Introduction / Context:
Two ratios at different times determine two linear relations that can be solved to find the present ages. We translate carefully and solve.
Given Data / Assumptions:
- After 6 years: (Pradhan + 6) = (3/7)(Father + 6).
- 10 years ago: (Pradhan − 10) : (Father − 10) = 1 : 5.
Concept / Approach:
Let present ages be p and f. Convert both statements to equations and solve for p, f. Both must be nonnegative integers in common test contexts.
Step-by-Step Solution:
Past ratio ⇒ Father − 10 = 5(Pradhan − 10) ⇒ f = 5p − 40.Future relation ⇒ p + 6 = (3/7)(f + 6).Substitute f: p + 6 = (3/7)(5p − 40 + 6) = (3/7)(5p − 34).7p + 42 = 15p − 102 ⇒ 8p = 144 ⇒ p = 18.f = 5*18 − 40 = 50.
Verification / Alternative check:
After 6 years: p = 24, f = 56 ⇒ 24 is 3/7 of 56 ✓10 years ago: p = 8, f = 40 ⇒ 8:40 = 1:5 ✓
Why Other Options Are Wrong:
- 40 and 56 fail one of the two time-shift relations when substituted.
- Data is adequate because two independent conditions uniquely determine ages.
Common Pitfalls:
- Interchanging the ratio direction.
- Forgetting to add/subtract years on both sides of each relation.
Final Answer:
50 years
Discussion & Comments