The ages of A and B are presently in the ratio of $5 : 6$ respectively. Six years hence, this ratio will become $6 : 7$ respectively. What was B's age 5 years ago?

### Concept & Strategy Like the previous problem, we leverage uniform ratio progression. If the parts of the ratio increment identically across a known time span, we can directly link the ratio unit to the years passed. ### Step-by-Step Solution * **Given:** * Present Ratio ($A : B$) = $5 : 6$ * Future Ratio ($+6$ years) = $6 : 7$ * **Calculation / Deduction:** * Let the present ages be $5x$ and $6x$. * After $6$ years, the equation becomes: * $$\frac{5x + 6}{6x + 6} = \frac{6}{7}$$ * Cross-multiply: * $$7(5x + 6) = 6(6x + 6)$$ * $$35x + 42 = 36x + 36$$ * $$36x - 35x = 42 - 36$$ * $$x = 6$$ * B's present age is $6x = 6(6) = 36$ years. * The question asks for B's age $5$ years ago. * B's age $5$ years ago = $36 - 5 = 31$ years. ### Exam Strategy & Shortcut Inspect the ratio change. A's ratio unit goes from $5 \rightarrow 6$ ($+1$ unit). B's ratio unit goes from $6 \rightarrow 7$ ($+1$ unit). This uniform $1$-unit change happens exactly over $6$ years. Thus, $1 \text{ unit} = 6 \text{ years}$. B's present age is $6 \text{ units} \times 6 = 36$ years. Subtract $5$ years to find the past age: $36 - 5 = 31$ years. ### Common Pitfall The most dangerous pitfall here is stopping at the present age ($36$) and marking option (d). Examiners specifically include intermediate calculation values in the options to catch students who don't read the final sentence of the question carefully ("What was B's age **5 years ago**?"). ### Final Answer **Therefore, the correct answer is 31 years.**