Hard GRE Quant Practice Test Practice Questions

Concept Explanation

A Hard GRE Quant Practice Test Practice Questions set evaluates a test-taker's ability to solve complex mathematical problems involving advanced algebra, geometry, data analysis, and number properties. These questions are designed to mirror the second, more difficult quantitative section of the GRE Prep journey, where the exam adapts to a higher difficulty level based on initial performance. Success on these problems requires more than basic arithmetic; it demands a deep understanding of mathematical theory, the ability to recognize patterns, and the agility to apply multiple concepts to a single problem. For instance, a single question might require you to use the Pythagorean theorem while simultaneously calculating probabilities or permutations.

According to ETS, the makers of the GRE, the quantitative reasoning section measures your ability to interpret and analyze quantitative information. At the "Hard" level, the exam often utilizes "distractors" or multi-step logic that can easily lead to errors if one is not meticulous. Utilizing an AI Exam Simulator can help students acclimate to the pacing and pressure of these high-difficulty tasks. Key areas often tested include prime factorization, coordinate geometry rotations, and shaded area problems involving circles and polygons.

Solved Examples

1. Number Properties: If $n$ is a positive integer and $n^2$ is divisible by 72, what is the smallest possible value of $n$?
   1. First, find the prime factorization of 72: $72 = 2^3 \times 3^2$.
   2. For $n^2$ to be divisible by 72, it must contain at least the same prime factors in at least the same powers. Thus, $n^2 = 2^a \times 3^b \times \dots$, where $a \geq 3$ and $b \geq 2$.
   3. Since $n^2$ is a perfect square, all exponents in its prime factorization must be even. The smallest even integer greater than or equal to 3 is 4, and the smallest even integer greater than or equal to 2 is 2.
   4. So, the smallest $n^2$ is $2^4 \times 3^2 = 16 \times 9 = 144$.
   5. Taking the square root, $n = \sqrt{144} = 12$. The answer is 12.
2. Geometry: A circle is inscribed in a square with a side length of 10. A smaller square is then inscribed inside that circle. What is the area of the smaller square?
   1. The diameter of the circle is equal to the side of the larger square, which is 10. Therefore, the radius $r = 5$.
   2. When a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle.
   3. Let the side of the smaller square be $s$. The diagonal is $s\sqrt{2}$.
   4. Set the diagonal equal to the diameter: $s\sqrt{2} = 10$.
   5. Solve for $s$: $s = \frac{10}{\sqrt{2}} = 5\sqrt{2}$.
   6. The area of the smaller square is $s^2 = (5\sqrt{2})^2 = 25 \times 2 = 50$.
3. Combinatorics: A committee of 3 people is to be chosen from a group of 5 men and 4 women. If the committee must contain at least one woman, how many different committees can be formed?
   1. Total number of ways to choose 3 people from 9 (5 men + 4 women) is $\binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$.
   2. Calculate the number of ways to choose a committee with *no* women (all men). This is $\binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10$.
   3. Subtract the "no women" cases from the total: $84 - 10 = 74$. There are 74 such committees.

Practice Questions

1. If $x$ and $y$ are integers such that $x^2 + y^2 = 100$, what is the maximum possible value of $|x - y|$?

2. 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 of 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. In the $xy$-coordinate plane, line $L$ passes through points $(0, 0)$ and $(4, 8)$. Line $M$ is perpendicular to line $L$ and passes through the point $(4, 8)$. What is the $x$-intercept of line $M$?

5. If $3^{x+1} = 9^y$ and $\log_2(x+y) = 3$, what is the value of $x$?

6. A cylindrical tank with a radius of 4 meters and a height of 9 meters is half full of water. If all the water is poured into a rectangular tank with a base of 6 meters by 8 meters, what will be the height of the water in the rectangular tank (in terms of $\pi$)?

7. For all numbers $n$, the function $f$ is defined by $f(n) = n^2 - n$. If $f(x) = f(x-2)$, what is the value of $x$?

8. A work crew of 10 people can complete a project in 12 days. If after 3 days, 2 people leave the crew, how many additional days will it take the remaining crew to finish the project?

