Question

Find the present ages of father and son. Statement (I): After two years the father's age is six times of his son's age.
Statement (II): Two years ago father's age was 26 times of son's age.

• A. Statement (I) alone is sufficient.
• B. Statement (II) alone is sufficient.
• C. Both statements (I) and (II) are sufficient.
• D. Neither statement is sufficient.

Hint:

Age problems with two time references (future/past) always require both statements to form a unique system of equations.

Answer 1

The Correct Option is C

Concept: Let \( F \) be the father's current age and \( S \) be the son's current age. We need to determine the specific values for \( F \) and \( S \).

Step 1: Translating Statement (I) "After two years": Father is \( F+2 \), son is \( S+2 \). Condition: \( F+2 = 6(S+2) \). Simplified: \( F+2 = 6S + 12 \implies F - 6S = 10 \). This is a linear equation with two variables. It is insufficient alone.

Step 2: Translating Statement (II) "Two years ago": Father was \( F-2 \), son was \( S-2 \). Condition: \( F-2 = 26(S-2) \). Simplified: \( F-2 = 26S - 52 \implies F - 26S = -50 \). This is also a linear equation with two variables. It is insufficient alone.

Step 3: Combining the statements We solve the system: (1) \( F - 6S = 10 \) (2) \( F - 26S = -50 \) Subtract (2) from (1): \( 20S = 60 \implies S = 3 \). Substitute \( S=3 \) into (1): \( F - 18 = 10 \implies F = 28 \). Both ages are determined. {Father 28, Son 3}