Medium GRE Problem Solving Questions Practice Questions

Concept Explanation

Medium GRE Problem Solving Questions are quantitative reasoning tasks that require multiple logical steps, a solid grasp of high school mathematics, and the ability to avoid common traps. These questions typically involve concepts from arithmetic, algebra, geometry, and data analysis, often combining two or more of these areas into a single problem. While the math itself is not at a collegiate level, the GRE tests your ability to apply these rules efficiently under time pressure. Success requires balancing speed with precision, as these questions often include "distractor" answers that result from common calculation errors.

To excel in this section, you must be comfortable with GRE Prep strategies like back-solving from answer choices or plugging in numbers. For instance, if a question involves variables but no fixed values, picking small prime numbers like 2 or 3 can simplify complex algebraic expressions. Additionally, understanding the properties of numbers—such as parity, divisibility, and prime factorization—is essential for solving problems without performing long-form calculations. You can enhance your skills by utilizing an AI Exam Simulator to mimic the actual testing environment.

Solved Examples

Review these worked examples to understand the logic required for medium-difficulty problems.

1. Algebra: If $3x - 5 = 7y + 2$, and $x = 3$, what is the value of $y$?
   1. Substitute the value of $x$ into the equation: $3(3) - 5 = 7y + 2$.
   2. Simplify the left side: $9 - 5 = 4$.
   3. Rewrite the equation: $4 = 7y + 2$.
   4. Subtract 2 from both sides: $2 = 7y$.
   5. Divide by 7: $y = \frac{2}{7}$.
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: $\text{Area} = 10 \times 10 = 100$.
   2. Identify the radius of the circle. Since it is inscribed, the diameter equals the side of the square (10), so the radius $r = 5$.
   3. Calculate the area of the circle: $\text{Area} = \pi \times 5^2 = 25\pi$.
   4. Subtract the circle's area from the square's area: $100 - 25\pi$.
3. Probability: 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 are red?
   1. The probability of the first marble being red is $\frac{4}{10}$.
   2. After one red marble is removed, 3 red marbles and 9 total marbles remain.
   3. The probability of the second marble being red is $\frac{3}{9}$.
   4. Multiply the probabilities: $\frac{4}{10} \times \frac{3}{9} = \frac{12}{90}$.
   5. Simplify the fraction: $\frac{2}{15}$.

Practice Questions

Test your knowledge with these Medium GRE Problem Solving Questions. Ensure you read each prompt carefully before calculating.

1. A car travels at a constant speed of 45 miles per hour. How many minutes will it take for the car to travel 15 miles?

2. If $a$ and $b$ are integers such that $a^2 + b^2 = 25$, which of the following could be the value of $a + b$?
I. 1
II. 5
III. 7

3. A rectangular garden has a perimeter of 40 feet. If the length is 4 feet longer than the width, what is the area of the garden in square feet?

4. In a group of 50 students, 28 are taking Spanish, 15 are taking French, and 8 are taking both. How many students are taking neither Spanish nor French?

5. If $x > 0$ and $\frac{x}{4} + \frac{x}{8} = 12$, what is the value of $x$?

6. The average (arithmetic mean) of five numbers is 20. If a sixth number, 32, is added to the set, what is the new average?

7. A shopkeeper increases the price of an item by 20% and then applies a 10% discount on the new price. What is the total percentage increase from the original price?

8. What is the value of $n$ if $2^{n+1} = 8^3$?

9. A cylinder has a height of 8 and a base radius of 3. What is its total surface area? (Use $\pi$ in your answer).

10. If $y$ is inversely proportional to $x$, and $y = 10$ when $x = 2$, what is the value of $y$ when $x = 5$?

Answers & Explanations

1. 20 minutes: Use the formula $\text{Time} = \frac{ \text{Distance}}{ \text{Speed}}$. $\text{Time} = \frac{15}{45} = \frac{1}{3}$ hours. Since there are 60 minutes in an hour, $\frac{1}{3} \times 60 = 20$.
2. I, II, and III: Possible integer pairs for $a^2 + b^2 = 25$ are $(0, 5), (3, 4), (-3, 4), (0, -5)$, etc. If $a=3, b=4$, sum is 7. If $a=0, b=5$, sum is 5. If $a=-3, b=4$, sum is 1. All three are possible.
3. 96 square feet: Let width be $w$. Length is $w+4$. Perimeter $2(w + w + 4) = 40$. $4w + 8 = 40 \rightarrow 4w = 32 \rightarrow w = 8$. Length is 12. Area is $8 \times 12 = 96$.
4. 15: Use the formula $n(A \cup B) = n(A) + n(B) - n(A \cap B)$. $28 + 15 - 8 = 35$. The number of students taking neither is $50 - 35 = 15$.
5. 32: Find a common denominator: $\frac{2x}{8} + \frac{x}{8} = 12 \rightarrow \frac{3x}{8} = 12$. Multiply by 8: $3x = 96$. Divide by 3: $x = 32$.
6. 22: The sum of the first five numbers is $5 \times 20 = 100$. Adding the sixth: $100 + 32 = 132$. New average: $132 / 6 = 22$.
7. 8%: Let the original price be $100. After a 20% increase, it is $120. A 10% discount on $120 is $12. New price: $120 - 12 = 108$. This is an 8% increase from $100.
8. 8: Rewrite 8 as a power of 2: $8^3 = (2^3)^3 = 2^9$. Thus, $2^{n+1} = 2^9$. Solving $n+1 = 9$ gives $n = 8$.
9. $66\pi$: Surface area formula is $2\pi r^2 + 2\pi rh$. $2\pi(3^2) + 2\pi(3)(8) = 18\pi + 48\pi = 66\pi$.
10. 4: Inversely proportional means $xy = k$. $10 \times 2 = 20$, so $k = 20$. When $x = 5$, $5y = 20$, so $y = 4$.

1. If a shirt originally priced at $40 is on sale for 25% off, and an additional 10% discount is applied to the sale price, what is the final cost?

• A$26
• B$27
• C$28
• D$30

Frequently Asked Questions

How is the difficulty of GRE problem solving questions determined?

Difficulty is determined by the complexity of the steps required and the statistical performance of previous test-takers. Medium questions often involve two-step logic or the application of multiple geometric or algebraic rules.

Can I use a calculator for medium-level math on the GRE?

Yes, an on-screen calculator is provided during the Quantitative Reasoning section. However, it is often faster to use mental math or simplification techniques for medium-difficulty problems than to type every step into the interface.

What are the most common topics in GRE Problem Solving?

The most frequent topics include percentage changes, ratios, linear equations, and basic geometry. Understanding these core concepts is vital for moving through the section efficiently. You can find more targeted help with GRE Practice Questions with Answers.

How much time should I spend on a medium difficulty question?

You should aim to spend approximately 1.5 to 2 minutes per question. If a problem takes longer, it may be beneficial to mark it and return later to ensure you complete the easier items first.

Are there penalties for guessing on the GRE?

There is no penalty for incorrect answers on the GRE. If you are stuck on a medium question, eliminate the obviously wrong choices and make an educated guess before moving to the next task.

How can I improve my accuracy on medium-difficulty problems?

Accuracy improves through identifying patterns and common traps, such as forgetting to convert units or misreading "except" in a prompt. Using tools like Retrieval Challenge can help reinforce the formulas and concepts needed for quick recall.