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Data Sufficiency

CSAT · Logical Reasoning

Introduction

Data sufficiency asks not for the answer itself but whether the given statements are enough to find a unique answer. It rewards disciplined thinking: test each statement alone, then together, and stop as soon as you know.

Key concepts

Standard answer choices (as used in this course)

  1. Statement I alone is sufficient (II alone is not).
  2. Statement II alone is sufficient (I alone is not).
  3. Both statements together are needed (neither alone is enough).
  4. Even both together are not sufficient.

(Some papers add a fifth option, 'each statement alone is sufficient'. The questions here avoid that case.)

Method

  1. Read the question and decide what a sufficient answer looks like – a single value or a definite yes/no.
  2. Test statement I alone, ignoring II. List the possibilities if needed.
  3. Test statement II alone, forgetting I.
  4. Only if neither works alone, combine them.

What 'sufficient' means

Sufficient means exactly one possible answer. 'N is a multiple of 9' gives ten two-digit values – not sufficient. A yes/no question is sufficient if the answer is always yes or always no.

Do not solve fully

You only need to know whether a unique answer exists, not to compute it. Two independent linear equations in two unknowns are sufficient – you can stop there.

Flow diagram: Test I alone → Test II alone → If neither, combine → Count unique answers → Pick the key
Sufficient = exactly one possible answer.
📐 Formula box
  • Sufficient ⇔ exactly one answer
  • Test I alone → II alone → together
  • Two independent linear equations → unique solution

Solved examples

Example 1

Question: What is the two-digit number N?

Statement I: Its two digits are equal.

Statement II: It lies between 76 and 79.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

✅ Worked solution
  1. I alone allows: 11, 22, 33, 44, 55, 66, 77, 88 … (9 values).
  2. II alone allows: 77, 78 (2 values).
  3. Together: 77.
  4. So: both statements together are needed.

Answer: (c) Both statements together are needed

Example 2

Question: What is the two-digit number N?

Statement I: It lies between 65 and 73.

Statement II: It is odd.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

✅ Worked solution
  1. I alone allows: 66, 67, 68, 69, 70, 71, 72 (7 values).
  2. II alone allows: 11, 13, 15, 17, 19, 21, 23, 25 … (45 values).
  3. Together: 67, 69, 71.
  4. So: even both statements together are not sufficient.

Answer: (d) Even both statements together are not sufficient

Example 3

Question: What is the two-digit number N?

Statement I: It lies between 79 and 87.

Statement II: It is even.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

✅ Worked solution
  1. I alone allows: 80, 81, 82, 83, 84, 85, 86 (7 values).
  2. II alone allows: 10, 12, 14, 16, 18, 20, 22, 24 … (45 values).
  3. Together: 80, 82, 84, 86.
  4. So: even both statements together are not sufficient.

Answer: (d) Even both statements together are not sufficient

Example 4

Question: What is the two-digit number N?

Statement I: The product of its digits is 9.

Statement II: It lies between 87 and 97.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

✅ Worked solution
  1. I alone allows: 19, 33, 91 (3 values).
  2. II alone allows: 88, 89, 90, 91, 92, 93, 94, 95 … (9 values).
  3. Together: 91.
  4. So: both statements together are needed.

Answer: (c) Both statements together are needed

Example 5

Question: What is the two-digit number N?

Statement I: It is even.

Statement II: The sum of its digits is 4.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

✅ Worked solution
  1. I alone allows: 10, 12, 14, 16, 18, 20, 22, 24 … (45 values).
  2. II alone allows: 13, 22, 31, 40 (4 values).
  3. Together: 22, 40.
  4. So: even both statements together are not sufficient.

Answer: (d) Even both statements together are not sufficient

Example 6

Question: What is the two-digit number N?

Statement I: The product of its digits is 8.

Statement II: The units digit is 2 more than the tens digit.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

✅ Worked solution
  1. I alone allows: 18, 24, 42, 81 (4 values).
  2. II alone allows: 13, 24, 35, 46, 57, 68, 79 (7 values).
  3. Together: 24.
  4. So: both statements together are needed.

Answer: (c) Both statements together are needed

Example 7

Question: What is the two-digit number N?

Statement I: It is odd.

Statement II: It lies between 65 and 76.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

✅ Worked solution
  1. I alone allows: 11, 13, 15, 17, 19, 21, 23, 25 … (45 values).
  2. II alone allows: 66, 67, 68, 69, 70, 71, 72, 73 … (10 values).
  3. Together: 67, 69, 71, 73, 75.
  4. So: even both statements together are not sufficient.

Answer: (d) Even both statements together are not sufficient

Example 8

Question: What is the two-digit number N?

Statement I: The sum of its digits is 8.

Statement II: It lies between 59 and 68.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

✅ Worked solution
  1. I alone allows: 17, 26, 35, 44, 53, 62, 71, 80 (8 values).
  2. II alone allows: 60, 61, 62, 63, 64, 65, 66, 67 (8 values).
  3. Together: 62.
  4. So: both statements together are needed.

