Hard GRE Problem Solving Questions Practice Questions

Concept Explanation

Hard GRE Problem Solving Questions are complex quantitative tasks that require a combination of advanced algebraic manipulation, geometric logic, and high-level reasoning to solve within strict time limits. Unlike easier questions that test a single formula, these problems often integrate multiple mathematical concepts—such as probability, number properties, and coordinate geometry—into a single multi-step prompt. To excel, students must move beyond rote memorization and develop the ability to deconstruct word problems into solvable equations. This involves identifying the underlying mathematical structure and using strategic shortcuts, such as plugging in numbers or back-solving from the answer choices, to maintain efficiency. For those looking to refine their approach to these challenging items, GRE Prep resources provide the necessary scaffolding to bridge the gap between basic arithmetic and the rigorous logic required for a high score. Understanding the nuances of GRE practice questions with explanations is essential for identifying common traps set by the Educational Testing Service (ETS).

Solved Examples

1. Example: Combinatorics and Probability
   A committee of 3 people is to be chosen from a group of 5 men and 4 women. What is the probability that the committee will contain exactly 2 women?
   1. Calculate the total number of ways to choose 3 people from 9: $$\binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$$
   2. Calculate the ways to choose exactly 2 women from 4: $$\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6$$
   3. Calculate the ways to choose the remaining 1 person from the 5 men: $$\binom{5}{1} = 5$$
   4. Multiply the favorable outcomes: $6 \times 5 = 30$
   5. Find the probability: $\frac{30}{84} = \frac{5}{14}$
2. Example: Rates and Work
   Machine A can complete a job in 6 hours. Machine B can complete the same job in 9 hours. If Machine A works alone for 2 hours and then both machines work together, how many additional hours will it take to finish the job?
   1. Determine the rates: Machine A = $\frac{1}{6}$ job/hr; Machine B = $\frac{1}{9}$ job/hr.
   2. Work done by A in 2 hours: $2 \times \frac{1}{6} = \frac{1}{3}$.
   3. Remaining work: $1 - \frac{1}{3} = \frac{2}{3}$.
   4. Combined rate: $\frac{1}{6} + \frac{1}{9} = \frac{3+2}{18} = \frac{5}{18}$ job/hr.
   5. Time to finish: $\frac{ \text{Work}}{ \text{Rate}} = \frac{2/3}{5/18} = \frac{2}{3} \times \frac{18}{5} = \frac{12}{5} = 2.4$ hours.
3. Example: Geometry and Algebra
   A circle is inscribed in a square with a side length of $s$. What is the ratio of the area of the square to the area of the circle?
   1. Area of the square: $s^2$.
   2. The diameter of the inscribed circle is equal to the side of the square, $s$. Therefore, the radius $r = \frac{s}{2}$.
   3. Area of the circle: $\pi r^2 = \pi (\frac{s}{2})^2 = \frac{\pi s^2}{4}$.
   4. Ratio: $\frac{s^2}{\frac{\pi s^2}{4}} = \frac{4}{\pi}$.

Practice Questions

1. If $x$ and $y$ are integers such that $2x + 3y = 31$, and $x > 4$ and $y > 2$, how many possible values are there for $x$?
2. A dresser contains 6 pairs of black socks and 4 pairs of white socks. If two socks are selected at random without replacement, what is the probability that they are the same color?
3. The average (arithmetic mean) of 7 consecutive integers is $k$. If the largest of these integers is removed, by how much does the average decrease?

4. A cylindrical tank with a radius of 4 feet and a height of 10 feet is filled with water at a rate of 5 cubic feet per minute. To the nearest minute, how long will it take to fill the tank if it is currently half full?
5. If $n$ is a positive integer and $n^2$ is divisible by 72, which of the following must be a factor of $n$? (A) 6, (B) 12, (C) 18, (D) 36.
6. In a group of 100 students, 60 study Spanish, 45 study French, and 20 study both. How many students study neither language?
7. A car travels from Town A to Town B at an average speed of 40 mph and returns from Town B to Town A at an average speed of 60 mph. What is the average speed for the entire round trip?
8. The sum of the first $n$ terms of a sequence is given by $S_n = n^2 + 2n$. What is the value of the 10th term of the sequence?

