6 years ago, the ratio of the ages of Kunal and Sagar was 6 : 5. Four years hence, the ratio of their ages will be 11 : 10. What is Sagar's age at present?

### Concept & Strategy For problems involving ratios of ages at two different points in time, the most efficient approach is the **Ratio Unit Method**. This method works because the time gap between the two scenarios is identical for both individuals, meaning the difference in their ratio units should ideally reflect that constant time gap. The total time gap between "6 years ago" and "4 years hence" is $6 + 4 = 10$ years. ### Step-by-Step Solution * **Given:** Let the ages of Kunal and Sagar 6 years ago be $6x$ and $5x$. * **Establish the Present Age:** * Kunal's present age = $6x + 6$ * Sagar's present age = $5x + 6$ * **Establish the Future Age:** Four years hence (from the present), their ages will be: * Kunal = $(6x + 6) + 4 = 6x + 10$ * Sagar = $(5x + 6) + 4 = 5x + 10$ * **Calculation:** The future ratio is given as $11 : 10$. $$ \frac{6x + 10}{5x + 10} = \frac{11}{10} $$ * Cross-multiply to solve for $x$: $$ 10(6x + 10) = 11(5x + 10) $$ $$ 60x + 100 = 55x + 110 $$ $$ 5x = 10 \implies x = 2 $$ * **Finding Target Age:** We need Sagar's *present* age. * Sagar's age 6 years ago = $5x = 5(2) = 10$ years. * Sagar's present age = $10 + 6 = 16$ years. ### Exam Strategy & Shortcut Use the Ratio Unit shift directly. Past Ratio (6 yrs ago) $\implies K : S = 6 : 5$ (Gap between ratios is 1 unit) Future Ratio (4 yrs hence) $\implies K : S = 11 : 10$ (Gap between ratios is 1 unit) Notice that for both Kunal and Sagar, the ratio increases from $6 \to 11$ and $5 \to 10$. That is exactly a 5-unit increase for both. This 5-unit increase corresponds exactly to the 10-year time gap ($6 \text{ past} + 4 \text{ future}$). $$ 5 \text{ units} = 10 \text{ years} \implies 1 \text{ unit} = 2 \text{ years} $$ Sagar's past age (5 units) = $5 \times 2 = 10$ years. Present age = $10 + 6 = 16$ years. Done in 10 seconds. ### Common Pitfall A very common mistake is finding the value of $x$ (which is 2) and simply plugging it into the past ratio to get 10, then selecting an option (if 10 were present) without adding the 6 years to arrive back at the *present* age. Always verify the timeline the question asks for. ### Final Answer **Therefore, the correct answer is 16 years.**