A classic age riddle with a time-shift: A says to B: “I am twice as old as you were when I was as old as you are.” The sum of their present ages is 63. Find the difference in their ages.
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A27 years
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B12 years
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C9 years
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D6 years
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ENone of these
Answer
Correct Answer: 9 years
Explanation
Introduction / Context:This celebrated age riddle asks you to translate a nested time-shift statement into algebra. Let A be older than B. Define the time t years ago when A's age equalled B's current age; then interpret “you were” accordingly. Solving yields a fixed ratio of their present ages.
Given Data / Assumptions:
- Let present ages be A (older) and B.
- At time t = A − B years ago, A − t = B (A was as old as B is now).
- Statement: A = 2 * (B − t).
- Sum: A + B = 63.
Concept / Approach:From t = A − B, substitute into A = 2(B − t) to get a linear relation between A and B. Combine with the sum to find both ages and hence their difference.
Step-by-Step Solution:
A = 2(B − (A − B)) = 2(2B − A) = 4B − 2A3A = 4B ⇒ A : B = 4 : 3A + B = 63 ⇒ parts sum 7 ⇒ one part = 9 ⇒ A = 36, B = 27Difference = 36 − 27 = 9Verification / Alternative check:t = A − B = 9. Nine years ago, B was 18; A now is 36 = 2 × 18, satisfying the riddle.
Why Other Options Are Wrong:27/12/6 do not match the 4:3 ratio with sum 63.
Common Pitfalls:Setting t incorrectly or mixing up which age corresponds to “you were” vs “I was.” Define t explicitly to avoid confusion.
Final Answer:9 years