1. Meaning and relevance to CSAT
A data sufficiency problem ordinarily contains a question followed by two or more statements. The task is to determine which statements, separately or together, permit a definite answer. The distinction from routine calculation is important: an equation may be solvable without giving the quantity asked for, while incomplete information about individual variables may still determine their sum, ratio or ordering.
Suppose the question asks for the average of five numbers. Knowing that their sum is 100 is sufficient, although the individual numbers remain unknown. Conversely, knowing four of the five numbers is generally insufficient unless another condition fixes the fifth. The relevant standard is not how much information is supplied, but whether that information resolves the precise target.
For UPSC aspirants, this topic develops an economy of reasoning. It draws on arithmetic, algebra, inequalities, geometry, sets, arrangements and comparisons, but the central operation is always the same: distinguish a logically warranted conclusion from an attractive assumption. Because CSAT is qualifying, accuracy and selection of manageable questions are more valuable than prolonged work on a single difficult item.
- A statement can be useful without being sufficient.
- A sufficient statement need not determine every unknown.
- An unequivocal negative answer is as sufficient as an unequivocal positive answer.
- Exam conditions reward a quick proof of sufficiency or a decisive counterexample to it.
2. The logical test: one target answer across all admissible cases
Begin by defining the target and the domain. Is the question asking for a number, a person, an ordering, or a yes-or-no judgment? Are the variables integers, positive integers, real numbers or quantities such as lengths? Retain all explicit conditions in the question stem while testing each statement. Do not introduce extra restrictions merely because they make the calculation easier.
A set of information is sufficient when every admissible case consistent with it gives the same answer to the target question. If two admissible cases give different answers, the information is insufficient. This counterexample test is especially efficient for inequalities, divisibility, ages, percentages and rankings. It replaces vague impressions with a clear logical standard.
For example, if the question asks whether x is positive, the condition x² = 9 is insufficient over the real numbers: x may be 3 or −3. The condition x > 2 is sufficient, even though it does not determine x exactly. Similarly, x < −2 is sufficient because the answer to the question is definitely no. A unique value is necessary only when the question itself asks for that value.
Statements are normally intended to be mutually consistent. If combining them produces a contradiction, first check arithmetic, interpretation and domain restrictions. An impossible set of cases is not a legitimate shortcut to a definite numerical answer; it may indicate an error in the reasoning or a defective question.
- Preserve stem conditions such as distinct, positive, integer, non-zero and exactly.
- Do not assume that an unspecified number is an integer or positive.
- In geometry, use stated properties rather than the apparent scale of a diagram.
- To demonstrate insufficiency, choose the simplest two cases that change the requested answer.
From question to sufficiency judgment
- 1. Identify the exact target and permitted domain.
- 2. Test Statement I with the stem alone.
- 3. Reset and test Statement II with the stem alone.
- 4. Combine statements if required.
- 5. Check alternative cases, roots and boundary values.
- 6. Match the logical outcome to the printed option.
3. A reliable solving sequence
First translate the question into a compact target, such as the value of x + y, whether a > b, or the identity of the person sitting at an end. Next test Statement I alone, using the question stem but ignoring Statement II. Record whether it is sufficient. Then reset completely and test Statement II alone. This reset prevents accidental borrowing of information.
Only after the independent tests should the statements be combined, where the answer choices require it. If neither statement alone is sufficient but their intersection determines the target, they are sufficient together. If both independently determine the target, each alone is sufficient. Finally map the result to the printed alternatives; never rely on a memorised letter code.
Consider a question asking for x + y. Statement I gives 2x + 2y = 18, while Statement II gives x − y = 3. Statement I alone is sufficient because the requested sum is 9. Statement II alone is insufficient. Solving for x and y using both statements would be unnecessary and could conceal the fact that the first statement already answers the question.
In a different example, a rectangle has perimeter 30 units and the question asks for its area. Knowing the perimeter alone allows many length–breadth combinations. Knowing only that its length is twice its breadth also allows many sizes. Together, 2(l + b) = 30 and l = 2b give b = 5 and l = 10, fixing the area at 50 square units.
- Write a brief status note: I sufficient or insufficient; II sufficient or insufficient.
- Stop calculating once uniqueness of the requested answer has been established.
- For arrangements, list only enough valid configurations to test whether the target changes.
