UPSC CSE · CSAT · Numeracy and data interpretation · TOPIC LESSON
Data sufficiency
Decide whether information fixes the requested answer, using separate tests and explicit counterexamples.
UPSC Civil Services40 min study guide8 indexed sections
BY THE END, YOU CAN
Distinguish target-value uniqueness from a determined yes/no answer.
Test statements alone and together without carrying assumptions across.
Use domains, tables, geometry and order constraints to prove sufficiency or insufficiency.
BEFORE YOU START
Solve simple equations and recognize alternative feasible models.
Original educational lesson. Work through the examples before revealing the practice answers. This original common-core lesson supports its declared paper scope. Optional subjects, qualifying language papers and the interview are outside this resource.
LEARN THROUGH ORIGINAL PRACTICE
Original UPSC-style worked applications and practice
All teaching examples, cases, prompts and model answers are original. Official exam questions and answer keys are not reproduced. This common-core lesson does not imply complete historical paper coverage or an official score.
01 / UNDERSTAND & APPLY
Sufficient for which question?
Data are sufficient when they determine one answer to the requested target within the stated model. They need not determine every variable. Knowing x²=9 does not uniquely determine real x, but it does determine x². Knowing x is either 2 or 5 does not fix x, but does answer yes to “Is x positive?”
Read the stem’s domain, units and distinctness conditions first. A positive real variable behaves differently from an unrestricted real or integer variable. Do not add a familiar restriction because it would make the problem easier.
This lesson uses verbal verdicts: I alone sufficient, II alone sufficient, both individually sufficient, together only sufficient, or still insufficient. These are educational response descriptions, not a claim that any official paper uses a particular option-letter code.
ORIGINAL WORKED EXAMPLE
Same data, different target
The stem says x is real and asks for x. Statement I: x²=9. Statement II: x>0. Test each alone and then together. How would the verdict for I change if the target were x² instead?
I alone allows x=3 and x=−3, giving different requested values, so it is insufficient for x.
Reset to the original stem. II alone allows many positive values such as 1 and 2, so it is insufficient.
Together, the roots from I are restricted by II to x=3, so the value is unique.
If the target is x², statement I already states its unique value 9 even though x itself remains ambiguous.
Answer For target x, together only is sufficient and x=3. For target x², I alone is sufficient.
ORIGINAL PRACTICE · 1
The target is whether x is positive. A statement gives x=2 or x=5. Is it sufficient?
Reveal answer & reasoning
Yes, for this yes/no target.
Both permitted values give the same answer, yes. Multiple values do not establish insufficiency unless they produce different answers to the actual question.
02 / UNDERSTAND & APPLY
Reset before testing the second statement
Test I using only the original stem and I. Then discard I and test II using only the original stem and II. Carrying a restriction from the first test into the second can falsely make II appear sufficient. Keep temporary assumptions labelled.
A statement that directly gives the requested sum may be sufficient without giving each addend. Conversely, knowing one variable may be insufficient for a product or sum involving another unknown, unless a special value makes the target invariant.
Look for a proof of uniqueness or two valid counterexamples. You do not always need to calculate an exact value, but doing so can be the clearest way to demonstrate that only one is possible.
ORIGINAL WORKED EXAMPLE
A sum without its parts
Real a and b are given, and the target is a+b. Statement I says a=2 and b=5. Statement II says a+b=7. Test each independently.
I alone supplies both parts, so their sum is 2+5=7.
Reset. II alone directly supplies the target 7, even though many pairs could produce it.
There is no need to combine statements to recover a or b because their separate values are not the target.
Answer Both statements are individually sufficient for the sum 7.
ORIGINAL PRACTICE · 1
The stem says x is an integer from 4 to 6 inclusive and asks for x. I says x is even; II says x>5. Test each alone.
Reveal answer & reasoning
I is insufficient; II alone is sufficient with x=6.
I leaves 4 and 6. Reset to the original integer domain for II: only 6 exceeds 5. Do not use I’s evenness in that second test; it is unnecessary.
