GRE Math Questions: Hardest Official Problem + 12 Practice
On this page, you'll find a super-tricky official GRE math practice question from ETS, and then you can practice further with 12 additional questions that will give you a pretty good sampling of things the GRE can do.
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Table of contents
- The hardest GRE math question (paraphrased)
- Why hard GRE math questions help you
- Solution (two methods, ELI5)
- The real takeaway (why ETS questions feel “weird”)
- 12 GRE math questions to practice (with answers)
- Where to find more official GRE math questions
- GRE math questions FAQ
- Need help? (tutoring)
You’re gonna get this hard GRE math question wrong.
Ok - I admit it - that line is clickbait. But also… it’s kind of true. I’ve worked with a LOT of GRE students, and this is one of those questions that smart people miss because it’s not “hard math.” It’s sneaky math.
Here’s the question (paraphrased from a real ETS-style prompt):
A bookcase contains 10 different books: 4 biographies and 6 novels. A student must select 4 books, and at least 2 of the selected books must be biographies.
How many different groups of 4 books could the student select?
Try it before you scroll. Seriously - even if you only spend 60 seconds on it, that effort matters.
Pro tip: After you read a new GRE math question, give yourself about 10 seconds to just think. Don’t rush to start calculating. Those 10 seconds are where you notice the “ohhh… it’s a counting question” shortcut.
P.S. - if you like practicing in short bursts, you should also check out my big “learn GRE math” guide (it includes my GRE Math Knight app): Vince’s Complete GRE Math Concepts Guide
Why hard GRE math questions help you (even if you hate them)
Before I solve it, here’s why questions like this are actually useful (and not just annoying):
- They force you to think. Not “apply a formula you memorized,” but actually think. That’s how your brain gets better at GRE Quant.
- They build confidence. When you crack a tough one on your own, your math anxiety takes a hit (in a good way).
- They train adaptability. The GRE loves disguising simple ideas in weird packaging. Getting good at adapting is basically the whole game.
One more important thing: don’t look up the answer the second you get stuck. If you always “tap out” early, you’re training yourself to quit. Give it an honest shot first.
GRE math question solution (ELI5)
We have 10 different books total. We want groups of 4 books. And the group must include 2 or more biographies.
Step 0: Translate the English into “math rules”
- There are 4 biographies (call them “bios”).
- There are 6 novels.
- We pick 4 books total.
- We must pick at least 2 bios → that means we could pick 2 bios, 3 bios, or 4 bios.
- Order doesn’t matter (a set of books is a set).
Method 1: Casework (2 bios, 3 bios, or 4 bios)
Case A: Pick 4 bios.
- How many ways to pick all 4 biographies out of 4? Just 1.
Case B: Pick 3 bios and 1 novel.
- Ways to pick 3 bios out of 4: C(4,3) = 4
- Ways to pick 1 novel out of 6: C(6,1) = 6
- Total for this case: 4 × 6 = 24
Case C: Pick 2 bios and 2 novels.
- Ways to pick 2 bios out of 4: C(4,2) = 6
- Ways to pick 2 novels out of 6: C(6,2) = 15
- Total for this case: 6 × 15 = 90
Final answer: add the three cases:
1 + 24 + 90 = 115
Method 2: Pick any 4 books… then subtract the “bad” groups
This is the “lazy genius” method. Count everything first, then remove what the rules don’t allow.
- Total ways to choose any 4 books from 10: C(10,4) = 210
- Bad groups with 0 bios (all novels): C(6,4) = 15
- Bad groups with exactly 1 bio: pick 1 bio out of 4 AND 3 novels out of 6: C(4,1) × C(6,3) = 4 × 20 = 80
- Good groups = total − bad: 210 − 15 − 80 = 115
ELI5: What does C(n, k) mean?
C(n, k) (said “n choose k”) means: How many different groups of size k can I make from n things? And the order does NOT matter.
Example: if you choose books A, B, C… that’s the same group as C, B, A. Same books. Same group.
If you like formulas, here it is:
C(n, k) = n! / (k! × (n − k)!)
But you don’t need to “memorize factorial math” to do most GRE counting problems. What you need is to recognize: “Oh, it’s groups, not order.”
If this question felt gross… you’re not alone.
This is exactly the kind of “simple concept, sneaky setup” problem that wrecks timing and confidence. If you want help fixing that (for real), we can talk.
The real takeaway (why ETS GRE math questions feel “weird”)
This question isn’t testing a brand-new math concept. Most people have heard of “combinations” before. The problem is that ETS presents familiar ideas in unfamiliar clothing.
- Knowing a concept isn’t enough - you need mastery.
- ETS wants you to notice structure: “This is 2-or-more… so I need cases or subtraction.”
- The fastest path to improvement is building a big mental database of official-style questions. The more ETS questions you’ve seen and truly solved, the less “random” the real test feels.
If you feel rusty on the basics, don’t start with “hard” problems. Start with concepts and build up: here’s my complete GRE math concepts guide.