- Re-read the alternatives before marking the answer, particularly when moving between practice sources.
| Statement I alone | Statement II alone | Together | Conclusion |
|---|---|---|---|
| Sufficient | Insufficient | Sufficient | I alone is sufficient |
| Insufficient | Sufficient | Sufficient | II alone is sufficient |
| Sufficient | Sufficient | Sufficient | Each alone is sufficient |
| Insufficient | Insufficient | Sufficient | Both are required |
| Insufficient | Insufficient | Insufficient | Even both together are insufficient |
4. Common mathematical and analytical patterns
In algebra, counting equations and unknowns is only a preliminary guide. Two equations may be dependent: x + y = 8 and 2x + 2y = 16 convey the same restriction. Conversely, a single equation may determine the requested expression even when it does not determine each variable. Non-linear equations require checking multiple roots and whether domain restrictions eliminate any of them.
For percentages and ratios, distinguish relative information from absolute quantities. A class with boys and girls in the ratio 3:2 need not have a fixed enrolment. An additional total of 40 fixes their numbers. A percentage increase without the starting quantity generally does not determine the absolute increase. Successive percentage changes must be applied to their respective bases.
In averages and mixtures, focus on totals and weights. Group averages cannot generally be combined into an overall average without group sizes or their ratio. In speed problems, distance, time and the relevant definition of average speed must be clear. Equal-distance and equal-time journeys produce different average-speed expressions.
In logical arrangements, distinguish uniqueness of the entire arrangement from uniqueness of the requested position. Several seating arrangements may exist while the same person occupies the middle seat in all of them. For set problems, check whether the universal set, intersections and exclusions are specified. Statements about some members do not justify conclusions about all members.
- Check zero, negative values, fractions and equality cases whenever the domain permits them.
- Treat distinctness as a stated condition, not an automatic assumption.
- Do not confuse a ratio with an absolute count or an average with a total.
- Check whether alternative configurations actually change the target rather than merely changing irrelevant details.
5. Error control and preparation strategy
The most frequent error is information contamination: using a fact from Statement I while supposedly testing Statement II alone. Other errors include assuming positive integers, overlooking a second root, mistaking two dependent equations for independent evidence, and rejecting a statement because it gives a definite no rather than a definite yes.
Use short mixed practice sets rather than practising only algebraic examples. Include inequalities, ratios, geometry, rankings and set-based reasoning. Maintain an error log with three headings: missed condition, incorrect independence test and unnecessary calculation. When reviewing an incorrect answer, write either a proof that all admissible cases agree or two valid cases that disagree.
Under examination conditions, a quick counterexample often saves more time than a full solution. Nevertheless, trying a few values and obtaining the same result is not a proof of sufficiency. Use substitution to disprove a claim, but use a general argument, a complete finite enumeration or a binding mathematical relation to establish it. The governing habit is disciplined restraint: conclude only what the supplied information warrants.
- Practise identifying the target before manipulating numbers.
- Time mixed sets, but prioritise correct independent testing over speed initially.
- Revisit mistakes caused by hidden assumptions, since these recur across topics.
- If a question remains uncertain after reasonable effort, move on and return later.
Real-world case studies
Election statistics: totals require a matching denominator
Election Commission of India statistical reports distinguish electors from votes polled. An absolute number of votes polled alone does not establish the turnout percentage; the corresponding electorate is also needed. Conversely, the electorate alone cannot determine turnout. This illustrates a combined-sufficiency structure, provided numerator and denominator refer to the same constituency, election and reporting definition.
Rainfall reporting: amount versus departure
India Meteorological Department reporting distinguishes recorded rainfall from its departure from normal. A rainfall total alone cannot determine percentage departure: the normal for the same location and period is also required. For example, 120 mm against a normal of 100 mm represents a 20 per cent excess. The case illustrates why a comparison needs both a measured quantity and a compatible reference.
Previous year questions
No source-linked PYQ is available here yet. Use the practice questions below.
Practice questions
Practice MCQ 1 · Easy
What is the arithmetic mean of five numbers? Statement I: Their sum is 125. Statement II: The largest of the five numbers is 35. Which of the following correctly describes the sufficiency of the statements?
- A. Statement I alone is sufficient, but Statement II alone is not sufficient.
- B. Statement II alone is sufficient, but Statement I alone is not sufficient.
- C. Both statements together are sufficient, but neither alone is sufficient.
