ConceptQuantitative

Data Sufficiency: Prove certainty without over-solving

Data Sufficiency asks whether the supplied information determines one answer to the question. It does not ask you to calculate every available value.

For a value question, “sufficient” means one unique value. For a yes/no question, it means the answer is always yes or always no across every allowed case.

The correct workflow

1. Define the target

Rewrite the stem as:

“Is there one value of x?”
“Is the answer always yes or always no?”
“Is this comparison fixed?”

2. Test each statement independently

Do not carry information from statement (1) into statement (2). Each receives its own clean world.

3. Search for two valid cases

If you can construct two cases satisfying the statement but producing different answers to the target, the statement is insufficient.

4. Combine only when required

If neither statement works alone, then test whether they work together.

Original example

Is integer n even?

(1) n is divisible by 4.

You do not need the value of n. Every integer divisible by 4 is even. Statement (1) fixes the yes/no answer as yes, so it is sufficient.

Now consider:

(2) n is divisible by 2 or by 3.

This permits n = 2, which is even, and n = 3, which is odd. The yes/no answer changes, so statement (2) is insufficient.

The two most expensive mistakes

Solving past sufficiency

Once a statement guarantees one answer, more algebra adds time and risk without adding information.

Accepting one successful example

Finding n = 2 and getting “yes” does not prove sufficiency. You must ask whether another permitted value changes the answer.

Data Sufficiency Very Hard · Weighted Average

An import shop sold a total of 60 bicycles, each of which was either a Roadster bicycle or a Commuter bicycle. Each Roadster sold for $r and each Commuter sold for $c. The average (arithmetic mean) selling price of the 60 bicycles was $150. How many Roadster bicycles were sold? (1) The Roadster bicycles sold for $20 less than the Commuter bicycles. (2) Either r=140 or c=140.

A Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D EACH statement ALONE is sufficient.
E Statements (1) and (2) TOGETHER are NOT sufficient.

Answer: C

Let n be the number of Roadster bicycles. Total revenue is 60×150=9,000. Statement (1) gives r=c−20, so revenue = n(c−20)+(60−n)c = 60c−20n. This leaves c and n variable, so (1) is insufficient. Statement (2) alone leaves the other price unknown. Combining: if r=140, then c=160 and 9600−20n=9000, so n=30. If c=140, then r=120 and 8400−20n=9000, so n=−30, impossible. Only the first branch is valid, so n=30.

Original GMAT-style question written for Premiergrad and checked by an independent blind solve. Not an official GMAT question.

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.

Common traps

Sufficiency vs. Solving: treating an exact value as the only acceptable result
Missing Case: failing to construct another valid world
Convenient-Case Generalization: testing one friendly number
Sign or Root Ambiguity: forgetting that a square may preserve two signs
Hidden Constraint: allowing a value the stem forbids

What mastery looks like

You can state the target before using the statements, keep the statements independent, stop once certainty is proven, and produce counterexamples deliberately.

Practice next with Hard GMAT Data Sufficiency Questions. Premiergrad’s explanations separate over-solving from false certainty because they require different corrections.

Practice Data Sufficiency →

Trap Necessary vs. Sufficient
A necessary condition must be present for a result to occur. A sufficient condition guarantees the result. The GMAT…

The logic underneath every sufficiency judgement.
Trap Hidden Constraint
A Hidden Constraint is a restriction implied by the wording, variable type, operation, or real world setting. If you…

What makes a statement narrower than it looks.
Practice Hard GMAT Data Sufficiency Questions
Hard Data Sufficiency is rarely hard because the arithmetic is long. It is hard because several cases remain possible…

Counterexample practice at the hard end.

Knowing it and recognising it are different

Most people who miss questions on this concept can explain it perfectly. What separates them is spotting it when the question does not name it. The diagnostic tells you which of the two you are.

Practise this concept Everything on this page stays free and open — no account needed to read it or to attempt the questions above.

Last reviewed 2026-09-17