12 GRE math questions to practice (with answers + ELI5 explanations)
These are GRE-style practice questions (not official ETS questions). They’re designed to feel like real GRE math questions: short, logical, and sometimes annoyingly sneaky.
How to use them: do the question, commit to an answer, then check the explanation. If you miss it, redo it the next day without looking. That’s how you actually improve.
Question 1 (Quantitative Comparison - Easy)
Let a be a positive number.
Quantity A: a + (1 / a)
Quantity B: 2
Answer + explanation
Answer: D (The relationship cannot be determined).
Here’s the deal: for positive a, the expression a + 1/a is always at least 2. But sometimes it’s exactly 2 and sometimes it’s bigger.
- If a = 1, then a + 1/a = 1 + 1 = 2 → equal.
- If a = 2, then a + 1/a = 2 + 0.5 = 2.5 → Quantity A is bigger.
Since the result changes depending on a, the answer is D.
Question 2 (Quantitative Comparison - Medium)
Let x be a real number and x > 2.
Quantity A: x / (x − 2)
Quantity B: 1
Answer + explanation
Answer: A (Quantity A is greater).
If x > 2, then (x − 2) is positive. And (x − 2) is smaller than x. So you’re doing: “a bigger number divided by a smaller positive number,” which makes the result bigger than 1.
Quick example: if x = 4, then 4 / (4 − 2) = 4 / 2 = 2, which is bigger than 1.
Question 3 (Multiple Choice - Medium)
A club has 6 seniors and 5 juniors. A committee of 4 must be formed with at least 1 senior and at least 1 junior. How many different committees are possible?
A) 210 B) 240 C) 300 D) 310 E) 330
Answer + explanation
Answer: D (310).
Start by counting all possible committees of 4 from 11 people total: C(11,4) = 330.
Now subtract the committees that break the rule:
- All seniors (no juniors): C(6,4) = 15
- All juniors (no seniors): C(5,4) = 5
So valid committees = 330 − 15 − 5 = 310.
Question 4 (Numeric Entry - Easy)
If x% of 250 is 45, what is x?
Answer + explanation
Answer: 18
“x% of 250” means (x/100) × 250. So:
(x/100) × 250 = 45
Multiply both sides by 100: x × 250 = 4500
Divide by 250: x = 4500 / 250 = 18.
Question 5 (Select All That Apply - Medium)
If an integer n is divisible by 30, which of the following must be true?
A) n divisible by 5
B) n divisible by 6
C) n divisible by 10
D) n divisible by 15
E) n divisible by 20
Answer + explanation
Answers: A, B, C, D
If n is divisible by 30, that means n = 30k. And 30 = 2 × 3 × 5.
- Divisible by 5? Yes, because 30 includes a 5.
- Divisible by 6? Yes, because 6 = 2 × 3, and 30 includes 2 and 3.
- Divisible by 10? Yes, because 10 = 2 × 5, and 30 includes 2 and 5.
- Divisible by 15? Yes, because 15 = 3 × 5, and 30 includes 3 and 5.
- Divisible by 20? Not necessarily. 20 = 2² × 5. You’d need two 2’s. 30 only guarantees one 2.
Question 6 (Multiple Choice - Medium)
A rectangle has perimeter 30 and area 56. What is the length of the longer side?
A) 6 B) 7 C) 8 D) 9 E) 10
Answer + explanation
Answer: C (8).
Let the sides be a and b. Perimeter 30 means: 2(a + b) = 30 → a + b = 15.
Area 56 means: ab = 56.
So we need two numbers that add to 15 and multiply to 56. 7 and 8 work because 7 + 8 = 15 and 7 × 8 = 56.
The longer side is 8.
Question 7 (Multiple Choice - Hard)
A data set contains: 4, 6, 6, 8, 10. One additional number x is added. After adding x, the mean, median, and mode of the 6-number set are all equal. What is x?
A) 2 B) 4 C) 6 D) 8 E) 12
Answer + explanation
Answer: A (2).
The mode right now is 6 (it shows up twice). The easiest way for “mean = median = mode” is to make them all 6.
If the mean is 6 for 6 numbers, the total sum must be 6 × 6 = 36.
Current sum is 4 + 6 + 6 + 8 + 10 = 34. So we need x to bring the sum to 36: x = 36 − 34 = 2.
Quick check: with x = 2, the numbers are 2, 4, 6, 6, 8, 10. Median is (3rd and 4th) = (6 and 6) → 6. Mode is 6. Mean is 36/6 = 6. Perfect.
Question 8 (Multiple Choice - Medium)
A bag contains 3 red balls, 2 blue balls, and 5 green balls. Two balls are selected at random without replacement. What is the probability that the two selected balls are the same color?
A) 7/45 B) 14/45 C) 1/3 D) 2/5 E) 7/15
Answer + explanation
Answer: B (14/45).
Total balls = 3 + 2 + 5 = 10. Total ways to pick 2 balls (in “group” form) is C(10,2) = 45.
Now count “same color” picks:
- Two reds: C(3,2) = 3
- Two blues: C(2,2) = 1
- Two greens: C(5,2) = 10
Favorable = 3 + 1 + 10 = 14. Probability = 14/45.
