Free Solutions and Explanations to GRE Quantitative MCQ (Multiple Answers) to Sample 1

Free Solutions and Explanations to Practice GRE Quantitative MCQ (Multiple Answers) to sample 1.

Which of the following would be the value of the digit B so that the number 3B324 is divisible by 6?

A) 0
B) 1
C) 2
D) 3
E) 4
F) 5
G) 6
H) 7
I) 8
J) 9

Solution

For a number to be divisible by 6, it must be divisible by 2 and 3. All the given numbers above are divisible by 2. We need to select values of B such the numbers are divisible by 3. For a number to be divisible 3, the sum of its digits must be divisible by 3. Hence

3 + B + 3 + 2 + 4 = 12 + B

The values of B that make 12 + B divisible by 3 are

0, 3, 6 and 9

The answer to the above question is

A , D , G , J

Which of these in NOT a prime number?

A) 27
B) 13
C) 43
D) 49
E) 119
F) 1111

Solution

A prime number is a number that has two factors only 1 and itself. Let us write the above numbers and list their factors, then decide which are not prime.

Factors of 27 are 1 , 3 , 9 , 27
Factors of 13 are 1 , 13
Factors of 43 are 1 , 43
Factors of 49 are 1 , 7 , 49
Factors of 119 are 1 , 7 , 17 , 119
Factors of 1111 are 1 , 11 , 101 , 1111

According to the definition above, 27, 49, 119 and 1111 are prime numbers. The answer to the above questions is:

A, D, E, F

If a, b, c and d are real numbers of a date set such that a < b < c < d, which of the following statement is always true?

A) The average (arithmetic mean) of a, b, c and d is smaller than d
B) The average of a, b, c and d is greater than d
C) The average of a, b, c and d is greater than a
D) Three times the average of a, b, c and d is greater than d

Solution

The average of a set of non equal numbers is always greater than the smallest data value in the set and smaller than the largest value in the set. Hence in this case the average of a, b, c and d is less than d and greater than a. Hence

Statement A is always true

Statement B is always false

Statement C is always true

Statement D is not always true. Explanation with counter example: Let

a = 0, b = 1, c = 3 and d = 100.

Find the mean m of a, b, c and d.

m = (0 + 1 + 3 + 100) / 4 = 26.

Three times the average = 3 * 26 = 78 and is not greater than d = 100.
Answer to above question:
A , C

If x is distant by 5 units from -1, what are the possible values of x?

A) 4
B) - 4
C) - 6
D) 6
E) 5

Solution

The distance on a number line between x and -1 is given by.

|x - (-1)| = |x + 1|

The values of x that make.

|x + 1| = 5 are x = 4 and x = -6

The answer to the above question is.

A , C

Which of these fractions are equivalent to 6 / 8?

A) 30 / 40
B) 4 / 5
C) 12 / 16
D) 3 / 4
E) 99 / 132
F) 4 / 3

Solution

We first reduce the fraction 6 / 8

6 / 8 = 3 / 4

We reduce the given fraction in order to find which is equivalent to 3 / 4

30 / 40 = 3 / 4 (divide numerator and denominator by 10)
4 / 5 cannot be further reduced
12 / 16 = 3 / 4 (divide numerator and denominator by 4)
3 / 4 cannot be further reduced
99 / 132 = 3 / 4 (divide numerator and denominator by 33)
4 / 3 cannot be further reduced

Answer to given question

A , C , D , E

Which of the triangles below, defined by their three sides, are right triangles?

A) 1 , 2 , 3
B) 6 , 8 , 10
C) 1.2 , 1.6 , 2.0
D) 3 , 5 , √34
E) 10 , 12 , 20

Solution

If a, b and c are the sides of a right triangle and c has the largest length, then a,b and c must satisfy Pythagora's theorem as follows

a^{2} + b^{2} = c^{2}

We now check if each of the given sides above satisfy Pythagora's theorem

The lengths of the sides AB and AC of triangle ABC are both equal to 10 and the size of the interior angle at vertex A of triangle ABC is equal to 30°. Which is true about triangle ABC?

A) Angles ABC and ACB are equal in size.
B) Angle ABC has a size of 30°
C) Angle ACB has a size of 75°
D) The area of triangle ABC is equal to 25

Solution

Triangle ABC has 2 sides of equal length is therefore isosceles and angles ABC and ACB are equal in size.

Statement A is true

The sum of all three angles ia 180°. Hence.

size of ABC + size of ACB + 30 = 180

angles ABC and ACB are equal in size, hence.

2(size of ABC) + 30 = 180

size of ABC = (180 - 30) / 2 = 75°

Statement B is false

angles ABC and ACB are equal in size, hence.

size of ACB = 75° , statement C is true

The area of a triangle may be calculated using two sides and the angle between them as follows.

(1/2) sin (30°) 10 * 10 = (1/2)(1/2)10*10 = 25 , statement D is true.

The answer to the above question is.

A , C , D

Which of the following is true?

A) x^{2} > 0 for all real values of x.
B) | x | > 0 for all real values of x.
C) x^{2} + 1 > 0 for all real values of x.
D) | x + 1| > 0 for all real values of x.
E) | x | + 1 > 0 for all real values of x.

Solution

For x = 0

x^{2} = 0 and | x | = 0 , and therefore both statement A and B are false.

The square of a real number is either positive or zero and if you add any positive number to a square such x^{2} + 1 it becomes greater than 0.

Statment C is true.

If x = -1, then

| x + 1| = 0, Statment D is false.

The absolute value of a real number is either positive or zero and if you add any positive number to an absolute value such | x | + 1 it becomes greater than 0.

Statment E is true.

The answer to the given question is.

C , E

Which of the following is true for all real numbers x and y?

A) |- x - y| = |x + y|
B) |x - y| = |x + y|
C) |y - x| = |x - y|
D) |- x + y| = |x - y|

Solution

|- x - y| may be written as

|- x - y| = |-(x + y)|

Apply the fact that | - x | = | x | to simplify

|- x - y| = |-(x + y)| = | x + y | , statement A is true

If x = 6 and y = 9, then

|x - y| = |6 - 9| = 3 and |x + y| = |6 + 9| = 15 , statement B in not true

|y - x| may be written as

|y - x| = |-(x - y) | = |x - y| , statement C is true

|- x + y| may be written as

|- x + y| = | -(x - y) | = |x - y| , statement D is true

The answer to the above question is

A , C , D

a, b, c, d, e, f, and h are numbers such that a × b × c = 100, a × d × e = 0 and b × f × h = 0. Which of the following could be true?

A) a = 0
B) d = 0 or f = 0
C) d × e not equal to 0
D) f = 0 or h = 0
E) f not equal to 0 and h not equal to 0

Solution

Since a × b × c = 100, none of the numbers a, b or c is equal to 0.

Statement A is false

Since a × d × e = 0 and b × f × h = 0, d may be equal to 0 or f may be equal to 0.

Statement B is true

Since a × d × e = 0 and a not equal to 0, d × e must be equal to 0.

Statement C is false

Since b × f × h = 0 and b not equal to 0, then either f equals 0 or h equals or both.