The sum of the present ages of a father and son is 45 years. Five years ago, the product of their ages was four times the father's age at that time. What are their present ages?
Verbal Reasoning
Problems on Ages
Difficulty: Medium
Choose an option
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A25 years, 10 years
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B36 years, 9 years
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C39 years, 6 years
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DNone of these
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E28 years, 17 years
Answer
Correct Answer: 36 years, 9 years
Explanation
Introduction / Context:This mixes a present sum condition with a past product condition. The structure simplifies after factoring the past product equation.
Given Data / Assumptions:
- Present ages: Father = f, Son = s
- f + s = 45
- Five years ago: (f − 5)(s − 5) = 4(f − 5)
Concept / Approach:Factor the past equation by taking (f − 5) common. This yields two possible cases; reject the infeasible one and use the sum to find both ages.
Step-by-Step Solution:
(f − 5)(s − 5) = 4(f − 5)(f − 5)[(s − 5) − 4] = 0 ⇒ (f − 5)(s − 9) = 0Either f = 5 (impossible for a father) or s = 9Thus s = 9 ⇒ f = 45 − 9 = 36Verification / Alternative check:
Five years ago: f = 31, s = 4 ⇒ product = 124; 4(f − 5) = 4×31 = 124 ✓Why Other Options Are Wrong:
- 25,10 and 39,6 do not satisfy both conditions.
- None of these and 28,17 are inconsistent with the factored constraint.
Common Pitfalls:Missing the factorization shortcut; choosing f = 5 despite being unrealistic; or misapplying the 5-year shift to only one age.
Final Answer:36 years, 9 years