Present ratio from past and future ratios: Ten years ago, Ram : Rahim = 1 : 3. Five years hence, Ram : Rahim = 2 : 3. Find their present age ratio.

Difficulty: Medium

1 : 2
3 : 5
3 : 4
2 : 5

Correct Answer: 3 : 5

Explanation:

Introduction / Context:
Two distinct time-based ratios can be reconciled to find a unique present ratio. Translate each ratio into a linear equation in present ages and solve the system carefully.

Given Data / Assumptions:

(R − 10) : (H − 10) = 1 : 3
(R + 5) : (H + 5) = 2 : 3

Concept / Approach:
Cross-multiply to form two linear equations in R (Ram) and H (Rahim). Solve to obtain their present ages; finally reduce to a ratio in simplest form.

Step-by-Step Solution:

3(R − 10) = 1(H − 10) ⇒ 3R − H = 203(R + 5) = 2(H + 5) ⇒ 3R − 2H = −5Subtract second from first: (3R − H) − (3R − 2H) = 20 − (−5) ⇒ H = 25Then 3R − 25 = 20 ⇒ 3R = 45 ⇒ R = 15Present ratio = 15 : 25 = 3 : 5

Verification / Alternative check:
Ten years ago: 5 : 15 = 1 : 3, and in five years: 20 : 30 = 2 : 3 ✓.

Why Other Options Are Wrong:
1 : 2 and 2 : 5 do not match the solved values; 3 : 4 is also inconsistent with both time-based constraints.

Common Pitfalls:
Arithmetic slips when moving constants; mixing up which person corresponds to which ratio term across times.

Present ratio from past and future ratios: Ten years ago, Ram : Rahim = 1 : 3. Five years hence, Ram : Rahim = 2 : 3. Find their present age ratio.

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