2.5 Data Sufficiency in Quantitative Aptitude

2.5 Data Sufficiency in Quantitative Aptitude

Data Sufficiency (DS) tests analytical reasoning, mathematical logic, and comprehension of constraints. Unlike standard Quantitative Aptitude questions where candidates must calculate a complete numerical answer, Data Sufficiency asks a fundamental meta-question: "Is the provided information sufficient to find a unique answer?"

In SBI PO Mains, Data Sufficiency carries heavy weightage (5 questions) and often features complex 3-statement formats across topics like Time & Work, Mixture Alligation, Mensuration, and Probability.

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The Standard 5-Option Structure for 2-Statement DS

Every 2-statement Data Sufficiency question presents a main question stem followed by Statement (I) and Statement (II). The standard options are:

1. Statement (I) ALONE is sufficient, but Statement (II) ALONE is not sufficient.
2. Statement (II) ALONE is sufficient, but Statement (I) ALONE is not sufficient.
3. EITHER Statement (I) ALONE or Statement (II) ALONE is sufficient.
4. BOTH Statement (I) and Statement (II) TOGETHER are necessary, but neither alone is sufficient.
5. Statements (I) and (II) TOGETHER are NOT sufficient (Additional data required).

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The 4-Step Elimination Algorithm (Flowchart Method)

Never evaluate both statements together at the beginning. Follow this strict sequential evaluation algorithm to eliminate options systematically:

[Start: Analyze Question Stem]
       │
       ▼
[Step 1: Test Statement I Alone]
  ├── SUFFICIENT ──► Eliminate Options 2, 4, 5 (Keep 1 or 3)
  └── INSUFFICIENT ─► Eliminate Options 1, 3   (Keep 2, 4, 5)
       │
       ▼
[Step 2: Test Statement II Alone (Cover Statement I)]
  ├── If Step 1 was SUFFICIENT:
  │     ├── Statement II SUFFICIENT   ──► Choice 3 (Either Alone)
  │     └── Statement II INSUFFICIENT ──► Choice 1 (Statement I Alone)
  │
  └── If Step 1 was INSUFFICIENT:
        ├── Statement II SUFFICIENT   ──► Choice 2 (Statement II Alone)
        └── Statement II INSUFFICIENT ──► Proceed to Step 3
       │
       ▼
[Step 3: Test Statements I + II Combined]
  ├── COMBINED SUFFICIENT   ──► Choice 4 (Both Together Necessary)
  └── COMBINED INSUFFICIENT ──► Choice 5 (Neither Sufficient / Data Inadequate)

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Categorization: "Value Questions" vs. "Yes/No Questions"

1. Value Questions ("What is the value of x?")

• A statement is Sufficient if and only if it produces exactly ONE unique numerical value.
• If a statement produces two possible answers (e.g., $x = 5$ or $x = -5$), it is INSUFFICIENT.

2. Yes/No Questions ("Is x an even integer?")

• A statement is Sufficient if it gives a definitive "ALWAYS YES" or a definitive "ALWAYS NO".
• If a statement yields "YES" under one case and "NO" under another, it is INSUFFICIENT.

[!IMPORTANT] A definitive "NO" answer to a Yes/No question means the statement IS SUFFICIENT! Candidates frequently confuse a logical "No" with data insufficiency.

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Worked Example 1: Arithmetic Value Question (Ratios & Ages)

Question Stem: What is the present age of Rahul?

• Statement I: Rahul is 4 years older than double his son's present age.
• Statement II: The ratio of the present ages of Rahul and his son is $5 : 2$.

Step-by-step Evaluation:

• Let Rahul's age be $R$ and son's age be $S$.
• Test Statement I alone: $R = 2S + 4$. This is a single linear equation in two unknown variables ($R, S$). Infinitely many solutions exist (e.g., $S=10 \implies R=24$; $S=15 \implies R=34$). INSUFFICIENT. (Eliminate Options 1 and 3).
• Test Statement II alone: $R / S = 5 / 2 \implies 2R = 5S$. This is another single ratio equation without absolute values. Infinitely many age pairs exist ($25:10, 50:20$). INSUFFICIENT. (Eliminate Option 2).
• Test Statements I + II Combined: We have a system of two independent linear equations:
  1. $R - 2S = 4$
  2. $2R - 5S = 0$ Substitute $R = 2.5S$ from eq (2) into eq (1): $2.5S - 2S = 4 \implies 0.5S = 4 \implies S = 8$. Then $R = 2.5 \times 8 = 20$. A single unique age ($R = 20$) is obtained. SUFFICIENT.
• Final Answer: BOTH Statement (I) and Statement (II) TOGETHER are necessary (Option 4).

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Worked Example 2: Yes/No Number System Question

Question Stem: Is the integer $n$ an even number?

• Statement I: $n^2 + 1$ is an odd integer.
• Statement II: $3n$ is an even integer.

Step-by-step Evaluation:

• Test Statement I alone: If $n^2 + 1 = \text{odd}$, then $n^2 = \text{odd} - 1 = \text{even}$. Since the square of an integer is even if and only if the integer itself is even, $n$ must be even.
  Yields a definitive "ALWAYS YES". SUFFICIENT.
• Test Statement II alone: If $3n = \text{even}$, since $3$ is an odd integer, the product can only be even if $n$ is even.
  Yields a definitive "ALWAYS YES". SUFFICIENT.
• Final Answer: EITHER Statement (I) ALONE or Statement (II) ALONE is sufficient (Option 3).

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Advanced 3-Statement Data Sufficiency (SBI PO Mains)

In SBI PO Mains, questions contain Statements I, II, and III. The options present combination patterns:

• Statements I and II together
• Statements II and III together
• Statements I and III together
• All three statements together

Worked Example 3: 3-Statement Work & Time Problem

Question Stem: In how many days can 10 men and 8 women complete a construction project?

• Statement I: 5 men can complete the project in 12 days.
• Statement II: 8 women can complete the project in 15 days.
• Statement III: 6 children can complete the project in 20 days.

Step-by-step Evaluation:

• Total Work required is constant ($W$).
• Statement I gives $1 \text{ man's daily work} = W / (5 \times 12) = W / 60$.
• Statement II gives $1 \text{ woman's daily work} = W / (8 \times 15) = W / 120$.
• Statement III gives $1 \text{ child's daily work} = W / (6 \times 20) = W / 120$.
• To answer the question stem, we need the combined daily rate of 10 men and 8 women.
• Combining Statement I and Statement II: $\text{Combined Rate} = 10 \times \left(\frac{W}{60}\right) + 8 \times \left(\frac{W}{120}\right) = \frac{W}{6} + \frac{W}{15} = \frac{7W}{30}$ $\text{Days required} = \frac{W}{7W/30} = \frac{30}{7} \text{ days}$
• Statement III (children) is entirely unnecessary.
• Final Answer: Statements (I) and (II) together are sufficient.

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Top 4 Candidate Traps in Data Sufficiency

1. Carryover Contamination Trap: Bringing data from Statement I into your evaluation of Statement II. Always mentally reset to zero before inspecting Statement II.
2. Over-Calculation Wasted Time: Calculating exact numerical values when data sufficiency is already established by linear independent equations.
3. Unstated Assumptions Trap: Assuming $x$ is positive or an integer. Unless specified, $x$ can be negative, zero, or a fraction.
4. Quadratic Multiplicity Trap: $x^2 = 25$ yields $x = 5$ or $x = -5$. This is INSUFFICIENT for value questions unless a constraint (e.g., $x > 0$) is provided.