The difference between the present ages of Arun and Deepak is 14 years. Seven years ago, the ratio of their ages was $5 : 7$ respectively. What is Deepak's present age?

### Concept & Logic The cardinal rule of age problems is that **the age difference between two people remains absolutely constant throughout their lives.** If the difference between Arun and Deepak is $14$ years today, it was also exactly $14$ years a decade ago, and will be $14$ years a century from now. ### Step-by-Step Solution * **Given:** * Present age difference = $14$ years. * $7$ years ago, Arun : Deepak = $5 : 7$. * **Calculation / Deduction:** * Because the age difference never changes, the difference $7$ years ago was also $14$ years. * Let their ages $7$ years ago be $5x$ (Arun) and $7x$ (Deepak). * Set up the equation for their difference: * $$7x - 5x = 14$$ * $$2x = 14$$ * $$x = 7$$ * Now, find Deepak's age $7$ years ago: * Deepak ($7$ years ago) = $7x = 7 \times 7 = 49$ years. * To find Deepak's present age, add the $7$ years back: * Deepak (Present) = $49 + 7 = 56$ years. ### Exam Strategy & Shortcut Look at the ratio gap. Seven years ago, the ratio was $5 : 7$. The gap in ratio units is $2$ units ($7 - 5 = 2$). We logically know this $2$-unit gap represents the constant $14$-year age difference. Therefore, $2 \text{ units} = 14 \text{ years} \Rightarrow 1 \text{ unit} = 7 \text{ years}$. Deepak was $7 \text{ units}$ ($7 \times 7 = 49 \text{ years old}$) in the past. Add $7$ to reach the present ($49 + 7 = 56$). This completely bypasses algebraic equations. ### Common Pitfall The most dangerous trap is solving for the past age ($49$) and selecting it from the options (Option C is specifically designed as a trap). Always perform a final check on the timeline the question is asking for ("What is Deepak's **present** age?"). ### Final Answer **Therefore, the correct answer is 56 years.**