The ages of Sulekha and Arunima are in the ratio 9 : 8 respectively. After 5 years, the ratio of their ages will be 10 : 9. What is the difference in their ages?

### Concept & Formula This is the most ideal scenario for the **Direct Ratio Unit Method**. When the difference in ratio units between two people is naturally balanced across time periods, the problem can be solved in a single step without equations. ### Step-by-Step Solution * **Given:** The present age ratio of Sulekha to Arunima is $9 : 8$. Let their ages be $9x$ and $8x$. * **Equation Setup:** After 5 years, the ratio is $10 : 9$. $$ \frac{9x + 5}{8x + 5} = \frac{10}{9} $$ * **Calculation:** $$ 9(9x + 5) = 10(8x + 5) $$ $$ 81x + 45 = 80x + 50 $$ $$ x = 5 $$ * **Finding Target Value:** The difference in their ages is $9x - 8x = x$. * Since $x = 5$, the difference in their ages is 5 years. ### Exam Strategy & Shortcut You can solve this instantly by visually inspecting the ratios. Present Ratio $\implies 9 : 8$ Future Ratio (+5 yrs) $\implies 10 : 9$ Notice that Sulekha's ratio unit increases from 9 to 10 (1 unit increase). Arunima's ratio unit increases from 8 to 9 (1 unit increase). Because the unit increase perfectly matches for both, we can directly equate it to the time passed: $$ 1 \text{ ratio unit} = 5 \text{ years} $$ The question asks for the difference in their ages. In the ratio (9:8), the difference is exactly $9 - 8 = 1$ ratio unit. Since 1 unit = 5 years, the age difference is 5 years. No algebra needed. ### Common Pitfall The only real pitfall here is overcomplicating it. Students often instinctively dive into cross-multiplication ($81x + 45 = 80x + 50$) instead of taking two seconds to realize the ratio units are already perfectly balanced. In a timed exam, recognizing this pattern saves crucial seconds. ### Final Answer **Therefore, the correct answer is 5 years.**