Data Sufficiency

Prerequisites

Tables and Bar and Line Graphs — DS questions often ask whether a growth rate or share can be found.
Averages and Ratio and Proportion — the arithmetic behind most DS statements.
Basic linear equations: one equation fixes one unknown; two independent equations fix two.

The Hook: The RTI Reply

Connect

A villager files an RTI (सूचना का अधिकार) application asking: “How much was spent on the new pond under the scheme in our panchayat?” The reply gives two documents: (I) the sanctioned amount, and (II) the percentage of the sanctioned amount released so far.

Before calculating anything, a sharp officer asks a different question: “Do these two documents together answer what was asked — and does either one alone?” Document I alone gives only the sanction; II alone gives only a percentage. Together, they give the amount released. (Whether “released” equals “spent” is yet another question — and that is the kind of gap DS questions test.)

The question this page answers: how do you decide, quickly and without solving fully, whether the information given is enough?

Why it matters for CGPSC: the Paper II syllabus names data sufficiency (आँकड़ों की पर्याप्तता) explicitly. These questions reward clear thinking more than calculation — a well-drilled candidate answers most of them in under a minute. The same discipline (“what exactly do I know?”) helps in Logical Reasoning and Decision Making.

The Standard Format

A DS question has a question stem and two statements, I and II. You must decide which statements are sufficient to answer the stem. The five standard choices are:

Choice	Meaning
I alone	Statement I alone is sufficient; II alone is not
II alone	Statement II alone is sufficient; I alone is not
Either	Each statement alone is sufficient
Both	Neither alone is sufficient, but I and II together are
Neither	Even I and II together are not sufficient

Four-option papers

A four-option MCQ paper cannot list all five choices in every question, so the options are usually written out in full in each question (and the set may change from question to question). Read the options every time — do not assume they appear in the same order as the table above.

The Decision Flow

Test each statement alone first, then together only if needed.

graph TD
    S[Read the stem: what exactly is asked?] --> I{Is I alone sufficient?}
    I -->|Yes| I2{Is II alone sufficient?}
    I2 -->|Yes| E[Either]
    I2 -->|No| A[I alone]
    I -->|No| J{Is II alone sufficient?}
    J -->|Yes| B[II alone]
    J -->|No| K{I and II together?}
    K -->|Yes| T[Both together]
    K -->|No| N[Neither]

You do not need the answer — only whether it can be found

Stop as soon as you know a unique answer exists. “Two independent linear equations in two unknowns” is enough; you need not solve them. This is where DS saves time.

Worked Examples by Answer Type

Both together

What is the value of \( x \)? I: \( 3x + 2y = 18 \). II: \( x + y = 7 \).

Each statement alone has two unknowns — not sufficient. Together they are two independent equations, so a unique \( x \) exists (it happens to be \( x = 4 \), \( y = 3 \)). Both together.

Either

Is the integer \( x \) even? I: \( 3x \) is even. II: \( x^2 \) is even.

I: 3 is odd, so \( 3x \) is even only if \( x \) is even — sufficient. II: the square of an odd number is odd, so \( x^2 \) even means \( x \) even — sufficient. Either statement alone.

I alone

The average of five numbers is 20. What is the fifth number? I: The average of the first four numbers is 18. II: The average of the last four numbers is 21.

Sum of all five = 100. I: first four sum to 72, so the fifth = 28 — sufficient. II: the last four sum to 84, which gives the first number (16), not the fifth — not sufficient. I alone.

II alone

What is the speed of the train? I: The train crosses an electric pole in 10 seconds. II: The train covers 180 km in 3 hours at a uniform speed.

I: crossing a pole means covering the train’s own length, which is not given — not sufficient. II: \( \frac{180}{3} = 60 \) km/h — sufficient. II alone.

Neither

What is Rohan’s age? I: Rohan is 4 years older than Meena. II: Meena’s age is one-third of her father’s age.

Together: Rohan = Meena + 4 and Meena = Father ÷ 3. Three unknowns, two relations — the father’s age is missing. Neither.

A DI-style DS question

What was the percentage increase in a society’s revenue in 2024? I: Revenue in 2024 was ₹75 lakh. II: Revenue in 2024 was ₹15 lakh more than in 2023.

I alone gives no earlier value; II alone gives the change but not the base. Together: 2023 revenue = ₹60 lakh, so growth = \( \frac{15}{60} = 25\% \). Both together.

“What” Questions vs “Is” Questions

Stem type	Sufficient when …
“What is …?” (value)	The statements fix one unique value
“Is …?” (yes/no)	The statements give a definite yes or a definite no

A definite “No” is sufficient. If a statement proves the answer is always “no”, that statement is sufficient.