ORIGINAL PRACTICE · 2
The stem says x is positive. Statement I gives x²=16. Is I sufficient for x?
Reveal answer & reasoning
Yes, x=4.
The equation has real roots±4, but the original stem already excludes the negative root. This is legitimate use of a stem restriction, not borrowing from another statement.
03 / UNDERSTAND & APPLY
Combine only after the separate verdicts
If neither statement alone is sufficient, combine them and test both feasibility and uniqueness. A second statement can eliminate alternatives, leave them unchanged or contradict the first. Count neither repeated information nor inconsistency as a solved valid case.
A unique answer requires at least one feasible model. Contradictory equations may make every ordinary implication formally vacuous, but in an applied sufficiency task they should be reported as inconsistent data, not a usable uniquely determined quantity.
ORIGINAL WORKED EXAMPLE
A rectangle’s area
A rectangle has positive length and width; its area is the target. I: perimeter 20 units. II: length 6 units. Test separately and together.
I gives length+width=10. Positive dimensions 1 and 9 give area 9, while 4 and 6 give 24, so I is insufficient.
Reset. II alone leaves width unknown; widths 1 and 2 give areas 6 and 12, so II is insufficient.
Together, length 6 and length+width 10 imply width 4, hence area 24. Both dimensions are positive.
Answer Together only is sufficient; area 24 square units.
ORIGINAL PRACTICE · 1
The target is a, with nonzero real a,b. I gives a/b=2; II gives b=3. Are the statements sufficient together, and is I alone sufficient?
Reveal answer & reasoning
Together they give a=6; I alone is insufficient.
I permits a=2,b=1 or a=4,b=2. Combining b=3 with the ratio gives a=2×3=6. The nonzero condition makes the ratio defined.
04 / UNDERSTAND & APPLY
Two valid models can settle insufficiency
To prove insufficiency, produce two complete models satisfying all available information but giving different target answers. If the target is yes/no, one model must give yes and the other no. Two distinct numerical values that both satisfy the same property do not prove that property uncertain.
Check every stated condition in each witness. A negative length is not a valid counterexample when positivity is supplied. A table with counts that do not sum to the stem total is also invalid. Counterexamples should reveal the actual freedom left by the data, not violate the problem.
ORIGINAL PRACTICE · 1
The target is ab for real a,b. A statement says a=0 but gives no value for b. Is it sufficient?
Reveal answer & reasoning
Yes: ab=0.
Every permitted real b gives the same product. The data need not determine b because its variation does not change the target.
05 / UNDERSTAND & APPLY
A domain can remove or preserve alternatives
Integer restrictions can reduce a continuous interval to one value or a small finite set. Positivity can eliminate a negative root. Nonzero restrictions permit division but do not determine magnitude. Distinctness excludes equality cases only when explicitly stated.
Geometry also needs supplied conditions. A drawn rectangle can justify perpendicular sides if the shape is stated as a rectangle; a roughly square-looking sketch does not establish equal sides. For probability, equally likely outcomes and independence likewise require a model, not a visual impression.
When solving equations, retain every candidate allowed by the stem before applying additional statements. Deleting an inconvenient root too early can falsely suggest sufficiency.
ORIGINAL PRACTICE · 1
Statement I says x=2 and statement II says x=3, with the same real x. Is their combination a uniquely sufficient answer?
Reveal answer & reasoning
No; the combined data are inconsistent.
No real x satisfies both statements. A usable answer requires a feasible model, not merely the absence of two different feasible models.
06 / UNDERSTAND & APPLY
Tables, shapes and orders need complete constraints
A table problem may give totals rather than every cell, but every claimed cell must follow from the supplied constraints. To show a target cell is not determined, construct two complete valid tables, including all complementary counts. Do not leave hidden negative or impossible cells.
In geometric area questions, a base and its perpendicular height suffice for a triangle even if the other sides are not fixed. The word perpendicular matters; an arbitrary side length is not automatically the height.