Answers & Explanations

1. Answer: 2. We test values for $y$ (since it has a larger coefficient). If $y=3$, $2x + 9 = 31 \rightarrow 2x = 22 \rightarrow x = 11$. If $y=5$, $2x + 15 = 31 \rightarrow 2x = 16 \rightarrow x = 8$. If $y=7$, $2x + 21 = 31 \rightarrow 2x = 10 \rightarrow x = 5$. If $y=9$, $2x + 27 = 31 \rightarrow 2x = 4 \rightarrow x = 2$ (fails $x>4$). The valid $x$ values are 11, 8, and 5. Wait, the condition is $x > 4$ and $y > 2$. The values are (11,3), (8,5), and (5,7). Total: 3 values.
2. Answer: 7/15. Total socks = 20. Black = 12, White = 8. P(BB) = $\frac{12}{20} \times \frac{11}{19} = \frac{132}{380}$. P(WW) = $\frac{8}{20} \times \frac{7}{19} = \frac{56}{380}$. Total = $\frac{188}{380} = \frac{47}{95}$.
3. Answer: 0.5. Let the integers be $n-3, n-2, n-1, n, n+1, n+2, n+3$. The mean is $n$. If $n+3$ is removed, the sum is $6n-3$. The new mean is $\frac{6n-3}{6} = n - 0.5$. The decrease is 0.5.
4. Answer: 50 minutes. Volume $V = \pi r^2 h = \pi (16)(10) = 160\pi \approx 502.65$ cu ft. Half full is $251.3$ cu ft. Time = $251.3 / 5 \approx 50.26$. Round to 50.
5. Answer: 12. $72 = 2^3 \times 3^2$. If $n^2$ is divisible by $2^3 \times 3^2$, then $n^2$ must at least be divisible by $2^4 \times 3^2$ to be a perfect square. Thus $n$ must be divisible by $2^2 \times 3^1 = 12$.
6. Answer: 15. Using the formula $|A \cup B| = |A| + |B| - |A \cap B|$, we get $60 + 45 - 20 = 85$. Students studying neither = $100 - 85 = 15$.
7. Answer: 48 mph. Average speed = $\frac{ \text{Total Distance}}{ \text{Total Time}}$. Let distance be 120 miles. Time A to B = 3 hrs. Time B to A = 2 hrs. Total distance = 240. Total time = 5. Speed = $240/5 = 48$.
8. Answer: 21. The 10th term $a_{10} = S_{10} - S_9$. $S_{10} = 10^2 + 2(10) = 120$. $S_9 = 9^2 + 2(9) = 81 + 18 = 99$. $120 - 99 = 21$.

1. If a printer can print 400 pages in 5 minutes, how many pages can it print in 12 minutes?

• A800
• B960
• C1000
• D1200

Frequently Asked Questions

How do I identify a hard GRE problem solving question?

Hard questions usually involve multiple steps, combine different math topics like geometry and probability, or provide information in a convoluted word-problem format. They often require you to recognize a pattern or use a logical shortcut rather than just applying a basic formula.

What is the best strategy for time management on hard questions?

If you cannot determine the first step within 30 seconds, mark the question and move on to ensure you answer all easier questions first. Returning to a hard problem with a fresh perspective later in the section can often reveal the solution more clearly.

Can I use a calculator for these problems?

Yes, the GRE provides an on-screen calculator for the quantitative section, but it is limited to basic functions. For many hard problems, relying too heavily on the calculator can be slower than using mental math or algebraic simplification.

Are these questions more common in the second quantitative section?

The GRE is section-level adaptive, meaning if you perform well on the first quantitative section, the second section will contain a higher density of hard problem-solving questions. This is why using an adaptive GRE practice test is vital for realistic preparation.

How can I improve my accuracy on multi-step problems?

Improving accuracy requires writing out each step of your work clearly to avoid simple calculation errors and verifying your answer by plugging it back into the original prompt. Utilizing AI-powered question generators can help you practice specific types of multi-step problems until the process becomes intuitive.