Worked example

Is the integer \( n \) greater than 10? I: \( n > 12 \). II: \( n^2 > 100 \).

I: every such \( n \) is above 10 — definite yes, sufficient. II: \( n \) could be 11 (yes) or \( -11 \) (no) — not sufficient. I alone.

Three-Statement DS

Some questions give statements I, II and III and ask which are sufficient or which can be dispensed with. Test pairs.

Worked example

What is the area of a rectangle? I: Its perimeter is 28 cm. II: Its length is 8 cm. III: Its diagonal is 10 cm.

I + II: \( l + b = 14 \), \( l = 8 \Rightarrow b = 6 \). Area = 48. Sufficient.
II + III: \( b = \sqrt{10^2 - 8^2} = 6 \). Area = 48. Sufficient.
I + III: \( l + b = 14 \) and \( l^2 + b^2 = 100 \). Since \( (l + b)^2 = l^2 + b^2 + 2lb \), we get \( lb = \frac{196 - 100}{2} = 48 \). Sufficient — without finding \( l \) and \( b \) separately.

So any two of the three statements are sufficient; no single one is.

Rank and Arrangement DS

Worked example

In a class of 40 students, what is Sunita’s rank from the top? I: Her rank from the bottom is 17th. II: Ravi is 19th from the top and Sunita is 5 ranks below him.

I: rank from top = \( 40 - 17 + 1 = 24 \) — sufficient. II: \( 19 + 5 = 24 \) — sufficient. Either. (The class size in the stem is information you are allowed to use.)

Memory Tricks

Anchor these

“I, then II, then together.” Never look at the two statements together until each has been tested alone.
“Two doors after the first check”: if I alone works, the answer can only be I alone or Either; if it does not, only II alone, Both or Neither remain. One check removes half the choices.
“Unique or useless.” A “what” question needs exactly one value; two possible values = not sufficient.
“No is an answer.” A definite no is as sufficient as a definite yes.
“Count unknowns, count equations.” \( n \) independent linear equations fix \( n \) unknowns.

Don’t Confuse

This	vs	That
Either — each statement alone works		Both — only the two together work
A definite “No” (sufficient)		“Sometimes yes, sometimes no” (not sufficient)
Data given in the stem — always usable		Outside assumptions (integers, positive values) — usable only if stated
Finding the answer		Knowing an answer exists (all DS needs)

Common Mistakes

Trap 1: Using one statement while testing the other

When you test Statement II “alone”, forget everything in Statement I. Carrying information across is the commonest reason candidates mark “II alone” when the answer is “Both”.

Trap 2: Assuming numbers are positive or whole

\( n^2 > 100 \) does not mean \( n > 10 \); \( n \) may be \( -11 \). Unless the stem says “positive integer”, test negatives, zero and fractions.

Trap 3: Marking 'Both' when one statement alone was enough

If I alone is sufficient, the answer is never “Both”, even though I and II together obviously also work. Always test each statement alone first.

Trap 4: Using outside knowledge

Answer only from the stem and the statements. Real-world facts you happen to know (a real train’s speed, a real district’s population) do not count.

Practice

All names and figures below are hypothetical.

Level 1: Foundation

If Statement I alone is sufficient and Statement II alone is also sufficient, which answer choice is correct?
For an “Is …?” question, is a statement that always gives the answer “No” sufficient?
How many independent linear equations are needed to find two unknowns?

Answers

Either statement alone is sufficient.
Yes — a definite “No” answers the question.
Two.

Level 2: Prelims MCQs

In Q1, Q2, Q4 and Q5, decide which statement(s) are sufficient to answer the question.

Q1. What is the value of \( x \)? I. \( 3x + 2y = 18 \) II. \( x + y = 7 \)

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Statements I and II together are sufficient, but neither alone is (d) Statements I and II together are not sufficient

Answer: (c)

Each statement has two unknowns, so neither alone fixes \( x \). Together they form two independent equations: from II, \( y = 7 - x \); substituting, \( 3x + 14 - 2x = 18 \), so \( x = 4 \).

Q2. Is the integer \( n \) greater than 10? I. \( n > 12 \) II. \( n^2 > 100 \)

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Either statement alone is sufficient (d) Statements I and II together are needed

Answer: (a)

I gives a definite “yes”. II allows \( n = 11 \) (yes) and \( n = -11 \) (no), so it is not sufficient. Option (c) is the trap for candidates who forget negative integers.

