Data Sufficiency

A Data Sufficiency (DS) question gives a problem and two (sometimes three) statements. You decide whether the statements are sufficient to answer the question — NOT what the answer is.

Standard 5-option format:
A. Statement I alone is sufficient; II alone is not.
B. Statement II alone is sufficient; I alone is not.
C. Both together are sufficient; neither alone is.
D. Each alone is sufficient.
E. Neither alone nor together is sufficient.

Solving steps:

1. Read the question and identify what you are asked.
2. Consider Statement I alone — can you answer the question uniquely?
3. Consider Statement II alone — can you answer uniquely? (Reset; ignore I.)
4. If neither alone works, consider both together.
5. Pick the option.

Key principle — uniqueness: an answer must be uniquely determined. If a statement allows two valid answers (e.g. x² = 16 gives x = ±4, two values), then it's NOT sufficient.

Common traps:

• Sufficient ≠ correct. You only need to know that some answer exists and is unique, not what it is.
• "Possible" values vs "definite" values: DS asks for definite.
• Statements may give redundant info — together they're still sufficient (if either alone is sufficient).

For numerical problems:

• Linear equation in one variable: sufficient.
• Linear equation in two variables: not sufficient (infinite solutions).
• Two linear equations in two variables: usually sufficient, unless they're equivalent (same line).
• Quadratic: may give two roots → check if one is rejected by context.

For relational/ranking problems (who is tallest, etc.):

• Each statement gives partial ordering. Combine to see if the order is fully determined.
• "X is taller than Y" + "Z is taller than X" gives Z > X > Y but doesn't fix Z vs Y of "W" if W is unmentioned.

Example 1:
Question: What is the value of x?
Statement I: x + 3 = 7.
Statement II: 2x = 8.
Analysis: I alone gives x = 4 ✓. II alone gives x = 4 ✓. Each alone is sufficient. Answer: D.

Example 2:
Question: Is x positive?
Statement I: x² > 9.
Statement II: x > −2.
Analysis: I: x² > 9 means x > 3 or x < −3. Could be positive or negative. NOT sufficient. II: x > −2 includes positive and small negative values. NOT sufficient alone. Both: x² > 9 AND x > −2 → x > 3 (positive). Sufficient together. Answer: C.

Example 3 (people in line):
Question: Who is third from the right among A, B, C, D, E in a line?
Statement I: A is two places to the right of E.
Statement II: B is at the rightmost end and C is to the left of A.
Analysis: I alone: positions of E and A are linked but not absolute. NOT sufficient. II alone: B is at position 5 (rightmost). C is left of A — but we don't know where. NOT sufficient. Together: B=5. E and A are 2 apart. C is left of A. Try A=4, E=2: then C is at 1, 2, or 3. D fills the rest. If we want a unique answer for the "3rd from right" (position 3), and the arrangement isn't uniquely fixed, NOT sufficient even together. Answer: E.

Example 4:
Question: How old is Ravi?
Statement I: Ravi is 5 years older than his sister.
Statement II: His sister will be 18 next year.
Analysis: I alone: only relative age. NOT sufficient. II alone: sister is 17 now, but Ravi's age not directly given. NOT sufficient. Together: Sister = 17, Ravi = 17 + 5 = 22. Sufficient. Answer: C.

Speed tip: don't compute the actual answer in DS — just confirm whether the statement would let you compute it. This saves substantial time.