The present ages of Amit and his father are in the ratio $2 : 5$ respectively. Four years hence, the ratio of their ages becomes $5 : 11$ respectively. What was the father's age five years ago?

Concept & Logic When given present and future ratios, establish an equation representing the timeline shift. The algebraic approach involves adding the time difference to the present age terms and setting them equal to the new ratio fraction. $$ \frac{2x + 4}{5x + 4} = \frac{5}{11} $$ Step-by-Step Solution * **Given:** The present age ratio of Amit and his father is $2 : 5$. * **Given:** Four years hence (in the future), the ratio becomes $5 : 11$. * **Calculation:** Let their present ages be $2x$ and $5x$. * Set up the equation for 4 years hence: $(2x + 4) / (5x + 4) = 5 / 11$. * Cross-multiply: $11(2x + 4) = 5(5x + 4)$. * Expand: $22x + 44 = 25x + 20$. * Isolate $x$: $25x - 22x = 44 - 20 \Rightarrow 3x = 24 \Rightarrow x = 8$. * The father's present age is $5x = 5(8) = 40$ years. * The father's age five years ago was $40 - 5 = 35$ years. Exam Strategy & Shortcut Use the **Age Difference Balancing Method**. Present ratio is $2 : 5$ (Difference = 3 parts). Future ratio (+4 years) is $5 : 11$ (Difference = 6 parts). Balance the differences by multiplying the present ratio by 2: $4 : 10$ (Difference = 6 parts). Compare the father's parts: Present (10 parts) vs. Future (11 parts). The difference is $11 - 10 = 1$ part, corresponding to the 4-year time shift. So, 1 part = 4 years. Father's present age = 10 parts $\times 4 = 40$ years. Father's age 5 years ago = $40 - 5 = 35$ years. Common Pitfall A standard trap is determining the father's present age (40) and assuming the problem is complete, but the prompt clearly requires stepping backward five years. Always highlight the specific timeline requested before finalizing your choice. Final Answer **Therefore, the correct answer is 35 years.**