- D. Even both statements together are not sufficient.
Practice MCQ 2 · Easy
For a real number x, is x positive? Statement I: x² = 16. Statement II: x < −1. Which of the following correctly describes the sufficiency of the statements?
- A. Statement I alone is sufficient, but Statement II alone is not sufficient.
- B. Statement II alone is sufficient, but Statement I alone is not sufficient.
- C. Both statements together are sufficient, but neither alone is sufficient.
- D. Even both statements together are not sufficient.
Practice MCQ 3 · Easy
What is the area of a rectangle? Statement I: Its perimeter is 36 cm. Statement II: Its length is twice its breadth. Which of the following correctly describes the sufficiency of the statements?
- A. Statement I alone is sufficient, but Statement II alone is not sufficient.
- B. Statement II alone is sufficient, but Statement I alone is not sufficient.
- C. Both statements together are sufficient, but neither alone is sufficient.
- D. Even both statements together are not sufficient.
Practice MCQ 4 · Easy
What is Arun's present age? Arun's and Bina's ages are positive integers measured in completed years. Statement I: Arun is 4 years older than Bina. Statement II: The sum of their present ages exceeds 30 years. Which of the following correctly describes the sufficiency of the statements?
- A. Statement I alone is sufficient, but Statement II alone is not sufficient.
- B. Statement II alone is sufficient, but Statement I alone is not sufficient.
- C. Both statements together are sufficient, but neither alone is sufficient.
- D. Even both statements together are not sufficient.
Practice MCQ 5 · Easy
A positive integer n is given. Is n divisible by 6? Statement I: n is a multiple of 12. Statement II: n is divisible by 18. Which of the following correctly describes the sufficiency of the statements?
- A. Statement I alone is sufficient, but Statement II alone is not sufficient.
- B. Each statement alone is sufficient.
- C. Both statements together are sufficient, but neither alone is sufficient.
- D. Even both statements together are not sufficient.
Practice MCQ 6 · Moderate
A vessel initially contains only milk and water, both in positive quantities. What is the initial total volume of the mixture? Statement I: The initial ratio of milk to water is 3 : 2. Statement II: Adding 10 litres of water, without removing anything, makes the quantities of milk and water equal. Which of the following correctly describes the sufficiency of the statements?
- A. Statement I alone is sufficient, but Statement II alone is not sufficient.
- B. Statement II alone is sufficient, but Statement I alone is not sufficient.
- C. Both statements together are sufficient, but neither alone is sufficient.
- D. Even both statements together are not sufficient.
Practice MCQ 7 · Moderate
Forty candidates are ranked by their scores, with no ties. What is Ritu's rank from the top? Statement I: The number of candidates ranked above Ritu is twice the number ranked below her. Statement II: At least 10 candidates are ranked below Ritu. Which of the following correctly describes the sufficiency of the statements?
- A. Statement I alone is sufficient, but Statement II alone is not sufficient.
- B. Statement II alone is sufficient, but Statement I alone is not sufficient.
- C. Both statements together are sufficient, but neither alone is sufficient.
- D. Even both statements together are not sufficient.
Practice MCQ 8 · Moderate
An article is sold after two successive discounts on its marked price. What is the overall discount percentage relative to the marked price? Statement I: The final selling price is ₹720. Statement II: The successive discounts are 20% and 10%, with the second applied to the price remaining after the first. Which of the following correctly describes the sufficiency of the statements?
- A. Statement I alone is sufficient, but Statement II alone is not sufficient.
- B. Statement II alone is sufficient, but Statement I alone is not sufficient.
- C. Both statements together are sufficient, but neither alone is sufficient.
- D. Even both statements together are not sufficient.
Practice MCQ 9 · Moderate
Five persons P, Q, R, S and T occupy five consecutive seats in a row, all facing north. Who occupies the middle seat? Statement I: P sits immediately to the left of Q. Statement II: R occupies the leftmost seat and S occupies the rightmost seat. Which of the following correctly describes the sufficiency of the statements?
- A. Statement I alone is sufficient, but Statement II alone is not sufficient.
- B. Statement II alone is sufficient, but Statement I alone is not sufficient.
- C. Both statements together are sufficient, but neither alone is sufficient.
- D. Even both statements together are not sufficient.
