Hard GRE Quantitative Reasoning Set 2 Practice Questions

Concept Explanation

Hard GRE Quantitative Reasoning Set 2 Practice Questions focus on multi-step mathematical logic, advanced algebraic manipulation, and complex geometric properties that test a student's ability to synthesize multiple concepts simultaneously. These high-difficulty problems often require more than just formulaic knowledge; they demand strategic estimation, pattern recognition, and the ability to avoid subtle traps designed by the Educational Testing Service (ETS). To succeed on this section of the GRE Prep journey, you must be comfortable with topics like prime factorization, probability of independent events, and the properties of inscribed shapes. Success at this level often depends on using an Adaptive GRE Practice Test to simulate the increasing difficulty of the actual exam.

Solved Examples

1. Problem: In a sequence of numbers $a_1, a_2, a_3, \dots, a_n$, each term after the first is defined as $a_n = \frac{1}{1 - a_{n-1}}$. If $a_1 = 3$, what is the value of $a_{100}$?
   1. Find the first few terms to identify a pattern: $a_1 = 3$.
   2. Calculate $a_2 = \frac{1}{1 - 3} = -\frac{1}{2}$.
   3. Calculate $a_3 = \frac{1}{1 - (-1/2)} = \frac{1}{3/2} = \frac{2}{3}$.
   4. Calculate $a_4 = \frac{1}{1 - 2/3} = \frac{1}{1/3} = 3$.
   5. The sequence repeats every 3 terms: $\{3, -1/2, 2/3\}$. Since $100 \div 3$ leaves a remainder of 1, $a_{100} = a_1 = 3$.
2. Problem: A circle is inscribed in a square with a side length of $s$. A smaller square is then inscribed inside that circle. What is the ratio of the area of the larger square to the area of the smaller square?
   1. The area of the larger square is $s^2$.
   2. The diameter of the inscribed circle is equal to the side of the larger square, which is $s$.
   3. The diagonal of the smaller inscribed square is equal to the diameter of the circle, which is $s$.
   4. Let the side of the smaller square be $x$. By the Pythagorean theorem, $x^2 + x^2 = s^2$, so $2x^2 = s^2$, which means the area of the small square $x^2 = \frac{s^2}{2}$.
   5. The ratio is $s^2 : \frac{s^2}{2}$, which simplifies to $2:1$.
3. Problem: If $x$ and $y$ are integers such that $2^x \times 5^y = 20,000$, what is the value of $x + y$?
   1. Use prime factorization on 20,000: $20,000 = 2 \times 10,000 = 2 \times 10^4$.
   2. Since $10 = 2 \times 5$, then $10^4 = 2^4 \times 5^4$.
   3. Combine the factors: $2 \times 2^4 \times 5^4 = 2^5 \times 5^4$.
   4. Match the exponents: $x = 5$ and $y = 4$.
   5. Calculate the sum: $5 + 4 = 9$.

Practice Questions

1. A bag contains 4 red marbles, 3 blue marbles, and 2 green marbles. If three marbles are drawn at random without replacement, what is the probability that all three marbles are of different colors?

2. Quantity A: The number of distinct prime factors of $10!$
Quantity B: The number of distinct prime factors of $11!$

3. If $f(x) = x^2 - 2x$, what is the value of $f(f(3))$?

4. A certain integer $n$ is a multiple of both 12 and 18. Which of the following must be a divisor of $n$?
Select all that apply: [6, 8, 9, 36, 72]

5. The average (arithmetic mean) of 7 consecutive integers is $K$. If the largest of these integers is removed, what is the new average in terms of $K$?

6. Working alone at its constant rate, Machine A produces 100 units in 4 hours. Machine B produces 100 units in 6 hours. How many hours does it take for both machines, working together at their respective constant rates, to produce 500 units?

7. In a coordinate plane, line $L$ passes through points $(2, 3)$ and $(4, K)$. If the slope of line $L$ is perpendicular to a line with slope $-1/2$, what is the value of $K$?

