Problems on Ages — “One year ago, the ratio of Yamini’s and Gamini’s ages was 6 : 7. Four years hence, the ratio will be 7 : 8. Find Gamini’s present age.”

Introduction / Context:Ratio-of-ages questions typically give relationships at different times (past/future). Converting each statement into linear equations lets us solve for the present ages directly without guesswork.

Given Data / Assumptions:

• One year ago: Yamini : Gamini = 6 : 7.
• Four years hence: Yamini : Gamini = 7 : 8.
• Let present ages be y (Yamini) and g (Gamini) in years; ages are positive integers.

Concept / Approach:Translate each ratio into an equation using cross-multiplication. For a:b = m:n we write n*a = m*b. Build two equations at the two time points and solve the simultaneous linear system in y and g.

Step-by-Step Solution:

Verification / Alternative check:One year ago: 35/ (Yamini?) From (1): 7y = 6*36 + 1 = 217 ⇒ y = 31. So one year ago y=30, g=35 → 30:35 = 6:7 ✓; four years hence 35:40 = 7:8 ✓.

Why Other Options Are Wrong:30/32/34/40 do not satisfy both time-shifted ratios simultaneously when paired with a consistent y.

Common Pitfalls:Forgetting to shift both ages by the same amount; mixing present ratios with shifted values; not cross-multiplying correctly.

Final Answer:36 years