PracticeQuantitative
Hard GMAT Data Sufficiency questions
Hard Data Sufficiency is rarely hard because the arithmetic is long. It is hard because several cases remain possible, a statement looks more informative than it is, or the solver computes past the point of certainty. This set trains those decisions.
What this set trains
- Finding the counterexample — the second case that survives a statement and destroys sufficiency
- Sign and root ambiguity, where an even power or a square root quietly doubles the candidate set
- Number-property constraints that narrow a statement far more than it first appears
- Stopping at certainty rather than solving — a statement is sufficient when the answer is fixed, not when you have the number in hand
- Statement independence, and the habit of forgetting statement (1) while evaluating (2)
The recurring traps are Sufficiency vs. Solving and Hidden Constraint. A wrong answer of C when the answer is A almost always means a case was missed; a wrong answer of E when the answer is C usually means the statements were never genuinely combined.
A clothing store sold a total of 75 jackets, each of which was either a Wool jacket or a Down jacket. Each Wool jacket sold for $w and each Down jacket sold for $d. The average (arithmetic mean) selling price of the 75 jackets was $300. How many Wool jackets were sold? (1) The Wool jackets sold for $50 less than the Down jackets. (2) Either w=280 or d=280.
Answer: C
Let n be the number of Wool jackets. Total revenue is 75×300=22500. Statement (1) gives w=d−50, so revenue = n(d−50)+(75−n)d = 75d−50n. This leaves d and n variable. Statement (2) alone leaves the other price unknown. Combining: if w=280, then d=330 and 24750−50n=22500, so n=45. If d=280, then w=230 and 21000−50n=22500, so n=−30, impossible. Thus n=45 is uniquely determined.
Original GMAT-style question written for Premiergrad and checked by an independent blind solve. Not an official GMAT question.
The total cost B(q), in dollars, of baking q cakes per day is given by B(q)=7,200 + mq + nq^2, where m and n are constants. What is the value of n? (1) The total cost of baking 600 cakes in a day is $9,600. (2) The average cost per cake when baking 1,800 cakes in a day is $4 more than the average cost per cake when baking 600 cakes in a day.
Answer: B
Statement (1) gives 7,200 + 600m + 360,000n = 9,600, so 600m + 360,000n = 2,400, one equation in m and n; it is not sufficient. Statement (2) compares average costs: B(1,800)/1,800 - B(600)/600 = 4. Since B(1,800)/1,800 = 4 + m + 1,800n and B(600)/600 = 12 + m + 600n, the difference is (4 + m + 1,800n) - (12 + m + 600n) = 1,200n - 8. Setting this equal to 4 gives 1,200n = 12, so n = 0.01. The linear coefficient m cancels, so statement (2) alone is sufficient.
Original GMAT-style question written for Premiergrad and checked by an independent blind solve. Not an official GMAT question.
A distribution center had a positive number of identical crates that could be loaded onto pallets in groups of 21 crates per pallet with no crates left over. After a truck delivered 35 additional crates, the total number could be loaded in groups of 25 crates per pallet with no crates left over. How many crates were at the distribution center before the delivery? (1) Before the delivery, there were fewer than 900 crates. (2) After the delivery, there were fewer than 720 crates.
Answer: B
Let N be the original number. N = 21k and N + 35 ≡ 0 mod 25, so N ≡ 15 mod 25. Thus 21k ≡ 15 mod 25. The inverse of 21 mod 25 is 6, so k ≡ 90 ≡ 15 mod 25. Therefore N = 21(25m + 15) = 525m + 315. Possible values are 315, 840, 1365, ... . (1) N < 900 allows 315 and 840, so not sufficient. (2) N + 35 < 720, or N < 685, allows only 315; sufficient. The correct answer is B.
Original GMAT-style question written for Premiergrad and checked by an independent blind solve. Not an official GMAT question.
The positive integers m, n, and p all have the same remainder when divided by 11. What is the value of p? (1) m + n = p + 7 (2) 500 ≤ p ≤ 510
Answer: C
Let R be the common remainder modulo 11. Statement (1) gives m≡n≡p≡R (mod 11). Reducing m+n=p+7 modulo 11 gives R+R≡R+7, so R≡7 (mod 11); since 0≤R≤10, R=7. Thus p≡7 mod 11. This alone is not sufficient: for example, p=29 (with m=n=18) and p=40 (with m=18,n=29) both satisfy (1). Statement (2) alone gives only the interval 500 to 510, containing several integers. Combining, the only integer in [500,510] with remainder 7 when divided by 11 is 502, so p=502. Both statements together are sufficient.
Original GMAT-style question written for Premiergrad and checked by an independent blind solve. Not an official GMAT question.
A list consists of five integers: 0, 19, m, n, and p, where m, n, and p are prime numbers. What is the range of the list? (1) m + n + p < 27 (2) m < n < p < 25
Answer: A
Statement (1): The minimum possible sum of m and n is 2 + 2 = 4. If p were 23, the smallest prime greater than 19, the total would be at least 2 + 2 + 23 = 27, contradicting m + n + p < 27. Thus p ≤ 19. Therefore the maximum of the set is max(19, p) = 19 and the minimum is 0, so the range is 19. Statement (2): p < 25 permits p = 23, giving range 23, and p = 7, giving range 19; not sufficient.
Original GMAT-style question written for Premiergrad and checked by an independent blind solve. Not an official GMAT question.
Your result on the public set
— / 5
5 of 17 questions in this collection are public.
- A raw score out of 5 tells you very little on its own — the useful signal is the shape of the misses, and that is what the notes under each question describe.
- If two or more of your wrong answers were correct work on a different target, that is a pattern rather than a bad day.
Reading your own result
Data Sufficiency is unusually diagnostic, because the five letters are not interchangeable. The direction of a wrong answer tells you what happened.
| Your answer | Correct answer | What it usually means |
|---|---|---|
| C | A or B | You did not test the weaker-looking statement hard enough on its own |
| E | C | You evaluated the statements separately and never combined them |
| A or B | C | You accepted a statement that a second case defeats |
| A or B | E | A constraint you assumed was never stated |
| D | A or B | You applied the first statement's reasoning to the second |
A run of the first row is a confidence problem about single statements. A run of the third is a counterexample habit. They are different weeks of work.
What Lumen looks for
Lumen reads the answer you chose, not just whether it was wrong. On a question like this it separates three things that all look identical in a score: a concept you have not met, a reasoning step that went sideways, and a correct method pointed at the wrong target. Those need three different next weeks, and only the third one is fixed by re-reading the topic.
A single answer is weak evidence and Lumen says so. It becomes a diagnosis when the same shape shows up across several questions — which is the part a page like this cannot do for you.
Related
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Concept
Data Sufficiency
Data Sufficiency asks whether the supplied information determines one answer to the question. It does not ask you to…
Why certainty, not calculation, is the target.
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Trap
Hidden Constraint
A Hidden Constraint is a restriction implied by the wording, variable type, operation, or real world setting. If you…
The restriction that decides half of these.
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Strategy
How to Review a Wrong GMAT Question
Review is not complete when the explanation makes sense.
How to review a DS miss so the letter means something.
Continue the full set
The questions above are the public part of this collection. An account adds the rest of the set, your attempts saved, and a diagnosis of the answer you actually chose rather than a general explanation.
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