Age ratio back in time: Harsh is 40 years old and Sumit is 60 years old at present. How many years ago were their ages in the ratio 3 : 5?

Introduction / Context:
This problem tests reverse-age ratio reasoning. We are given two present ages and asked to find how many years back their ages had a specified ratio. The method is to write a linear equation in a single variable representing “x years ago” and use the definition of a ratio (cross-multiplication) to solve for x.

Given Data / Assumptions:

• Harsh's present age = 40 years.
• Sumit's present age = 60 years.
• Some x > 0 years ago, (40 − x) : (60 − x) = 3 : 5.
• Ages increase uniformly with time; x must be less than both current ages.

Concept / Approach:
For a ratio a : b, we use a/b. So, set (40 − x)/(60 − x) = 3/5 and solve. Cross-multiplication is the quickest consistent method for such linear age questions.

Step-by-Step Solution:

Verification / Alternative check:
10 years ago, ages would be 30 and 50, which indeed are in the ratio 3 : 5 (since 30/50 = 3/5).

Why Other Options Are Wrong:

• 15 yr, 20 yr, 37 yr produce age pairs that do not simplify to 3 : 5.
• None of these is invalid because 10 years works exactly.

Common Pitfalls:
Reversing the ratio order or forgetting that the same x is subtracted from both ages. Always check feasibility: x must be less than 40 and 60, which 10 is.

Final Answer:
10 yr