Order problems use the same uniqueness test. Specify distinct people, one position per person and the exact target, such as first person or complete order. A constraint may fix the first person while leaving the rest interchangeable; that can still suffice for the narrower target.
ORIGINAL WORKED EXAMPLE
Two possible class tables
There are 40 students in two classes A and B, each student belonging to exactly one class and being counted as either girl or boy for this hypothetical table. The target is the number of girls in A. I: total girls in both classes=22. II: class A has 20 students. Test the combination.
Together the statements make B’s size 20 and total boys 18, but do not allocate girls between classes.
Model 1: A has 10 girls and 10 boys; B has 12 girls and 8 boys. Totals are 40 students and 22 girls, with A size 20.
Model 2: A has 12 girls and 8 boys; B has 10 girls and 10 boys. The same totals and class sizes hold.
The target girls-in-A differs 10 versus 12, so even the combined information is insufficient. Each statement alone is therefore also insufficient.
Answer Still insufficient. Both complete nonnegative tables satisfy the givens but give different target values.
ORIGINAL WORKED EXAMPLE
Who is first?
Three distinct people A,B,C occupy three row positions, one per position. The target is who is first. I: A is before B. II: C is after B. Test separately and together.
I alone permits ABC, with A first, and CAB, with C first: insufficient.
Reset. II alone permits ABC, with A first, and BAC, with B first: insufficient.
Together the constraints are A before B before C, forcing the complete order ABC.
Answer Together only is sufficient; A is first.
ORIGINAL PRACTICE · 1
A triangle’s area is the target. I gives a base of 10 units; II gives its perpendicular height as 6 units. Test each alone and together.
Reveal answer & reasoning
Neither alone is sufficient; together the area is 30 square units.
A base alone leaves the perpendicular height variable, and a height alone leaves the base variable. Together, area=(1/2)×10×6=30. Other side lengths need not be determined.
07 / UNDERSTAND & APPLY
More statements may mean no more information
A second equation can be a multiple or reformulation of the first. It then adds no independent restriction. Knowing the total of two parts does not ordinarily fix each part, even if the total is repeated in a different form.
Test dependence algebraically or by maintaining two distinct feasible examples. If both examples survive the added statement, it has not resolved the target. Conversely, a statement that looks broad may directly fix the requested aggregate, as the sum example shows.
ORIGINAL PRACTICE · 1
Real x,y satisfy x+y=8 and 2 x+2 y=16. Is x uniquely determined?
Reveal answer & reasoning
No.
The second equation doubles the first. For example,(x,y)=(1,7) and(3,5) satisfy both but give different x. The added equation is redundant for this target.
08 / UNDERSTAND & APPLY
Keep a compact proof record
Use a consistent record: stem domain and target; I-alone verdict and proof or witnesses; reset; II-alone verdict and proof or witnesses; then, if needed, joint verdict. State whether the result is a value, a determined property or an inconsistency.
Do not confuse difficulty with insufficiency. A long but uniquely solvable calculation is sufficient; an easy-looking equation with several permitted roots may not be. Nor should you solve every variable merely because it is available: stop once the target is uniquely established.
The strongest explanations are short and checkable. A pair of valid countermodels can settle a claim of insufficiency more clearly than a vague statement that “more data are needed”. A sufficiency proof should identify the condition that removes the last alternative.
KEEP THE BIG IDEAS
Quick revision
Sufficiency is relative to the requested target.
Reset to the stem before testing each statement alone.
Use two valid models with different target answers to prove insufficiency.
Domains and explicit geometric conditions can change the verdict.
Redundant information adds no constraint; inconsistent data are not a solved feasible case.
The teaching notes, passages, worked explanations and practice are original. Official references identify the assessment framework and skill scope; they do not endorse this lesson or license official question reproduction.
The 2026 official notice was checked on 12 October 2026 for the broad skill scope. The public landing page did not respond reliably during that check. All teaching, passages, scenarios and solutions here are original educational material; no official questions, answer keys or future-topic guarantees are supplied.