Q3. Question: “How many girls are there in a class?” Statement I: The ratio of boys to girls is 3 : 2. Statement II: There are 10 more boys than girls.

Consider the following statements about this question:

Statement I alone is sufficient to answer it.
Statement II alone is not sufficient to answer it.
Statements I and II together show that there are 20 girls.

Which of the statements given above is/are correct?

(a) 1 and 2 only (b) 2 and 3 only (c) 3 only (d) 1, 2 and 3

Answer: (b) 2 and 3 only

A ratio alone never gives an actual number, so 1 is wrong. II alone gives a difference but not the total, so 2 is correct. Together: the difference of ratio parts is \( 3 - 2 = 1 \) part = 10 students, so girls = \( 2 \times 10 = 20 \); 3 is correct.

Q4. What is the speed of a train? I. The train crosses an electric pole in 10 seconds. II. The train covers 180 km in 3 hours at a uniform speed.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Either statement alone is sufficient (d) Statements I and II together are needed

Answer: (b)

I needs the train’s length, which is not given. II gives \( \frac{180}{3} = 60 \) km/h directly. Combining them would give the length (\( \frac{60 \times 1000}{3600}\times 10 \approx 167 \) m), but that is not what was asked.

Q5. In a class of 40 students, what is Sunita’s rank from the top? I. Her rank from the bottom is 17th. II. Ravi is 19th from the top, and Sunita is 5 ranks below Ravi.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Either statement alone is sufficient (d) Statements I and II together are needed

Answer: (c)

I: \( 40 - 17 + 1 = 24 \)th from the top. II: \( 19 + 5 = 24 \)th from the top. Each works alone.

Level 3: Speed Drill — Solve in 60 Seconds

For each, choose: I alone / II alone / Either / Both / Neither.

What is the value of \( a \)? I: \( a^2 = 49 \). II: \( a < 0 \).
Is the integer \( k \) divisible by 6? I: \( k \) is divisible by 3. II: \( k \) is divisible by 2.
What was the population of village X in 2025? I: It grew 10% from 2024 to 2025. II: Its population in 2023 was 4,800.
How old is Arun today? I: Arun is twice as old as his son. II: Six years ago, Arun was 30.
Is triangle ABC right-angled? I: Its angles are in the ratio 1 : 2 : 3. II: Its sides are 5 cm, 12 cm and 13 cm.
What is the profit percentage on an article? I: Its selling price is 1.125 times its cost price. II: The profit is ₹50.

Answers with the reason

Both — I gives \( a = 7 \) or \( -7 \); II alone gives no value; together \( a = -7 \).
Both — divisible by 2 and by 3 (co-prime) means divisible by 6; neither alone is enough.
Neither — the 2024 figure is missing; 2023 does not help.
II alone — \( 30 + 6 = 36 \). I alone gives only a relation.
Either — I: angles 30°, 60°, 90°. II: \( 5^2 + 12^2 = 25 + 144 = 169 = 13^2 \).
I alone — profit = 12.5% of cost. II alone lacks the cost price.

Quick Revision Table

Idea	Rule
Five choices	I alone / II alone / Either / Both / Neither
Order of testing	I alone → II alone → together
“What” question	Sufficient only if one unique value
“Is” question	Definite yes or definite no
Unknowns	\( n \) independent equations fix \( n \) unknowns
A ratio alone	Never gives actual numbers
Three statements	Test pairs; find the smallest sufficient set

Key Takeaways

DS asks whether an answer can be found, not what it is — stop calculating once you know.
Test each statement strictly alone before combining; never carry information across.
A “what” question needs a unique value; an “is” question needs a definite yes or no.
Do not assume positivity or whole numbers unless the stem says so.
In four-option papers, the options are written out — read them each time.

What’s Next

Data Interpretation — Quick Revision — the whole chapter on one page.
Logical Reasoning — statements, assumptions and conclusions use the same “what do I really know?” discipline.
General Mental Ability — rank, direction and arrangement puzzles that DS questions borrow from.
Tables — back to the start of the chapter for a timed revision.
Basic Numeracy — averages and ratios behind most DS statements.

Prerequisites

The Hook: The RTI Reply

The Standard Format

The Decision Flow

Worked Examples by Answer Type

“What” Questions vs “Is” Questions

Three-Statement DS

Rank and Arrangement DS

Memory Tricks

Don’t Confuse

Common Mistakes

Practice

Level 1: Foundation

Level 2: Prelims MCQs

Level 3: Speed Drill — Solve in 60 Seconds

Quick Revision Table

Key Takeaways

What’s Next

Keep going in this chapter