One of the most valuable habits a reasoning paper rewards is deciding whether a question can be answered at all. "Cannot be determined" is a real option, and it is the correct one whenever two different situations both satisfy everything you were told but give different answers.
The habit pays twice. It wins the data-sufficiency items directly, and it protects you from inventing information in ordinary questions — assuming a rate is constant, assuming a total is 100, assuming a figure is drawn to scale.
Ask: do the facts pin down exactly one answer? If two different answers are compatible with everything stated, the data is insufficient.
The test for sufficiency is the two-number test: try a simple value, then try a second, deliberately awkward value (zero, one, a negative, a fraction, or a very large number). If both fit the statements but give different answers, the data is insufficient — and you have proved it, rather than suspected it.
Data-sufficiency items give a question followed by two statements, marked (1) and (2), and five fixed answer choices:
| Choice | Meaning |
|---|---|
| A | Statement (1) alone is sufficient, but (2) alone is not |
| B | Statement (2) alone is sufficient, but (1) alone is not |
| C | Both statements together are sufficient, but neither alone is |
| D | Each statement alone is sufficient |
| E | Neither statement, even together, is sufficient |
Three rules of procedure make this format mechanical:
Statement (1) alone gives a relationship but not a value: x could be 6 or 100, depending on y. Statement (2) alone says nothing about x. Together they give x = 6, so the answer is C.
(1) alone gives an average with no count. (2) alone gives a total with no average. Together, count = total ÷ average = 1,200 ÷ 60 = 20. The answer is C. This pair is worth memorising as the classic example of two facts that are useless separately and sufficient together.
(1) alone is sufficient: every multiple of 6 is even, so the answer is yes whatever n is. (2) alone is not: 11 and 12 are both greater than 10 and have different parity. Since (1) alone suffices and (2) does not, the answer is A.
Two situations fit — A = 1 with B = 2, and A = 2 with B = 3 — and they give different answers. The correct choice is cannot be determined. Notice how the two-number test settled it in five seconds.
What is the value of x? (1) x + y = 10 (2) y = 4
How many students are in the class? (1) The average mark is 60. (2) The sum of all marks is 1,200.
Is n an even number? (1) n is a multiple of 6. (2) n is greater than 10.
When you assess whether statement (1) is sufficient, what may you use?
Without looking back: what is the two-number test, and which single pattern almost always signals insufficient information?
The answer: try two different values that both fit the facts and see whether the answers differ; and an average without a count (or a ratio without a total) is almost always insufficient.
Data is sufficient when every situation consistent with it gives the answer. The test tries two values that both fit and checks whether the answers differ. An average without a determines no total, and when assessing a statement you judge it .
Decide whether each question can be answered from what is given: (1) "A rectangle has a perimeter of 20. What is its area?" (2) "A rectangle has an area of 24. What is its perimeter?" (3) "A square has a perimeter of 20. What is its area?"
(1) Cannot be determined. Perimeter 20 means length + width = 10, and that admits 6 by 4 (area 24), 9 by 1 (area 9) and 5 by 5 when it is a square (area 25). The two-number test finds different areas for the same perimeter.
(2) Cannot be determined either: area 24 admits 6 by 4 (perimeter 20), 12 by 2 (perimeter 28) and 8 by 3 (perimeter 22). Again, different answers from the same fact.
(3) Determined. A square with perimeter 20 has side 5, so its area is 25 — one answer only, because the word "square" removes the freedom the earlier questions had.
The lesson generalises: your job in these questions is not to compute but to establish whether the constraints leave any freedom. When the wording removes the freedom, the data becomes sufficient, and often the computation is trivial.
Data sufficiency trains you to notice what is missing. The next lesson works with data that is fully present: tables, charts and graphs, where the task is to read the right number and resist the chart's attempts to mislead.