The ages of Samina and Suhana are in the ratio of 7 : 3 respectively. After 6 years, the ratio of their ages will be 5 : 3. What is the difference in their ages?

### Concept & Strategy When given a present ratio and a future ratio, the most reliable method is to equalize the ratio differences. Because two people age at the exact same rate, the *difference* in their age in ratio units must be identical in both the past and the future. ### Step-by-Step Solution * **Method 1: Algebraic Setup** * **Given:** Let present ages be $7x$ (Samina) and $3x$ (Suhana). * **Establish Future Ages:** After 6 years, their ages will be $7x + 6$ and $3x + 6$. * **Calculation:** The future ratio is $5 : 3$. $$ \frac{7x + 6}{3x + 6} = \frac{5}{3} $$ * Cross-multiply: $$ 3(7x + 6) = 5(3x + 6) $$ $$ 21x + 18 = 15x + 30 $$ $$ 6x = 12 \implies x = 2 $$ * **Finding the Difference:** The present ages are $7(2) = 14$ and $3(2) = 6$. * The difference in their ages is $14 - 6 = 8$ years. ### Exam Strategy & Shortcut Use the **Ratio Balancing Shortcut** to skip the algebra entirely. Present Ratio $\implies 7 : 3$ (Difference between them is 4 units) Future Ratio $\implies 5 : 3$ (Difference between them is 2 units) Since age difference must be constant, balance the units by multiplying the future ratio by 2: Adjusted Future Ratio $\implies 10 : 6$ (Difference is now 4 units) Now, look at the progression: Samina goes from 7 to 10 units, Suhana from 3 to 6 units. Both increased by exactly 3 units. This 3-unit increase represents the 6 years that passed. $$ 3 \text{ units} = 6 \text{ years} \implies 1 \text{ unit} = 2 \text{ years} $$ We want the difference in their ages, which is 4 units. $$ \text{Difference} = 4 \text{ units} \times 2 = 8 \text{ years} $$ ### Common Pitfall A common mistake is forgetting that "difference in ages" is constant and can be calculated immediately from the ratio difference ($7x - 3x = 4x$). Students will calculate the two individual ages (14 and 6), which takes slightly longer. Another pitfall is not multiplying the future ratio carefully when trying to balance units. ### Final Answer **Therefore, the correct answer is 8 years.**