About Data Sufficiency
A data sufficiency question gives a question together with two statements of information, and asks not for the answer but for whether the statements are enough to answer it. The four options describe which statement, or which pair, is sufficient, so each statement has to be tested on its own before the two are tested together.
What you need to understand
- Test each statement alone first, then the two together. Starting with the pair hides which statement was doing the work.
- For a ratio question, information that fixes one quantity is not enough, because one number does not give a ratio.
- One linear statement about two unknowns rarely fixes either of them, while two independent statements usually fix both.
- Two statements that say the same thing are worth no more than one of them, however many there are.
- A statement that fixes the ratio on its own settles the question, whatever the other statement then adds.
- The fourth option, that even together the statements are not enough, is a real answer rather than a failure to solve.
How to work through these questions
- Read the question and write down exactly what is being asked for.
- Cover statement II and ask whether statement I alone answers the question.
- Cover statement I and ask the same of statement II.
- Only if neither works on its own, put them together and ask again.
- Watch for one statement restating the other, since together they then add nothing.
- Match the pattern you found to one option, and check it against the other three before committing.
Mistakes that cost marks
- Solving the question and then forgetting to test each statement on its own.
- Assuming two statements must be better than one, when two versions of the same fact are worth one fact.
- Treating information about one quantity as though it fixed the other, or fixed their ratio.
- Reading statement II while testing statement I, which makes statement I look more useful than it is.
- Overlooking the fourth option because the question feels as though it ought to be answerable.
- Doing more arithmetic than the question needs. Sufficiency is about what is determined, not about the value itself.
Worked example
A and B are two positive integers. What is the ratio of A to B? Statement I: The sum of A and B is 30. Statement II: A is 6 more than B.
- Statement I alone gives A + B = 30. That allows 1 : 29, 2 : 28 and very many others, so it does not fix the ratio. Statement I is not sufficient.
- Statement II alone gives A - B = 6. That allows 7 : 1, 8 : 2 and many others, so it is not sufficient either.
- Together they give A + B = 30 and A - B = 6, so A = 18 and B = 12, and the ratio is 3 : 2.
- Neither statement alone fixes the ratio, but the two together leave only one possibility, so the third option is the one that applies.
Answer: Both statements together are sufficient, but neither statement alone is
Practice questions with answers
A few Data Sufficiency questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
A and B are two positive integers. What is the ratio of A to B?
Statement I: Twice B is 32.
Statement II: B is 16.
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A
Statement II alone is sufficient, but statement I alone is not
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B
Both statements together are sufficient, but neither statement alone is
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C
Statement I alone is sufficient, but statement II alone is not
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D
Neither statement alone nor both statements together are sufficient
Answer: Option D — with explanation
Only the ratio of A to B is asked for, so information that fixes one of them is not enough. Every ratio the statements allow is still consistent with them. Possible ratios include 1 : 16, 1 : 8, 3 : 16, 1 : 4, so the ratio is not determined.
Common mistakes
- treated information that fixes one quantity as though it fixed their ratio: Statement II alone is sufficient, but statement I alone is not
- treated information that fixes one quantity as though it fixed their ratio: Both statements together are sufficient, but neither statement alone is
- treated information that fixes one quantity as though it fixed their ratio: Statement I alone is sufficient, but statement II alone is not
Question 2
A and B are two positive integers. What is the ratio of A to B?
Statement I: Twice A added to B gives 81.
Statement II: A and B are in the ratio 4 : 1.
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A
Both statements together are sufficient, but neither statement alone is
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B
Statement II alone is sufficient, but statement I alone is not
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C
Statement I alone is sufficient, but statement II alone is not
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D
Neither statement alone nor both statements together are sufficient
Answer: Option B — with explanation
Only the ratio of A to B is asked for, so information that fixes one of them is not enough. Statement II on its own already fixes the ratio.
Common mistakes
- treated information that fixes one quantity as though it fixed their ratio: Neither statement alone nor both statements together are sufficient
- treated information that fixes one quantity as though it fixed their ratio: Statement I alone is sufficient, but statement II alone is not
- treated information that fixes one quantity as though it fixed their ratio: Both statements together are sufficient, but neither statement alone is
Question 3
A and B are two positive integers. What is the ratio of A to B?
Statement I: Twice the sum of A and B is 70.
Statement II: Twice B is 42.
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A
Neither statement alone nor both statements together are sufficient
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B
Statement I alone is sufficient, but statement II alone is not
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C
Both statements together are sufficient, but neither statement alone is
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D
Statement II alone is sufficient, but statement I alone is not
Answer: Option C — with explanation
Only the ratio of A to B is asked for, so information that fixes one of them is not enough. Neither statement alone fixes the ratio, because each allows many, but together they allow only one.
Common mistakes
- treated information that fixes one quantity as though it fixed their ratio: Neither statement alone nor both statements together are sufficient
- treated information that fixes one quantity as though it fixed their ratio: Statement II alone is sufficient, but statement I alone is not
- treated information that fixes one quantity as though it fixed their ratio: Statement I alone is sufficient, but statement II alone is not
Question 4
A and B are two positive integers. What is the ratio of A to B?
Statement I: A is 18 more than B.
Statement II: B is 19.
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A
Statement I alone is sufficient, but statement II alone is not
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B
Both statements together are sufficient, but neither statement alone is
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C
Neither statement alone nor both statements together are sufficient
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D
Statement II alone is sufficient, but statement I alone is not
Answer: Option B — with explanation
Only the ratio of A to B is asked for, so information that fixes one of them is not enough. Neither statement alone fixes the ratio, because each allows many, but together they allow only one.
Common mistakes
- treated information that fixes one quantity as though it fixed their ratio: Neither statement alone nor both statements together are sufficient
- treated information that fixes one quantity as though it fixed their ratio: Statement I alone is sufficient, but statement II alone is not
- treated information that fixes one quantity as though it fixed their ratio: Statement II alone is sufficient, but statement I alone is not
Question 5
A and B are two positive integers. What is the ratio of A to B?
Statement I: B is 11.
Statement II: A is 30 more than B.
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A
Both statements together are sufficient, but neither statement alone is
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B
Statement II alone is sufficient, but statement I alone is not
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C
Statement I alone is sufficient, but statement II alone is not
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D
Neither statement alone nor both statements together are sufficient
Answer: Option A — with explanation
Only the ratio of A to B is asked for, so information that fixes one of them is not enough. Neither statement alone fixes the ratio, because each allows many, but together they allow only one.
Common mistakes
- treated information that fixes one quantity as though it fixed their ratio: Statement II alone is sufficient, but statement I alone is not
- treated information that fixes one quantity as though it fixed their ratio: Statement I alone is sufficient, but statement II alone is not
- treated information that fixes one quantity as though it fixed their ratio: Neither statement alone nor both statements together are sufficient
Frequently asked questions
Should I work out the answer to the question?
Only as far as you need to. The question asks which statements are sufficient, so the arithmetic matters only where it settles whether the information determines the quantity.
Should I test the statements together first?
No. Test each on its own first, covering the other. Testing them together first hides which statement was carrying the information.
What if both statements are sufficient on their own?
Then the question cannot be asked in this form, because the four standard options cannot express it. A fair question is written so that at most one statement is sufficient alone.
How do I spot statements that add nothing together?
Look for one statement restating the other, perhaps with both sides doubled or the same relationship worded differently. Two versions of one fact are worth one fact.
Is the fourth option ever the right answer?
Yes. Sometimes even the two statements together leave the quantity undetermined, and recognising that is the whole point of the question.
