GRE Quant Challenge: 15 Hard Questions with Full Solutions

Introduction

If you've been preparing for the GRE, you already know that Quant isn't just about knowing formulas—it's about applying them under pressure. The harder questions often combine multiple concepts, test logical reasoning, and reward strategic thinking over brute-force calculation.

This challenge is designed for aspirants who've covered the basics and are ready to test themselves with tougher problems. Each question here mirrors the difficulty level you might encounter in the upper score ranges, and every solution is explained step-by-step to help you understand not just the answer, but the approach.

Let's dive in.

Question 1: Probability and Counting

A bag contains 5 red balls and 7 blue balls. If two balls are drawn at random without replacement, what is the probability that both are red?

Solution:

Total balls = 12

Probability of first red ball = 5/12

After drawing one red ball, remaining red balls = 4, total balls = 11

Probability of second red ball = 4/11

Combined probability = (5/12) × (4/11) = 20/132 = 5/33

Answer: 5/33

Question 2: Algebra – Quadratic Equations

If x² – 5x + 6 = 0, what are the possible values of x?

Solution:

Factorize: (x – 2)(x – 3) = 0

So x = 2 or x = 3

Answer: x = 2 or 3

Question 3: Geometry – Circle Properties

A circle has a radius of 7 cm. What is the area of a sector with a central angle of 60°?

Solution:

Area of full circle = πr² = π(7)² = 49π

Sector represents 60°/360° = 1/6 of the circle

Area of sector = (1/6) × 49π = 49π/6 ≈ 25.66 cm²

Answer: 49π/6 cm²

Question 4: Number Properties

What is the remainder when 7⁵⁰ is divided by 10?

Solution:

Find the pattern of last digits of powers of 7:

7¹ = 7, 7² = 49, 7³ = 343, 7⁴ = 2401

Pattern of last digits: 7, 9, 3, 1 (repeats every 4 powers)

50 ÷ 4 = 12 remainder 2

So 7⁵⁰ has the same last digit as 7² = 9

Answer: 9

Question 5: Data Interpretation

The average of five numbers is 24. If one number is removed, the average becomes 22. What is the removed number?

Solution:

Sum of 5 numbers = 24 × 5 = 120

Sum of 4 numbers = 22 × 4 = 88

Removed number = 120 – 88 = 32

Answer: 32

Question 6: Coordinate Geometry

What is the distance between points A(3, 4) and B(7, 1)?

Solution:

Distance formula = √[(x₂ – x₁)² + (y₂ – y₁)²]

= √[(7 – 3)² + (1 – 4)²]

= √[16 + 9] = √25 = 5

Answer: 5

Question 7: Ratios and Proportions

If a:b = 2:3 and b:c = 4:5, find a:b:c.

Solution:

Make b equal in both ratios.

a:b = 2:3 = 8:12 (multiply by 4)

b:c = 4:5 = 12:15 (multiply by 3)

So a:b:c = 8:12:15

Answer: 8:12:15

Question 8: Exponents and Roots

Simplify: (16)^(3/4)

Solution:

16 = 2⁴

(2⁴)^(3/4) = 2^(4 × 3/4) = 2³ = 8

Answer: 8

Question 9: Percentage Change

A product's price increased by 20% and then decreased by 20%. What is the net percentage change?

Solution:

Let original price = 100

After 20% increase = 120

After 20% decrease on 120 = 120 – 24 = 96

Net change = 96 – 100 = –4

Answer: 4% decrease

Question 10: Word Problem – Work Rate

A can complete a task in 6 days, B in 8 days. How long will they take together?

Solution:

A's rate = 1/6 per day

B's rate = 1/8 per day

Combined rate = 1/6 + 1/8 = 4/24 + 3/24 = 7/24 per day

Time = 1 ÷ (7/24) = 24/7 ≈ 3.43 days

Answer: 24/7 days

Question 11: Inequalities

If 3x + 7 > 22, what is the smallest integer value of x?

Solution:

3x > 15

x > 5

Smallest integer = 6

Answer: 6

Question 12: Statistics – Standard Deviation

Which set has a higher standard deviation: {2, 4, 6} or {1, 4, 7}?

Solution:

Both have mean = 4

First set deviations: –2, 0, 2 (evenly spaced)

Second set deviations: –3, 0, 3 (more spread)

Second set has higher standard deviation.

Answer: {1, 4, 7}

Question 13: Functions

If f(x) = 2x + 3, find f(f(2)).

Solution:

f(2) = 2(2) + 3 = 7

f(7) = 2(7) + 3 = 17

Answer: 17

Question 14: Divisibility

How many integers between 1 and 100 are divisible by both 3 and 5?

Solution:

Numbers divisible by both 3 and 5 are divisible by 15.

Multiples of 15 between 1 and 100: 15, 30, 45, 60, 75, 90

Answer: 6

Question 15: Combinatorics

In how many ways can 3 books be arranged on a shelf?

Solution:

Number of arrangements = 3! = 3 × 2 × 1 = 6

Answer: 6

Final Thoughts

Working through difficult Quant questions isn't just about getting the right answer—it's about building the mental stamina and strategic thinking you'll need on test day. If you found yourself stuck on a few of these, that's completely normal. Revisit the concepts, practice similar problems, and track your improvement over time.

Some aspirants find it easier to stay consistent using structured practice tools like PrepAiro, which offer topic-wise drills and performance tracking. But regardless of how you prepare, the key is regular, focused practice.

Keep challenging yourself, and remember—every tough question you solve today is one less obstacle on exam day.

FAQ Section

Q1: How difficult are these questions compared to the actual GRE?

These questions reflect the moderate-to-hard difficulty range you'd encounter in the GRE Quant section, especially if you're aiming for a score above 160. They combine conceptual understanding with problem-solving efficiency.

Q2: Should I time myself while solving these questions?

Yes. Ideally, aim to solve each question in about 1.5 to 2 minutes. Timing yourself helps simulate test conditions and improves your decision-making under pressure.

Q3: What if I got most of these wrong?

Don't worry. These are intentionally challenging. Review the solutions carefully, identify which topics tripped you up, and revisit those fundamentals before attempting similar problems again.

Q4: Can I use a calculator for GRE Quant?

Yes, the GRE provides an on-screen calculator. However, it's basic, so mental math and estimation skills still play a big role in saving time.

Q5: How often should I practice hard Quant questions?

Aim to include 5–10 challenging questions in your study sessions 3–4 times a week. Balance them with concept revision and easier questions to build both speed and accuracy.

Q6: Are there topics that repeat more often in hard Quant questions?

Yes. Probability, number properties, coordinate geometry, and word problems involving rates or ratios tend to show up frequently in harder question sets. Focus extra time on these areas.