Five years ago, a father’s age was 5 times his son’s age; two years from now, it will be 3 times the son’s age. What is the ratio of their present ages?

Introduction / Context:
This age problem provides two separate time-based conditions. We convert each into an equation and solve the system to find present ages and their ratio.

Given Data / Assumptions:

• 5 years ago: Father was 5 times the son.
• In 2 years: Father will be 3 times the son.
• Find present age ratio Father : Son.

Concept / Approach:
Translate conditions into equations for present ages F and S. Solve the two linear equations to find F and S, then form the ratio F : S.

Step-by-Step Solution:
From 5 years ago: F − 5 = 5(S − 5) ⇒ F = 5S − 20. From 2 years ahead: F + 2 = 3(S + 2) ⇒ F = 3S + 4. Equate: 5S − 20 = 3S + 4. 2S = 24 ⇒ S = 12; F = 3*12 + 4 = 40. Present ratio = 40 : 12 = 10 : 3.

Verification / Alternative check:
5 years ago: 35 and 7 (35 = 5*7). In 2 years: 42 and 14 (42 = 3*14). Both conditions hold.

Why Other Options Are Wrong:
5 : 2, 5 : 3, 11 : 5 do not match the derived 40 : 12 ratio.

Common Pitfalls:
Sign errors (adding instead of subtracting years) or equating the wrong expressions. Always write both equations clearly before solving.

Final Answer:
10 : 3