The age of a father 10 years ago was thrice the age of his son. 10 years hence, the father's age will be twice that of his son. The ratio of their present ages is

### Concept & Formula The core concept is translating past and future time conditions into a single algebraic variable based on the present age. When given "ago" and "hence" conditions, set up linear equations by adding or subtracting the given years from the present age variables. ### Step-by-Step Solution * **Given:** * $10$ years ago: Father = $3 \times$ Son * $10$ years hence (future): Father = $2 \times$ Son * **Calculation / Deduction:** * Let the ages $10$ years ago be: Son = $x$, Father = $3x$. * This means their *present* ages are: Son = $x + 10$, Father = $3x + 10$. * Their ages $10$ years hence (from the present) will be: * Son = $(x + 10) + 10 = x + 20$ * Father = $(3x + 10) + 10 = 3x + 20$ * The problem states that $10$ years hence, the father is twice the son's age: * $$3x + 20 = 2(x + 20)$$ * $$3x + 20 = 2x + 40$$ * $$3x - 2x = 40 - 20$$ * $$x = 20$$ * Now, substitute $x$ back into the present age expressions: * Present age of Son = $20 + 10 = 30$ years. * Present age of Father = $3(20) + 10 = 70$ years. * Required present ratio = Father : Son = $70 : 30 = 7 : 3$. ### Exam Strategy & Shortcut Use the Cross-Multiplication Method for age ratios. Past ratio ($10$ yrs ago) = $3 : 1$ Future ratio ($10$ yrs hence) = $2 : 1$ Total time gap = $20$ years. Difference in future ratio parts = $2 - 1 = 1$. Cross product difference = $(3 \times 1) - (2 \times 1) = 1$. Value of 1 part = $\frac{20 \times 1}{1} = 20$. Past age of son = $20$. Past age of father = $60$. Present ages = $70$ and $30$. Ratio is $7 : 3$. ### Common Pitfall The most common mistake is forgetting to add $10$ to find the present age after solving for $x$. Students often calculate the past ages ($60$ and $20$) and look for a $3:1$ option, or use the future ages ($80$ and $40$) resulting in $2:1$. Always track exactly which point in time your variable $x$ represents. ### Final Answer **Therefore, the correct answer is 7 : 3.**