Answer: (c) Both statements together are needed

💡 Shortcut / exam tip
  • List the possibilities for small domains (two-digit numbers, days of the week).
  • Look for statements that are really the same information.
  • A definite 'no' is also sufficient for a yes/no question.
⚠️ Common traps
  • Carrying information from statement I into the test of statement II.
  • Using outside assumptions (e.g. assuming a number is positive).
  • Solving completely and wasting time.

Practice set

Attempt all 10 questions (about 90 seconds each), then check the worked solutions below.

Q1. Question: What is the two-digit number N?

Statement I: It lies between 28 and 33.

Statement II: It is a multiple of 3.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

Q2. Question: What is the two-digit number N?

Statement I: Its two digits are equal.

Statement II: It is a multiple of 11.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

Q3. Question: What is the two-digit number N?

Statement I: The product of its digits is 35.

Statement II: It lies between 55 and 59.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

Q4. Question: What is the two-digit number N?

Statement I: It lies between 58 and 63.

Statement II: It is a multiple of 3.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

Q5. Question: What is the two-digit number N?

Statement I: The tens digit is 2 more than the units digit.

Statement II: It lies between 93 and 101.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

Q6. Question: What is the two-digit number N?

Statement I: The product of its digits is 10.

Statement II: It is odd.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

Q7. Question: What is the two-digit number N?

Statement I: The sum of its digits is 12.

Statement II: It lies between 42 and 52.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

Q8. Question: What is the two-digit number N?

Statement I: It lies between 87 and 98.

Statement II: The product of its digits is 18.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

Q9. Question: What is the two-digit number N?

Statement I: The product of its digits is 64.

Statement II: The sum of its digits is 16.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

Q10. Question: What is the two-digit number N?

Statement I: Its two digits are equal.

Statement II: It is a multiple of 7.

(a) Statement I alone is sufficient (b) Statement II alone is sufficient (c) Both statements together are needed (d) Even both statements together are not sufficient

✅ Answers and worked solutions

Q1 – (c) Both statements together are needed. I alone allows: 29, 30, 31, 32 (4 values). II alone allows: 12, 15, 18, 21, 24, 27, 30, 33 … (30 values). Together: 30. So: both statements together are needed.

Q2 – (d) Even both statements together are not sufficient. I alone allows: 11, 22, 33, 44, 55, 66, 77, 88 … (9 values). II alone allows: 11, 22, 33, 44, 55, 66, 77, 88 … (9 values). Together: 11, 22, 33, 44, 55, 66, 77, 88 …. So: even both statements together are not sufficient.

Q3 – (c) Both statements together are needed. I alone allows: 57, 75 (2 values). II alone allows: 56, 57, 58 (3 values). Together: 57. So: both statements together are needed.

Q4 – (c) Both statements together are needed. I alone allows: 59, 60, 61, 62 (4 values). II alone allows: 12, 15, 18, 21, 24, 27, 30, 33 … (30 values). Together: 60. So: both statements together are needed.

Q5 – (c) Both statements together are needed. I alone allows: 20, 31, 42, 53, 64, 75, 86, 97 (8 values). II alone allows: 94, 95, 96, 97, 98, 99 (6 values). Together: 97. So: both statements together are needed.

Q6 – (c) Both statements together are needed. I alone allows: 25, 52 (2 values). II alone allows: 11, 13, 15, 17, 19, 21, 23, 25 … (45 values). Together: 25. So: both statements together are needed.

Q7 – (c) Both statements together are needed. I alone allows: 39, 48, 57, 66, 75, 84, 93 (7 values). II alone allows: 43, 44, 45, 46, 47, 48, 49, 50 … (9 values). Together: 48. So: both statements together are needed.

Q8 – (c) Both statements together are needed. I alone allows: 88, 89, 90, 91, 92, 93, 94, 95 … (10 values). II alone allows: 29, 36, 63, 92 (4 values). Together: 92. So: both statements together are needed.

Q9 – (a) Statement I alone is sufficient. I alone allows: 88 (1 value). II alone allows: 79, 88, 97 (3 values). Together: 88. So: statement i alone is sufficient.

Q10 – (c) Both statements together are needed. I alone allows: 11, 22, 33, 44, 55, 66, 77, 88 … (9 values). II alone allows: 14, 21, 28, 35, 42, 49, 56, 63 … (13 values). Together: 77. So: both statements together are needed.

⚡ Quick revision
  • Unique answer = sufficient
  • I alone, II alone, then together
  • Don't compute fully
  • No outside assumptions

Revision note

One-page note / PDF
  • Unique answer = sufficient
  • I alone, II alone, then together
  • Don't compute fully
  • No outside assumptions
Revision card summarising Data Sufficiency
One-page visual summary of the key points to revise.
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