primes in its
factorization, and clearly any positive integer whose prime factorization has only even powers of primes is
a perfect square. Any factor of N is the product of some or all of the primes contained in 34
53
7; the
largest such product containing only even powers of is 34
52
= (9 5)2
. The answer is (D).
18. 50% of x equals the sum of y and 20. Expressing this as an equation yields
(50/100)x = y + 20
x/2 = y + 20
x = 2y + 40
x – 2y = 40
The answer is (B).
19. Column B has 1,000 pounds of coal, and there are 16 ounces in 1 pound. So, Column B has 1,000
pounds = 1,000(16 ounces) = 16,000 ounces. Hence, each column weighs 16,000 ounces. The answer is
(C).
20. Let x and y be the numbers of gold coins the two sons received. Since we are given that the difference
between the squares of the two numbers is 36 times the difference between the two numbers, we have the
equation
x2
– y2
= 36(x – y)
(x – y)(x + y) = 36(x – y) by the Difference of Squares formula a2
– b2
= (a – b)(a + b)
x + y = 36 by canceling (x – y) from both sides
Hence, the total number of gold coins the old man had, namely x + y, equals 36. The answer is (D).
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GRE Math Tests
108
21. Let j be the total number of marbles in the jar. Then 60%j must be red (given), and the remaining 40%j
must be green (given). Now,
Column A equals 40% of the red marbles = 40%(60%j) = .40(.60j) = .24j.
Column B equals 60% of the green marbles = 60%(40%j) = .60(.40j) = .24j.
Since both columns equal .24j, the answer is (C).
22. The number of attendees at the meeting is 750 of which 450 are female. Hence, the number of male
attendees is 750 – 450 = 300. Half of the female attendees are less than 30 years old. One half of 450 is
450/2 = 225. Also, one-fourth of the male attendees are less than 30 years old. One-fourth of 300 is 300/4 =
75.
Now, the total number of (male and female) attendees who are less than 30 years old is 225 + 75 = 300.
So, out of the total 750 attendees 300 attendees are less than 30 years old. Hence, the probability of
randomly selecting an attendee less than 30 years old (equals the fraction of all the attendees who are less
than 30 years old) is 300/750 = 2/5. The answer is (D).
23. Let A, B, C stand for the three colors, and let W, X, Y, Z stand for the four luxury features. There are
three ways of selecting the colors:
A B A C B C
There are four ways of selecting the luxury features:
W X Y W Y Z W X Z X Y Z
Hence, there are 3 4 = 12 ways of selecting all the features. The answer is (D).
24. Since the sum of the digits is 4, x must be 13, 22, 31, or 40. Further, since the difference of the digits is
4, x must be 40, 51, 15, 62, 26, 73, 37, 84, 48, 95, or 59. We see that 40 and only 40 is common to the two
sets of choices for x. Hence, x must be 40. The answer is (C).

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