Ages (two-time multiples; ratio asked 8 years ago): Sixteen years ago Sudha's grandfather was “eight times older” than her, and eight years from now he will be three times her age. What was the ratio Sudha : Grandfather eight years ago?
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A11 : 50
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B3 : 8
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C1 : 5
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D1 : 2
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E2 : 7
Answer
Correct Answer: 1 : 5
Explanation
Introduction / Context:Statements like “eight times older” are often intended as “grandfather's age equals 9 times Sudha's age” at that snapshot (older by 8 times Sudha's age). Interpreting English phrasing correctly is vital. Here, two time-separated multiplicative constraints uniquely determine the present ages and allow the requested ratio eight years ago.
Given Data / Assumptions:
- Let present ages be S (Sudha) and G (Grandfather).
- Sixteen years ago: G − 16 = 9(S − 16) (interpreting “8 times older” as 9 times as old).
- Eight years hence: G + 8 = 3(S + 8).
Concept / Approach:Solve the two linear equations for S and G. Then compute their ages eight years ago and simplify the ratio S : G at that time.
Step-by-Step Solution:
From future: G = 3S + 16. From past: G = 9S − 128. Equate: 9S − 128 = 3S + 16 ⇒ 6S = 144 ⇒ S = 24; then G = 3*24 + 16 = 88. Eight years ago: S = 16, G = 80 ⇒ ratio = 16 : 80 = 1 : 5.Verification / Alternative check:Check the 16-years-ago relation: 88 − 16 = 72 and 9(24 − 16) = 72 (✓). Check the 8-years-hence relation: 88 + 8 = 96 and 3(24 + 8) = 96 (✓).
Why Other Options Are Wrong:Other ratios do not simultaneously satisfy both time-based multiplicative conditions when reverse-engineered.
Common Pitfalls:Treating “8 times older” as “8 times as old,” which gives inconsistent (non-integer) ages here; or forming a ratio at the wrong time slice.
Final Answer:1 : 5