Practice MCQ 10 · Moderate
For positive real numbers a and b, is a + b at least 10? Statement I: ab = 25. Statement II: a² + b² = 100. Which of the following correctly describes the sufficiency of the statements?
- A. Statement I alone is sufficient, but Statement II alone is not sufficient.
- B. Statement II alone is sufficient, but Statement I alone is not sufficient.
- C. Both statements together are sufficient, but neither alone is sufficient.
- D. Each statement alone is sufficient.
Practice MCQ 11 · Difficult
A positive integer N is less than 2,000. What is N? Statement I: N leaves remainder 2 when divided by 7 and remainder 3 when divided by 5. Statement II: N is divisible by 9 and leaves remainder 2 when divided by 4. Which of the following correctly describes the sufficiency of the statements?
- A. Statement I alone is sufficient, but Statement II alone is not sufficient.
- B. Statement II alone is sufficient, but Statement I alone is not sufficient.
- C. Both statements together are sufficient, but neither alone is sufficient.
- D. Even both statements together are not sufficient.
Practice MCQ 12 · Difficult
Among 150 students, 80 study subject A, 75 study subject B and 65 study subject C. Students may study any number of these three subjects, including none. How many students study exactly one of the three subjects? Statement I: Exactly 20 students study none of the three subjects. Statement II: The sum of the numbers studying both A and B, both B and C, and both C and A is 110. Each of these three counts includes students studying all three subjects. Which of the following correctly describes the sufficiency of the statements?
- A. Even both statements together are not sufficient.
- B. Both statements together are sufficient, but neither alone is sufficient.
- C. Statement I alone is sufficient, but Statement II alone is not sufficient.
- D. Statement II alone is sufficient, but Statement I alone is not sufficient.
Practice MCQ 13 · Difficult
Five distinct positive integers are arranged as a < b < c < d < e. Their arithmetic mean is 18. What is their median c? Statement I: b = 12 and d = 24. Statement II: The difference between the largest and smallest integers is 36. Which of the following correctly describes the sufficiency of the statements?
- A. Statement I alone is sufficient, but Statement II alone is not sufficient.
- B. Statement II alone is sufficient, but Statement I alone is not sufficient.
- C. Both statements together are sufficient, but neither alone is sufficient.
- D. Even both statements together are not sufficient.
Practice MCQ 14 · Difficult
A trader buys two batches of rice having equal actual weights. The cost per kilogram of the second batch is 25% higher than that of the first. Both batches are sold completely, with no wastage. What is the trader's overall profit percentage on the total purchase cost? Statement I: For the first batch, the marked price per billed kilogram is 40% above its purchase cost per actual kilogram. The trader allows a 10% discount on this marked price but supplies only 900 grams for every kilogram billed. The second batch is sold using correct weights at a 20% loss on its own purchase cost. Statement II: The total sales revenue from the two batches is ₹4,800. Which of the following correctly describes the sufficiency of the statements?
- A. Statement I alone is sufficient, but Statement II alone is not sufficient.
- B. Statement II alone is sufficient, but Statement I alone is not sufficient.
- C. Both statements together are sufficient, but neither alone is sufficient.
- D. Even both statements together are not sufficient.
Practice MCQ 15 · Difficult
A bag contains only red and blue balls, with at least one ball of each colour. Two balls are drawn uniformly at random without replacement. How many blue balls are in the bag? Statement I: The probability that the two balls have different colours is 1/2. Statement II: The number of red balls exceeds the number of blue balls by 6. Which of the following correctly describes the sufficiency of the statements?
- A. Statement I alone is sufficient, but Statement II alone is not sufficient.
- B. Both statements together are sufficient, but neither alone is sufficient.
- C. Statement II alone is sufficient, but Statement I alone is not sufficient.
- D. Even both statements together are not sufficient.
Further reading
- UPSC Civil Services Examination notification: Preliminary Examination scheme, syllabus and negative-marking rules, upsc.gov.in.
- UPSC official previous question papers: Civil Services Preliminary Examination, General Studies Paper II, upsc.gov.in.
- NCERT Mathematics, Class X: Pair of Linear Equations in Two Variables; Quadratic Equations; Statistics.
- NCERT Mathematics, Class XI: Sets; Linear Inequalities.
- Election Commission of India: election statistical reports, eci.gov.in.
- India Meteorological Department: rainfall statistics and departure-from-normal reporting, mausam.imd.gov.in.
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