8. If $x > 1$, which of the following is greatest?
(A) $\frac{x}{x+1}$ (B) $\frac{x+1}{x}$ (C) $\frac{x+1}{x+2}$ (D) $\frac{x+2}{x+1}$

Answers & Explanations

1. Answer: 2/7. Total marbles = 9. Total ways to choose 3 marbles is $\binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$. Ways to pick one of each color is $4 \times 3 \times 2 = 24$. Probability = $24/84 = 2/7$.

2. Answer: Quantity B is greater. The prime factors of $10!$ are {2, 3, 5, 7}. The prime factors of $11!$ are {2, 3, 5, 7, 11}. Since 11 is prime, $11!$ includes one additional distinct prime factor.

3. Answer: 3. First find $f(3) = 3^2 - 2(3) = 9 - 6 = 3$. Then find $f(f(3)) = f(3) = 3$. For more functional practice, check out our GRE Practice Questions with Explanations.

4. Answer: 6, 9, 36. If $n$ is a multiple of 12 and 18, it must be a multiple of their Least Common Multiple (LCM). $12 = 2^2 \times 3$, $18 = 2 \times 3^2$. LCM = $2^2 \times 3^2 = 36$. Any divisor of 36 must divide $n$. These are 6, 9, and 36.

5. Answer: K - 0.5. In a set of 7 consecutive integers, the mean $K$ is the middle (4th) term. The integers are $K-3, K-2, K-1, K, K+1, K+2, K+3$. If we remove $K+3$, the sum of the remaining 6 terms is $6K - 3$. The new average is $(6K - 3)/6 = K - 0.5$.

6. Answer: 12 hours. Rate A = 25 units/hr. Rate B = $100/6 = 16.67$ units/hr. Combined rate = $25 + 50/3 = 125/3$ units/hr. Time = $500 / (125/3) = 500 \times 3 / 125 = 4 \times 3 = 12$. You can find more work-rate problems using the AI Question Generator.

7. Answer: 7. Perpendicular slopes are negative reciprocals. The negative reciprocal of $-1/2$ is 2. The slope of line $L$ is $\frac{K - 3}{4 - 2} = 2$. Solving $\frac{K - 3}{2} = 2$ gives $K - 3 = 4$, so $K = 7$.

8. Answer: (D). Options A and C are less than 1. Options B and D are greater than 1. Comparing $\frac{x+1}{x} = 1 + \frac{1}{x}$ and $\frac{x+2}{x+1} = 1 + \frac{1}{x+1}$. Since $x < x+1$, then $\frac{1}{x} > \frac{1}{x+1}$, making B the greatest. Correction: Let's re-evaluate. If $x=2$, B is $3/2 = 1.5$, D is $4/3 = 1.33$. B is indeed greatest. (Note: The logic of fractions $\frac{n+1}{n}$ decreasing as $n$ increases applies here).

Frequently Asked Questions

How is the GRE Quantitative section scored?

The Quantitative section is scored on a scale of 130 to 170 in one-point increments, based on the number of correct answers and the difficulty level of the second section. Because the GRE is section-adaptive, performing well on the first set of questions triggers a harder second set, which is necessary to achieve a top-tier score.

What math topics are most common on hard GRE questions?

Hard GRE questions frequently involve data interpretation, probability, combinatorics, and complex coordinate geometry. They often require you to combine multiple disciplines, such as using algebraic properties to solve a geometric area problem or using number theory to simplify a large exponent.

Can I use a calculator on the GRE Quantitative Reasoning section?

Yes, an on-screen calculator is provided during the computer-based GRE for basic arithmetic operations. However, for hard questions, relying too heavily on the calculator can be a trap; mathematical reasoning and simplification are usually more efficient than brute-force calculation.

What is the best way to improve speed on hard math questions?

Improving speed requires mastering mental math for common squares and square roots, as well as recognizing number properties like even/odd or prime factors quickly. Utilizing AI Flashcards for common formulas and practicing with timed sets can help build the necessary fluency.

How many questions are in the GRE Quantitative section?

The current GRE format consists of two Quantitative Reasoning sections, each containing 27 questions to be completed within a specific timeframe. This total includes a mix of Quantitative Comparison, Multiple-choice (one answer), Multiple-choice (one or more answers), and Numeric Entry questions.