Ages (sum and past product condition): The sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages was four times the father's age at that time. Find their present ages.

Introduction / Context:
This item blends a present-time linear relation (the sum of ages) with a past-time multiplicative relation. Such problems are standard in algebraic age reasoning and test the ability to translate verbal statements into equations and apply factor reasoning efficiently.

Given Data / Assumptions:

• Father's present age = F years; Son's present age = S years.
• F + S = 45.
• Five years ago: (F − 5)(S − 5) = 4(F − 5).
• Ages are non-negative integers and F > S in typical father–son contexts.

Concept / Approach:
Use the product condition from five years ago. If (F − 5)(S − 5) = 4(F − 5), then either (F − 5) = 0 or (S − 5) = 4. The first case would give F = 5, which cannot fit a realistic father–son pair with sum 45. Hence use the second case to determine S, then use the sum to obtain F.

Step-by-Step Solution:

Verification / Alternative check:
Consider (F − 5) = 0 ⇒ F = 5. Then S = 40, inconsistent for a father–son pair and contradicts typical expectations, so discard. The chosen pair satisfies both constraints exactly.

Why Other Options Are Wrong:
Each alternative pair either violates the sum 45 or fails the past product relation when tested numerically.

Common Pitfalls:
Missing the factor trick and expanding fully (creating unnecessary algebra), or accepting the unrealistic F = 5 branch without checking plausibility.

Final Answer:
36 years, 9 years