← Maths

Data Sufficiency

Competitive Mathematics · Data Interpretation

6 Part2 worked examples0 practice questions

A question comes with two or three statements. You do not have to find the answer — only decide whether the statements are enough to find it: one statement alone, both together, or not even both. There is no new topic here; you need the old knowledge of numbers, ages, percents, profit and loss, speed. The test is a single one: “do these statements leave exactly one answer to the question?” — exactly one answer = sufficient; more than one possible answer (or none) = not sufficient.

The test of sufficiency
1

How to test sufficiency

Statement I → Statement II → both together

The order of testing, the five options and the commonest confusions.

The three-step method

Test the statements separately

Solve every question in this order.

  1. Test Statement I alone — does it give exactly one answer? (Forget II completely.)
  2. Test Statement II alone — forgetting I. These two results settle the option almost completely.
  3. Test both together only if neither is sufficient alone. If together they give one answer, “both needed”; if not, “not even both”.
I aloneII aloneTogetherDecision
sufficientnot sufficient—I alone
not sufficientsufficient—II alone
sufficientsufficient—either alone
not sufficientnot sufficientsufficientboth needed
not sufficientnot sufficientnot sufficientmore data needed
  • “Sufficient” does not mean computing the value — it means the value is fixed to one. For a yes/no question like “is the number even?” sufficient = “always yes” or “always no”.
  • Take the statements as always true and not contradictory — if no number fits both, the question itself would be wrong; this does not happen in exams.
  • Use your own knowledge: a statement like “the number is divisible by 7” leaves limited possibilities; ask how many remain among the two-digit numbers.
Example 1 · Basic
What is the value of ? I. . II. .
Solution, step by step
  1. 1I alone: — is unknown, so many answers (not sufficient). II alone: nothing about (not sufficient).
  2. 2Together: — exactly one answer.
Answer Both together are needed
Example 2 · Basic
Is an even integer? I. is even. II. is a multiple of 3.
Solution, step by step
  1. 1I alone: a square is even only when the number is even, so is even — always “yes” (sufficient).
  2. 2II alone: (odd) and (even) are both possible — not sufficient.
Answer I alone is sufficient
2

Two-statement questions

Numbers, ages, profit and loss, trains

Four families of questions and how to handle each.

Numbers and ages

List the possibilities

Questions on two-digit numbers and ages: count the possibilities left by each statement.

🔒

Profit and loss, and trains

Count the unknowns in the formula

Where a formula has several variables — how many values are needed?

🔒
3

Three-statement questions

Which pair is enough?

Three statements: a particular pair, either of two, all three or none.

Testing the pairs

I+II, I+III, II+III

In a three-statement question every pair is tested separately.

🔒
4

Traps and fast tricks

Slips · tricks

Where marks go in data-sufficiency questions.

The commonest mistakes

Seven traps

The slips that cost marks.

🔒

Fast tricks

Count, do not solve

Habits for speed.

🔒
5

The practice ladder

Basic → standard → advanced

New questions of every family, in three levels.

Level 1 — Basic

One statement enough, either enough

Questions that test directly.

🔒

Level 2 — Standard

Both needed, or not even both

Exam-level questions.

🔒

Level 3 — Advanced

Three statements and tough pairs

Questions at the hard end of competitive exams.

🔒
6

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

🔒