Three years ago, the ratio of the ages of Amisha and Nimisha was 8 : 9 respectively. 3 years hence, the ratio of their ages will be 11 : 12 respectively. What is the present age of Amisha?

### Concept & Logic This problem bridges two time periods without explicitly stating the present age ratio. The strategy is to find the **total time gap** between the two given ratios and use the ratio unit progression to solve it quickly. The total time gap between "3 years ago" and "3 years hence" is $3 + 3 = 6$ years. ### Step-by-Step Solution * **Given:** Let their ages 3 years ago be $8x$ and $9x$. * **Establish Future Ages:** The gap between "3 years ago" and "3 years hence" is 6 years. So, we add 6 to their past ages. * Amisha's age 3 years hence = $8x + 6$ * Nimisha's age 3 years hence = $9x + 6$ * **Calculation:** The future ratio is given as $11 : 12$. $$ \frac{8x + 6}{9x + 6} = \frac{11}{12} $$ * Cross-multiply to solve for $x$: $$ 12(8x + 6) = 11(9x + 6) $$ $$ 96x + 72 = 99x + 66 $$ $$ 99x - 96x = 72 - 66 $$ $$ 3x = 6 \implies x = 2 $$ * **Finding Target Age:** We need Amisha's *present* age. * Amisha's age 3 years ago = $8x = 8(2) = 16$ years. * Amisha's present age = $16 + 3 = 19$ years. ### Exam Strategy & Shortcut Use visual ratio tracking. Past Ratio (3 yrs ago) $\implies 8 : 9$ Future Ratio (3 yrs hence) $\implies 11 : 12$ Look at the vertical jump: Amisha goes $8 \to 11$ (3 units). Nimisha goes $9 \to 12$ (3 units). The ratio jumps by exactly 3 units for both. This 3-unit jump covers the entire 6-year gap ($3 \text{ past} + 3 \text{ future}$). $$ 3 \text{ units} = 6 \text{ years} \implies 1 \text{ unit} = 2 \text{ years} $$ Amisha's age 3 years ago was 8 units $\implies 8 \times 2 = 16$ years. Her present age is $16 + 3 = 19$ years. ### Common Pitfall The most frequent error is miscalculating the time gap. Students will often set up the equation as $\frac{8x + 3}{9x + 3} = \frac{11}{12}$, mistakenly thinking the jump from "3 years ago" to "3 years hence" is only 3 years instead of 6 years. Always map the timeline explicitly. ### Final Answer **Therefore, the correct answer is 19 years.**