Data sufficiency

Data Sufficiency – CMAT Study Notes

(Data Interpretation & Logical Reasoning – CMAT)

Goal:

1. Understand the Data Sufficiency (DS) format.
2. Master the systematic approach to solve DS questions.
3. Memorise key patterns and common pitfalls.

1. Simple Explanation

In a Data Sufficiency question you are given two independent statements (Statement 1 and Statement 2).
You must decide whether each statement, taken alone or together, is enough to answer the question (e.g., the value of (x), the sign of an expression, etc.).

Answer choices

• (A) Only Statement 1 is sufficient.
• (B) Only Statement 2 is sufficient.
• (C) Both statements together are sufficient.
• (D) Statements 1 and 2 together are not sufficient.
• (E) The question cannot be answered even with both statements.

Important: You cannot use the statements together in a way that would create a new statement not given.

2. Key Points & Rules

• Exact value: often requires solving an equation or inequality.
• Sign: check if the expression can be positive/negative/zero.
• Range: determine bounds. | | 8 | Avoid guessing | Each choice is mutually exclusive. | | 9 | Time-saving techniques |
• Symbolic algebra (e.g., let (x = y - 1)).
• Bounding arguments (e.g., if (x + y = 5) and both are non‑negative, then (0 ≤ x ≤ 5)). | |10 | Check for contradictions | If a statement implies a contradiction, it cannot be true. |

3. Systematic Approach (Step‑by‑Step)

1. Read the question carefully → Identify what exactly is being asked (value, sign, comparison, etc.).
2. Write down each statement → Label them as (1) & (2).
3. Analyze Statement 1 alone
   • Can you answer the question using only (1)?
   • If yes → Answer = (A) and stop.
4. Analyze Statement 2 alone
   • If (1) was not enough, check (2).
   • If (2) alone gives the answer → Answer = (B).
5. Combine statements
   • If neither alone is enough, combine them.
   • Solve the system or use the combined information.
   • If combined gives a definite answer → Answer = (C).
6. If still ambiguous → Answer = (D).
7. If the question cannot even be answered (e.g., the statement set is contradictory or incomplete) → Answer = (E).

4. Worked Example

Question:

Let (x) and (y) be real numbers satisfying:
Statement 1: (x + y = 10)
Statement 2: (x - y = 4)
Which of the following can be determined?
(i) The value of (x)
(ii) The sign (positive/negative) of (x)

Answer Choices:
A. Only Statement 1 is sufficient.
B. Only Statement 2 is sufficient.
C. Both statements together are sufficient.
D. Statements 1 and 2 together are not sufficient.
E. The question cannot be answered even with both statements.

Step 1: Identify what is asked

• (i) We need the exact value of (x).
• (ii) We need just whether (x) is positive, negative, or zero.

Step 2: Analyze Statement 1 alone

• Statement 1: (x + y = 10)
• This gives a linear relation but no unique solution for (x).
• Without any bound on (y), (x) can be any real number ≤ 10 (if (y ≥ 0)) or > 10 (if (y < 0)).
• Conclusion: Statement 1 alone is not sufficient for either part.

Step 3: Analyze Statement 2 alone

• Statement 2: (x - y = 4)
• Similar issue: infinite solutions.
• Conclusion: Statement 2 alone is not sufficient for either part.

Step 4: Combine statements

Solve the system:
[ \begin{cases} x + y = 10 \ x - y = 4 \end{cases} ]

Add the equations: (2x = 14 \Rightarrow x = 7).
Subtract: (2y = 6 \Rightarrow y = 3).

• (i) Value of (x): Exactly 7 → determined.
• (ii) Sign of (x): Positive (since (x = 7)) → determined.

Thus both statements together are sufficient for both parts.

Step 5: Choose answer

Answer