Answers & Explanations

1. 14: Possible integer pairs for $x^2 + y^2 = 100$ are $(0, 10), (6, 8), (8, 6), (10, 0)$. To maximize $|x - y|$, choose $x=10, y=0$ which gives 10, or consider negative integers like $x=10, y=-0$ (still 10) or $x=8, y=-6$. If $x=8$ and $y=-6$, $x^2+y^2=64+36=100$. Then $|8 - (-6)| = 14$.
2. 7/15: Total ways to pick two: $\binom{10}{2} = 45$. Ways to pick two red: $\binom{4}{2} = 6$. Ways to pick two blue: $\binom{6}{2} = 15$. Total favorable: $6 + 15 = 21$. Probability: $21/45 = 7/15$.
3. 0.5: Let the integers be $n-3, n-2, n-1, n, n+1, n+2, n+3$. The average $K = n$. If $n+3$ is removed, the sum of the remaining 6 integers is $6n - (n+3) = 6n - 3$. The new average is $\frac{6n-3}{6} = n - 0.5$. The decrease is 0.5.
4. 20: Slope of $L = \frac{8-0}{4-0} = 2$. Slope of $M = -1/2$. Equation of $M$: $y - 8 = -1/2(x - 4)$. For $x$-intercept, set $y=0$: $-8 = -1/2(x-4) \rightarrow 16 = x - 4 \rightarrow x = 20$.
5. 5: From $3^{x+1} = (3^2)^y$, we get $x+1 = 2y$. From $\log_2(x+y) = 3$, we get $x+y = 2^3 = 8$. Substitute $y = 8-x$ into the first equation: $x+1 = 2(8-x) \rightarrow x+1 = 16 - 2x \rightarrow 3x = 15 \rightarrow x = 5$.
6. $1.5\pi$: Volume of water = $0.5 \times \pi \times 4^2 \times 9 = 72\pi$. Volume in rectangular tank = $6 \times 8 \times h = 48h$. Set $48h = 72\pi$, so $h = 72\pi / 48 = 1.5\pi$.
7. 1.5: $x^2 - x = (x-2)^2 - (x-2)$. Expand RHS: $x^2 - x = x^2 - 4x + 4 - x + 2$. Simplify: $-x = -5x + 6 \rightarrow 4x = 6 \rightarrow x = 1.5$.
8. 11.25: Total work = $10 \times 12 = 120$ man-days. Work done in 3 days = $10 \times 3 = 30$ man-days. Remaining work = $120 - 30 = 90$ man-days. Remaining crew = 8 people. Time = $90 / 8 = 11.25$ days.

1. If the length of a rectangle is increased by 20% and the width is decreased by 20%, what is the net percentage change in the area?

• ANo change
• B4% increase
• C4% decrease
• D2% decrease

Frequently Asked Questions

What makes a GRE Quant question "hard"?

Hard questions typically involve multiple steps, combine different mathematical concepts, or use abstract variables that require algebraic manipulation. They often include trap answers that result from common calculation errors or misinterpretations of the prompt.

How can I improve my speed on Hard GRE Quant Practice Test Practice Questions?

Improving speed requires mastering mental math and recognizing common patterns like Pythagorean triples or quadratic identities. Additionally, using tools like Adaptive GRE Practice Test Practice Questions helps you get used to the specific pacing required for high-scoring sections.

Are there calculators allowed on the GRE Quant section?

Yes, a basic on-screen calculator is provided for the GRE General Test, but it is limited to addition, subtraction, multiplication, division, and square roots. Relying too heavily on it can slow you down, so it is better used for verification rather than every step.

How many hard questions appear on the GRE?

The GRE is section-adaptive, meaning if you perform well on the first quantitative section, the second section will contain a higher density of difficult questions. In a hard second section, you might encounter 10-15 questions categorized as level 4 or 5 difficulty.

What is the best way to study for advanced probability and counting?

Focus on understanding the fundamental counting principle and the difference between permutations and combinations. Practicing with Unlimited GRE Practice Questions Practice Questions allows you to see various iterations of these problems until the logic becomes intuitive.

Should I skip hard questions and come back to them?

Yes, because all questions within a section are worth the same amount of points, it is strategic to answer easier questions first. Mark difficult ones and return to them once you have secured the points from the more straightforward problems.