Data Sufficiency

 🔹 What Is Data Sufficiency?

Data Sufficiency is a logical reasoning-based question type where you are required to determine whether the given data (in the form of statements) is sufficient to answer a question—not to actually solve it.

The primary objective is to assess analytical ability, logical reasoning, and the ability to evaluate information effectively.

You are not required to compute the final answer; you must simply determine whether the provided information is sufficient to answer the question.

🔹 Structure of a Data Sufficiency Question

Each question consists of:

• A main question or problem.

• Two (sometimes more) statements labeled (1) and (2).

• Multiple-choice options about whether the statements are sufficient to answer the question.

Standard answer options:

A. Statement (1) alone is sufficient, but statement (2) alone is not.
B. Statement (2) alone is sufficient, but statement (1) alone is not.
C. Both statements together are sufficient, but neither alone is sufficient.
D. Each statement alone is sufficient.
E. Statements (1) and (2) together are not sufficient.

🔹 Key Skills Tested

• Arithmetic (percentages, averages, ratios, etc.)

• Algebra (equations, variables)

• Number properties (divisibility, prime numbers)

• Geometry (angles, shapes, area)

• Logical analysis and decision-making

🔹 Solving Approach & Tips

✔ Understand what is being asked in the main question.

✔ Analyze each statement independently first:

• Does statement (1) alone give enough information?

• Does statement (2) alone give enough information?

✔ If neither is sufficient alone, check whether the statements combined are sufficient.

✔ Never assume extra information beyond what is stated.

✔ Do not solve the question fully—just assess whether it can be solved with the given data.

✔ Be cautious of traps involving:

• Hidden variables

• Ambiguous or insufficient conditions

• Multiple possible values

🔹 Example Questions

Example 1 (Arithmetic):
Q: What is the value of x?

Statement (1): 2x + 3 = 11
Statement (2): x² = 16

A. Statement (1) alone is sufficient.
B. Statement (2) alone is sufficient.
C. Both statements together are sufficient.
D. Each statement alone is sufficient.
E. Neither alone nor together is sufficient.

Analysis:

• (1) gives x = 4 clearly → Sufficient

• (2) gives x = ±4 → Not unique → Not sufficient
  Answer: A

Example 2 (Geometry):
Q: Is triangle ABC a right-angled triangle?

Statement (1): One angle measures 90°
Statement (2): Two angles are 45° and 45°

A. Statement (1) alone is sufficient.
B. Statement (2) alone is sufficient.
C. Both statements together are sufficient.
D. Each statement alone is sufficient.
E. Neither alone nor together is sufficient.

Analysis:

• (1) directly answers the question → Sufficient

• (2) implies third angle is 90° → Sufficient
  Answer: D

Example 3 (Number Properties):
Q: Is n divisible by 6?

Statement (1): n is divisible by 2
Statement (2): n is divisible by 3

A. Statement (1) alone is sufficient.
B. Statement (2) alone is sufficient.
C. Both statements together are sufficient.
D. Each statement alone is sufficient.
E. Neither alone nor together is sufficient.

Analysis:

• (1) alone: Divisible by 2 is not enough

• (2) alone: Divisible by 3 is not enough

• Together: Divisible by both 2 and 3 ⇒ Divisible by 6
  Answer: C

🔹 Common Pitfalls

❌ Solving the question instead of checking sufficiency.
❌ Assuming information not explicitly stated.
❌ Forgetting to evaluate each statement independently first.

❌ Misinterpreting “sufficiency” as “completeness.””

🔹 Practice Set Ideas

You can build a set of 10–20 practice questions covering:

• Simple equations (e.g., find x or y)

• Ratios and averages

• Ages and mixtures

• Number theory (even/odd, factors, multiples)

• Geometry (lengths, angles, areas)

📘 Data Sufficiency – Sample Questions with Options and Clear Answers

1. What is the value of x?

(1) 3x + 5 = 14
(2) x² – 9 = 0

A. Statement (1) alone is sufficient, but statement (2) alone is not.
B. Statement (2) alone is sufficient, but statement (1) alone is not.
C. Both statements together are sufficient, but neither alone is sufficient.
D. Each statement alone is sufficient.
E. Statements (1) and (2) together are not sufficient.

✔️ Answer: A
Explanation: (1) gives a unique x = 3 → Sufficient.
(2) gives x = ±3 → Not unique → Not sufficient.

