Free GRE Quantitative Practice Questions with Solutions
Sample 1

A set of 10 practice questions similar to those found on the GRE Quantitative Reasoning section is presented below. Each question includes options and a step-by-step solution provided below.

Question 1

A car covered 130 miles using 4 gallons of diesel fuel. What distance would the same car cover, under similar conditions, using 6.7 gallons?

A) 77

B) 260.25

C) 520

D) 217.75

E) 871

View Solution and Explanation

Set up a direct proportion based on fuel consumption:

\[ \dfrac{130 \text{ miles}}{4 \text{ gallons}} = \dfrac{x \text{ miles}}{6.7 \text{ gallons}} \]

Solve for \(x\):

\[ x = \dfrac{130 \times 6.7}{4} = \dfrac{871}{4} = 217.75 \]

Answer: D) 217.75

Question 2

If plotted on the same rectangular coordinate system, the graphs of

\[ f(x) = \left||x| - 4\right| \quad \text{and} \quad g(x) = 2 \]

will have how many points of intersection?

A) No points of intersection

B) 1 point of intersection

C) 2 points of intersection

D) 3 points of intersection

E) 4 points of intersection

View Solution and Explanation

Set the two equations equal to each other to find points of intersection:

\[ \left||x| - 4\right| = 2 \]

This gives two possibilities for the inner absolute value:

\[ |x| - 4 = 2 \quad \text{or} \quad |x| - 4 = -2 \]
  1. If \(|x| - 4 = 2\), then \(|x| = 6\), which yields \(x = 6\) or \(x = -6\) (2 solutions).
  2. If \(|x| - 4 = -2\), then \(|x| = 2\), which yields \(x = 2\) or \(x = -2\) (2 solutions).

In total, there are 4 distinct points of intersection.

Answer: E) 4 points of intersection

Question 3

Which of the following numbers is the largest?

A) \(125\%\)

B) \(1.25\)

C) \(1 + \dfrac{1}{3}\)

D) \( \dfrac{4}{3}\)

E) \( \dfrac{0.0015}{0.001}\)

View Solution and Explanation

Convert each option into decimal format for comparison:

  • A) \(125\% = 1.25\)
  • B) \(1.25\)
  • C) \(1 + \dfrac{1}{3} \approx 1.333\)
  • D) \( \dfrac{4}{3} \approx 1.333\)
  • E) \( \dfrac{0.0015}{0.001} = 1.5\)

Comparing the values, \(1.5\) (Option E) is the largest.

Answer: E) \( \dfrac{0.0015}{0.001}\)

Question 4

Evaluate the expression:

\[ \left|\left|-10 - 19\right| - 20\right| \]

A) 9

B) 49

C) \(-49\)

D) 19

E) 11

View Solution and Explanation

Evaluate step-by-step from the innermost absolute value outward:

  • \(-10 - 19 = -29\)
  • \(\left|-29\right| = 29\)
  • \(29 - 20 = 9\)
  • \(\left|9\right| = 9\)

Answer: A) 9

Question 5

The algebraic expression

\[ \dfrac{x}{x + 2} \]

is undefined for which value of \(x\)?

A) 0

B) 1

C) 2

D) \(-2\)

E) \(-1\)

View Solution and Explanation

A rational expression is undefined where its denominator equals zero:

\[ x + 2 = 0 \implies x = -2 \]

Answer: D) \(-2\)

Question 6

Evaluate:

\[ (\sqrt{5} - \sqrt{7})(\sqrt{5} + \sqrt{7}) \]

A) 12

B) 2

C) 24

D) \(-24\)

E) \(-2\)

View Solution and Explanation

Use the difference of squares identity \((a - b)(a + b) = a^2 - b^2\):

\[ (\sqrt{5})^2 - (\sqrt{7})^2 = 5 - 7 = -2 \]

Answer: E) \(-2\)

Question 7

In the figure below, the sides \(AB\) and \(AC\) of triangle \(ABC\) have equal lengths. Find the measure of angle \(\angle ABC\).

Isosceles triangle GRE question

A) \(46^\circ\)

B) \(67^\circ\)

C) \(134^\circ\)

D) \(136^\circ\)

E) Cannot be calculated using the given information

View Solution and Explanation

Since \(AB = AC\), triangle ABC is isosceles. The angles opposite to the equal sides are also equal. Using triangle angle sum properties with the given vertex angle yields \(67^\circ\).

Answer: B) \(67^\circ\)

Question 8

If \(y = 10^\circ\) in the figure below, what is the value of \(x\)?

Angle relationships GRE question

A) \(20^\circ\)

B) \(30^\circ\)

C) \(10^\circ\)

D) \(5^\circ\)

E) \(15^\circ\)

View Solution and Explanation

Using the property that the sum of all interior angles in a triangle is equal to \(180^\circ\), we sum the given angle expressions:

\[ (4x + 4y) + (x + 3y) + (x + 2y) = 180^\circ \]

Combine like terms to simplify the equation:

\[ 6x + 9y = 180^\circ \]

Substitute the known value \(y = 10^\circ\) into the equation:

\[ 6x + 9(10^\circ) = 180^\circ \] \[ 6x + 90^\circ = 180^\circ \] \[ 6x = 90^\circ \implies x = \dfrac{90^\circ}{6} = 15^\circ \]

Answer: E) \(15^\circ\)

Question 9

By what percent will the volume of a rectangular solid increase if its length, width, and height each increase by \(25\%\)?

A) 25%

B) 95%

C) 74%

D) 15.6%

E) 50%

View Solution and Explanation

Let original dimensions be \(L\), \(W\), and \(H\). Original Volume \(V_1 = LWH\).

Each dimension increases by \(25\%\), becoming \(1.25L\), \(1.25W\), and \(1.25H\).

New Volume \(V_2 = (1.25L)(1.25W)(1.25H) = (1.25)^3 LWH = 1.953125 \, LWH\).

Percentage increase = \((1.953125 - 1) \times 100\% = 95.3125\%\), which rounds to \(95\%\).

Answer: B) 95%

Question 10

What is the average of all prime numbers between 20 and 40?

A) 20

B) 25

C) 30

D) 35

E) 40

View Solution and Explanation

Identify all prime numbers between 20 and 40:

\[ 23, \quad 29, \quad 31, \quad 37 \]

Calculate their sum:

\[ 23 + 29 + 31 + 37 = 120 \]

Calculate the average by dividing the sum by the count (4):

\[ \text{Average} = \dfrac{120}{4} = 30 \]

Answer: C) 30

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