Question 9 (Multiple Choice - Easy/Medium)
Solve for x: (x − 3) / (x + 3) = 1/2
A) 3 B) 6 C) 9 D) 12 E) 15
Answer + explanation
Answer: C (9).
Cross-multiply (this is just “undoing the fraction”):
2(x − 3) = 1(x + 3)
2x − 6 = x + 3
Subtract x from both sides: x − 6 = 3
Add 6 to both sides: x = 9.
Question 10 (Multiple Choice - Medium)
The line x + y = 10 intersects the x-axis at point A and the y-axis at point B. What is the distance AB?
A) 10 B) 10√2 C) 5√2 D) 20 E) 20√2
Answer + explanation
Answer: B (10√2).
Intercepts are easy:
- On the x-axis, y = 0 → x = 10 → A = (10, 0)
- On the y-axis, x = 0 → y = 10 → B = (0, 10)
Distance formula: √[(10 − 0)² + (0 − 10)²] = √[100 + 100] = √200 = 10√2.
Question 11 (Numeric Entry - Medium)
A solution is 30% salt. You have 10 liters of the solution. How many liters of water must be added to make the solution 20% salt?
Answer + explanation
Answer: 5
The key idea: adding water changes the total amount of liquid, but it does NOT change the amount of salt.
Salt amount now = 30% of 10 liters = 0.30 × 10 = 3 liters of salt.
After adding water, we want salt to be 20% of the total. So:
3 / (total volume) = 0.20
Total volume = 3 / 0.20 = 15 liters.
We started with 10 liters, so water added = 15 − 10 = 5.
Question 12 (Data Interpretation - Easy)
A student tracked practice questions completed:
| Student | Week 1 | Week 2 |
|---|---|---|
| A | 40 | 50 |
| B | 30 | 45 |
| C | 20 | 60 |
Which student had the greatest percent increase from Week 1 to Week 2?
A) A B) B C) C D) A and B (tie) E) B and C (tie)
Answer + explanation
Answer: C.
Percent increase is: (new − old) / old. Think “how big is the increase compared to where you started?”
- A: (50 − 40) / 40 = 10/40 = 25%
- B: (45 − 30) / 30 = 15/30 = 50%
- C: (60 − 20) / 20 = 40/20 = 200%
Student C has the biggest percent increase.
Where to find more official GRE math questions (and the best practice)
If you take only one thing away from this page, let it be this: ETS questions matter most. Other companies can be fine for repetition, but ETS is the company that actually writes the GRE.
Official ETS practice (start here)
- ETS GRE Mini Quiz (includes real GRE Quant questions)
- ETS POWERPREP + POWERPREP PLUS practice tests (two free tests + paid tests; pricing can vary by region)
- ETS official books: Official Guide + Official Quant Questions (best for realistic practice sets)
My favorite “buy it once and use it forever” resources
Note: The links below include Amazon affiliate links. If you purchase through them, I may earn a small commission at no extra cost to you. I recommend these because they’re genuinely useful, not because of the commission.
- ETS Super Power Pack (best for realism): buy it here
- GMAT Official Guides (great “cross-training” if you’re aiming high, just skip Data Sufficiency): buy it here
- Free GMAT practice tests: GMAT Official Starter Kit (free)
If you want a step-by-step plan for building up to harder GRE math questions, start here: Complete GRE Math Concepts Guide.
GRE math questions FAQ
What types of GRE math questions are there?
You’ll see Quantitative Comparison (Quantity A vs. Quantity B), multiple-choice (select one), multiple-choice (select one or more), numeric entry, and Data Interpretation sets (several questions tied to one chart/table).
How many GRE math (Quant) questions are on the test?
On the current GRE, Quant is two sections: 12 questions in 21 minutes, then 15 questions in 26 minutes. That’s 27 total Quant questions and 47 minutes total.
Is there calculus or trigonometry on GRE math?
No. GRE math is mostly arithmetic, algebra, geometry, and basic stats/probability. It’s a reasoning test disguised as math, not a calculus test.
Where can I find official GRE math questions?
ETS is your best source: the official books, POWERPREP (free), POWERPREP PLUS (paid), and the ETS GRE Mini Quiz. If your goal is a high Quant score, you eventually want a lot of ETS exposure.
What’s the fastest way to improve on GRE math questions?
First get accurate (untimed). Then get fast (timed). Build fundamentals until they’re automatic, drill by topic, move into official ETS questions, and review mistakes like it’s your job (because it’s literally your score).
When should I get a GRE math tutor?
If you’ve been practicing but your score won’t move, if timing is crushing you, if your fundamentals still take effort, or if you keep missing the same “logic-y” question types, tutoring can save you a ton of time.
Need help? (GRE math tutoring)
We offer one-on-one tutoring online and in San Diego. The first step is a free 15-minute call. Tell us your goal score, your timeline, and what you’ve tried so far - and we’ll tell you what to do next.
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That’s it! If you got the “hardest” question right on your own, please tell me - I will be impressed.
- Vince Kotchian