Sum now and product 5 years ago: The sum of father and son’s present ages is 45. Five years ago, the product of their ages equalled four times the father’s age at that time. Find their present ages (father, son).

Introduction / Context:
This mixes a present-time sum with a past-time product constraint. Often one factor cancels, simplifying the algebra. Careful translation into an equation avoids quadratic expansions in many such cases.

Given Data / Assumptions:

• Let father F, son S, with F + S = 45.
• Five years ago: (F − 5)(S − 5) = 4(F − 5).

Concept / Approach:
Factor out (F − 5). Either (F − 5) = 0 (impossible) or (S − 5) = 4. Use the viable factor to derive S and then F using the sum constraint.

Step-by-Step Solution:

Verification / Alternative check:
Five years ago: 31 and 4 ⇒ product 124; 4 × father then = 4 × 31 = 124, consistent.

Why Other Options Are Wrong:
Other pairs do not satisfy both the sum and the product condition.

Common Pitfalls:
Expanding to a quadratic unnecessarily; recognize the common factor (F − 5) to simplify.

Final Answer:
36, 9