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?

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:

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