2. What is the average of three numbers a, b, and c?

(1) a + b + c = 72
(2) a : b : c = 2 : 3 : 1

A. Statement (1) alone is sufficient.
B. Statement (2) alone is sufficient.
C. Both statements together are sufficient, but neither alone is sufficient.
D. Each statement alone is sufficient.
E. Statements (1) and (2) together are not sufficient.

✔️ Answer: C
Explanation: Either alone is not enough, but together you can find a, b, c using the ratio and total.

3. What is Ramesh’s current age?

(1) Five years ago, Ramesh was three times as old as his son.
(2) The sum of their current ages is 48.

A. Statement (1) alone is sufficient.
B. Statement (2) alone is sufficient.
C. Both statements together are sufficient, but neither alone is sufficient.
D. Each statement alone is sufficient.
E. Statements (1) and (2) together are not sufficient.

✔️ Answer: C
Explanation: Together give two equations with two variables → Sufficient.

4. Is n divisible by 12?

(1) n is divisible by 3 and 4.
(2) n is a multiple of 6.

A. Statement (1) alone is sufficient.
B. Statement (2) alone is sufficient.
C. Both statements together are sufficient.
D. Each statement alone is sufficient.
E. Statements together are not sufficient.

✔️ Answer: A
Explanation: LCM of 3 and 4 = 12 ⇒ (1) is enough.
(2) alone: Not necessarily divisible by 12.

5. Is n an even number?

(1) n² is even.
(2) n is divisible by 4.

A. Statement (1) alone is sufficient.
B. Statement (2) alone is sufficient.
C. Both statements together are sufficient.
D. Each statement alone is sufficient.
E. Statements together are not sufficient.

✔️ Answer: D
Explanation: Either statement guarantees n is even.

6. What is the area of triangle ABC?

(1) Base = 10 cm, height = 5 cm
(2) Sides are 10 cm, 10 cm, and 12 cm

A. Statement (1) alone is sufficient.
B. Statement (2) alone is sufficient.
C. Both together are sufficient.
D. Each statement alone is sufficient.
E. Neither alone nor together is sufficient.

✔️ Answer: D
Explanation: You can compute area with (1) using ½ × base × height, and (2) using Heron’s formula.

7. Is angle ABC a right angle?

(1) Sides of triangle ABC are 5 cm, 12 cm, and 13 cm.
(2) One angle in triangle ABC is 90°.

A. Statement (1) alone is sufficient.
B. Statement (2) alone is sufficient.
C. Both together are sufficient.
D. Each statement alone is sufficient.
E. Neither alone nor together is sufficient.

✔️ Answer: D
Explanation: (1) satisfies Pythagoras → right triangle.
(2) clearly mentions 90° angle → sufficient.

8. What is the value of y?

(1) 2y + 3 = 15
(2) y² – 16 = 0

A. Statement (1) alone is sufficient.
B. Statement (2) alone is sufficient.
C. Both together are sufficient.
D. Each statement alone is sufficient.
E. Neither alone nor together is sufficient.

✔️ Answer: A
Explanation:
(1) → 2y = 12 ⇒ y = 6 → Unique ⇒ Sufficient.
(2) → y² = 16 ⇒ y = ±4 ⇒ Two values ⇒ Not sufficient.

9. What is the average of four numbers?

(1) Their sum is 100.
(2) Each number is a multiple of 5.

A. Statement (1) alone is sufficient.
B. Statement (2) alone is sufficient.
C. Both statements together are sufficient.
D. Each statement alone is sufficient.
E. Neither alone nor together is sufficient.

✔️ Answer: A
Explanation:
(1) ⇒ average = 100 ÷ 4 = 25 ⇒ Sufficient.
(2) tells nothing about the actual values or sum ⇒ Not sufficient.

10. Is the integer n divisible by 5?

(1) n ends in 0.
(2) n is divisible by 10.

A. Statement (1) alone is sufficient.
B. Statement (2) alone is sufficient.
C. Both statements together are sufficient.
D. Each statement alone is sufficient.
E. Neither alone nor together is sufficient.

✔️ Answer: D
Explanation:
(1) If n ends in 0 ⇒ divisible by 10 ⇒ also divisible by 5 ⇒ Sufficient.
(2) If n divisible by 10 ⇒ divisible by 5 ⇒ Sufficient.