GRE Problem Solving Questions Practice Questions with Answers

Approximately 50% of the Quantitative Reasoning section consists of standard multiple-choice items known as GRE Problem Solving questions. These tasks require you to apply basic mathematical concepts to solve for a specific value or relationship. Success on this portion of the exam is not just about raw calculation; it involves recognizing patterns and choosing the most efficient path to the solution. By engaging with a structured GRE Prep routine, you can familiarize yourself with the common pitfalls and logic traps set by the test makers.

Concept Explanation

GRE Problem Solving questions are standard multiple-choice items that require selecting a single correct answer from five options based on arithmetic, algebra, geometry, or data analysis. These questions test your ability to synthesize information and apply mathematical formulas in both abstract and real-world contexts. Unlike Quantitative Comparison questions, which focus on the relative size of two quantities, Problem Solving questions focus on finding a definitive numerical or logical result. The Educational Testing Service (ETS) designs these problems to ensure that while the math is generally at a high school level, the application requires sophisticated critical thinking. To prepare effectively, many students use a AI Question Generator to encounter a wide variety of scenarios, from rate-time-distance problems to complex probability sets.

Solved Examples

1. Algebraic Manipulation: If $3x + 7 = 22$, what is the value of $x^2 - 5$?
   1. Subtract 7 from both sides of the equation: $3x = 15$.
   2. Divide by 3 to find $x$: $x = 5$.
   3. Substitute $x = 5$ into the target expression: $5^2 - 5 = 25 - 5 = 20$.
   4. The final answer is 20.
2. Geometry: A circle is inscribed in a square with a side length of 10. What is the area of the region inside the square but outside the circle?
   1. Calculate the area of the square: $10 \times 10 = 100$.
   2. Identify the diameter of the circle, which is equal to the side of the square (10). Thus, the radius $r = 5$.
   3. Calculate the area of the circle: $\pi r^2 = \pi(5^2) = 25\pi$.
   4. Subtract the circle's area from the square's area: $100 - 25\pi$.
3. Averages/Statistics: The average of five numbers is 20. If a sixth number, 44, is added to the set, what is the new average?
   1. Find the sum of the original five numbers: $5 \times 20 = 100$.
   2. Add the sixth number to the total sum: $100 + 44 = 144$.
   3. Divide the new sum by the new count of numbers: $144 / 6 = 24$.
   4. The new average is 24.

Practice Questions

1. A car travels at an average speed of 60 miles per hour for 3 hours and then at an average speed of 40 miles per hour for 2 hours. What is the average speed of the car for the entire 5-hour trip?

2. If $\frac{2}{x} + \frac{3}{y} = \frac{5}{6}$ and $xy = 12$, what is the value of $2y + 3x$?

3. A rectangular box has a length of 8, a width of 6, and a height of 5. If all dimensions are increased by 50%, what is the percentage increase in the volume of the box?

4. In a group of 80 people, 45 speak French, 30 speak Spanish, and 10 speak both languages. How many people in the group speak neither language?

5. If $2^n = 128$, what is the value of $n^2 - 2n$?

6. A shopkeeper sells a product for $120, which is a 20% profit over the cost price. What was the original cost price to the shopkeeper?

7. If the area of an equilateral triangle is $16\sqrt{3}$, what is the perimeter of the triangle?

8. A bag contains 4 red marbles and 6 blue marbles. If two marbles are drawn at random without replacement, what is the probability that both marbles are red?

9. Solve for $z$ if $3(z - 4) = 2(z + 1) + 5$.

10. The ratio of the interior angles of a quadrilateral is 2:3:4:6. What is the measure of the largest angle in degrees?

Answers & Explanations

1. 52 mph: Total distance = $(60 \times 3) + (40 \times 2) = 180 + 80 = 260$ miles. Total time = 5 hours. Average speed = $260 / 5 = 52$.
2. 10: Combine the fractions: $\frac{2y + 3x}{xy} = \frac{5}{6}$. Since $xy = 12$, substitute it in: $\frac{2y + 3x}{12} = \frac{5}{6}$. Multiply both sides by 12: $2y + 3x = 10$.
3. 237.5%: Original volume $V_1 = 8 \times 6 \times 5 = 240$. New dimensions: 12, 9, 7.5. New volume $V_2 = 12 \times 9 \times 7.5 = 810$. Increase = $810 - 240 = 570$. Percentage increase = $(570/240) \times 100 = 237.5\%$.
4. 15: Total speakers = $( \text{French} + \text{Spanish}) - \text{Both} = (45 + 30) - 10 = 65$. Neither = $80 - 65 = 15$.
5. 35: Since $2^7 = 128$, $n = 7$. Target expression: $7^2 - 2(7) = 49 - 14 = 35$.
6. $100: Let cost be $C$. $1.20C = 120$. $C = 120 / 1.2 = 100$.
7. 24: Area of equilateral triangle = $\frac{s^2\sqrt{3}}{4} = 16\sqrt{3}$. Solving for $s$: $s^2 = 64$, so $s = 8$. Perimeter = $3 \times 8 = 24$.
8. 2/15: Probability of first red = 4/10. Probability of second red = 3/9. Total probability = $\frac{4}{10} \times \frac{3}{9} = \frac{12}{90} = \frac{2}{15}$.
9. 19: Expand: $3z - 12 = 2z + 2 + 5$. Simplify: $3z - 12 = 2z + 7$. Subtract $2z$ and add 12: $z = 19$.
10. 144: Sum of angles in a quadrilateral is 360. Let parts be $2x, 3x, 4x, 6x$. $15x = 360$, so $x = 24$. Largest angle = $6 \times 24 = 144$.

Frequently Asked Questions

How much time should I spend on each GRE Problem Solving question?

You should aim to spend approximately 1.5 to 2 minutes per question to ensure you complete all 20 questions in the 35-minute section. Budgeting time for reviewing tricky answers at the end is a highly effective strategy for high scorers.

Can I use a calculator on GRE Problem Solving questions?

Yes, an on-screen calculator is provided for the computer-delivered GRE, featuring basic arithmetic functions and a square root key. However, it is often faster to use mental math or estimation for simpler calculations to save time.

What math subjects are most frequently tested in this section?

The GRE focuses heavily on arithmetic, basic algebra, geometry, and data analysis, including topics like ratios, percentages, and coordinate geometry. You will not encounter advanced calculus or trigonometry on the exam.

Are these questions harder than the Quantitative Comparison questions?

Hardness is subjective, but Problem Solving questions are often considered more straightforward because they require a single numerical answer rather than a comparison of two values. However, they can involve more multi-step calculations that increase the chance of simple errors.

How can I avoid making careless mistakes in GRE math?

Double-checking your work by plugging the answer back into the original equation and carefully reading the units requested (e.g., minutes vs. hours) are the best ways to minimize errors. Utilizing a AI Exam Simulator can also help build the mental stamina needed to stay focused.