Introduction / Context:
This is a two-condition ratio problem across different times (one year ago and one year ahead). By setting present ages as variables and applying the conditions properly (subtracting or adding 1), we solve a pair of rational equations.
Given Data / Assumptions:
- One year ago: (S − 1) : (A − 1) = 4 : 3.
- One year later: (S + 1) : (A + 1) = 5 : 4.
- Find S + A (present sum).
Concept / Approach:
Convert ratios into equations and solve the system. Cross-multiplication clears denominators and yields linear equations in S and A.
Step-by-Step Solution:
(S − 1)/(A − 1) = 4/3 ⇒ 3(S − 1) = 4(A − 1) ⇒ 3S − 3 = 4A − 4 ⇒ 3S = 4A − 1. … (1)(S + 1)/(A + 1) = 5/4 ⇒ 4(S + 1) = 5(A + 1) ⇒ 4S + 4 = 5A + 5 ⇒ 4S = 5A + 1. … (2)From (2) and (1): (4S) − (3S) = (5A + 1) − (4A − 1) ⇒ S = A + 2.Substitute into (2): 4(A + 2) = 5A + 1 ⇒ 4A + 8 = 5A + 1 ⇒ A = 7 ⇒ S = 9.Present sum = 9 + 7 = 16.
Verification / Alternative check:
One year ago: 8 : 6 = 4 : 3; one year later: 10 : 8 = 5 : 4 ✔.
Why Other Options Are Wrong:
12 or 15 fail one of the time-shifted ratios; “Cannot be determined” is incorrect.
Common Pitfalls:
Using present ages directly in ratios for past/future without adjusting ±1.Swapping numerator/denominator during cross-multiplication.
Final Answer:
16 years
Discussion & Comments