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 The fundamental rule in age problems is that the **age difference between two people remains constant** over time. By utilizing cross-multiplication on the two given ratios and the time gap, we can quickly determine the value of a single ratio unit without heavy algebra. ### Step-by-Step Solution * **Given:** * Present ratio (Samina : Suhana) = $7 : 3$ * Future ratio ($+6$ years) = $5 : 3$ * **Calculation / Deduction:** * Let the present ages be $7x$ and $3x$. * After $6$ years, their ages will be $7x + 6$ and $3x + 6$. * Set up the future ratio equation: * $$\frac{7x + 6}{3x + 6} = \frac{5}{3}$$ * Cross-multiply to solve for $x$: * $$3(7x + 6) = 5(3x + 6)$$ * $$21x + 18 = 15x + 30$$ * $$21x - 15x = 30 - 18$$ * $$6x = 12$$ * $$x = 2$$ * The present ages are $7(2) = 14$ and $3(2) = 6$. * Difference in ages = $14 - 6 = 8$ years. ### Exam Strategy & Shortcut Use the Ratio Difference Equalization technique. Present Ratio: $7 : 3$ (Difference is $4$ units) Future Ratio: $5 : 3$ (Difference is $2$ units) Since the age difference must be constant, equalize the unit differences by multiplying the future ratio by $2$. Adjusted Future Ratio: $(5 \times 2) : (3 \times 2) = 10 : 6$ Now observe Samina's ratio change: from $7$ to $10$, which is an increase of $3$ units. This $3$-unit increase corresponds to the $6$-year time gap. $$3 \text{ units} = 6 \text{ years} \Rightarrow 1 \text{ unit} = 2 \text{ years}$$ The present age difference is $4$ units ($7 - 3$). Difference = $4 \times 2 = 8$ years. ### Common Pitfall Students often waste time calculating the individual present ages of both people and their future ages just to subtract them at the end. Once you find the value of $x$ (or $1$ unit), simply multiply it by the ratio difference ($7 - 3 = 4$) to get the answer instantly. ### Final Answer **Therefore, the correct answer is 8 years.**