The ages of Shakti and Kanti are in the ratio of $8 : 7$ respectively. After 10 years, the ratio of their ages will be $13 : 12$. What is the difference between their ages?

Concept & Logic When given a present and future ratio, if the proportional increase for both individuals is identical, you can bypass complex algebra. The age difference stays strictly constant throughout a person's lifetime. $$ \frac{8x + 10}{7x + 10} = \frac{13}{12} $$ Step-by-Step Solution * **Given:** The present age ratio of Shakti and Kanti is $8 : 7$. * **Given:** After 10 years, their age ratio becomes $13 : 12$. * **Calculation:** Let their present ages be $8x$ and $7x$. * Set up the equation for 10 years later: $(8x + 10) / (7x + 10) = 13 / 12$. * Cross-multiply to solve for $x$: $12(8x + 10) = 13(7x + 10)$. * Expand the brackets: $96x + 120 = 91x + 130$. * Isolate $x$: $96x - 91x = 130 - 120 \Rightarrow 5x = 10 \Rightarrow x = 2$. * The difference between their ages is $8x - 7x = x$. * Substitute $x = 2$ to find the difference directly: 2 years. Exam Strategy & Shortcut Use the **Ratio Constant Difference** technique. Notice that the difference between the ratio parts for Shakti and Kanti is $8 - 7 = 1$ part originally, and $13 - 12 = 1$ part after 10 years. Because the internal difference is perfectly balanced, we compare the timeline shift directly. Shakti went from 8 parts to 13 parts, an increase of exactly 5 parts. This 5-part increase corresponds to the 10 years elapsed. Therefore, 5 parts = 10 years $\Rightarrow 1$ part = 2 years. The difference in their ages is exactly 1 part ($8 - 7$), which equals 2 years. Common Pitfall A common inefficiency is solving for $x = 2$, computing Shakti's age (16) and Kanti's age (14), and then subtracting them to get 2. While correct, it wastes precious seconds. Recognize that finding the difference in $x$ terms ($8x - 7x = x$) gives you the answer immediately once $x$ is known. Final Answer **Therefore, the correct answer is 2 years.**