Ten years hence, the respective ratio between Simmi's age and Niti's age will be 7 : 9. Two years ago, the respective ratio between Simmi's age and Niti's age was 1 : 3. If Abhay is 4 years older to his sister Niti, what is Abhay's present age? (in years)

### Concept & Formula This problem uses the classic "Cross-Multiplication" or "Ratio Gap" concept for age problems. By defining the ages at a specific point in time using the given ratio, we can fast-forward to another timeline to set up a linear equation. ### Step-by-Step Solution * **Given:** Two years ago, the ratio of Simmi to Niti was $1:3$. Let their ages 2 years ago be $x$ and $3x$ respectively. This means their **present ages** are: Simmi = $x + 2$ Niti = $3x + 2$ * **Calculation (Finding the Multiplier):** Ten years hence (which is 12 years from the "two years ago" timeline): Simmi's age will be $x + 12$ Niti's age will be $3x + 12$ The ratio at this future time is given as $7:9$. $$\frac{x + 12}{3x + 12} = \frac{7}{9}$$ Cross-multiply to solve for $x$: $$9(x + 12) = 7(3x + 12)$$ $$9x + 108 = 21x + 84$$ $$108 - 84 = 21x - 9x$$ $$24 = 12x \implies x = 2$$ * **Calculation (Finding Present Ages):** Now substitute $x$ back to find Niti's present age: Niti's present age = $3x + 2 = 3(2) + 2 = 8$ years. The problem states Abhay is 4 years older than his sister Niti: Abhay's present age = $8 + 4 = 12$ years. ### Exam Strategy & Shortcut Use the **Ratio Gap Method** to avoid $x$ entirely. Ratio 2 years ago (Past) = $1 : 3$ (Gap is 2) Ratio 10 years hence (Future) = $7 : 9$ (Gap is 2) Since the age gap between two people never changes, and the ratio gaps are conveniently already equal ($3-1=2$ and $9-7=2$), we can compare the units directly. The timeline difference from "2 years ago" to "10 years hence" is $12$ years. For Niti, the ratio unit changed from 3 to 9, an increase of 6 units. So, 6 units = 12 years $\implies$ 1 unit = 2 years. Niti's age 2 years ago (3 units) = $3 \times 2 = 6$ years. Niti's present age = $6 + 2 = 8$ years. Abhay's present age = $8 + 4 = 12$ years. This method takes 15 seconds! ### Common Pitfall Students frequently calculate Niti's present age (8) and immediately select option (a) in a rush. Always re-read the final sentence of the question before marking your answer to ensure you are answering what is actually being asked (Abhay's age, not Niti's). ### Final Answer **Therefore, the correct answer is 12.**