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

### Concept & Strategy This is a classic "double ratio" age problem. The best approach is the **Cross-Multiplication Method** or equalizing the ratio differences, as the time gap between the two given scenarios is fixed. The total time gap between "10 years ago" and "10 years hence" is $10 + 10 = 20$ years. ### Step-by-Step Solution * **Method 1: Algebraic Setup** * Let the son's age 10 years ago be $x$. The man's age 10 years ago was $3x$. * Their present ages are $x + 10$ and $3x + 10$. * Their ages 10 years hence (from the present) will be $x + 10 + 10 = x + 20$ and $3x + 10 + 10 = 3x + 20$. * **Calculation:** The problem states that 10 years hence, the man will be twice his son's age. $$ 3x + 20 = 2(x + 20) $$ $$ 3x + 20 = 2x + 40 $$ $$ x = 20 $$ * **Finding Present Ages:** * Son's present age = $x + 10 = 20 + 10 = 30$. * Man's present age = $3(20) + 10 = 60 + 10 = 70$. * Ratio of their present ages (Man : Son) = $70 : 30 = 7 : 3$. ### Exam Strategy & Shortcut Use the Ratio Balance technique: Past Ratio (10 yrs ago) $\implies M : S = 3 : 1$ (Gap = 2 units) Future Ratio (10 yrs hence) $\implies M : S = 2 : 1$ (Gap = 1 unit) To make the age difference gaps equal, multiply the future ratio by 2: Adjusted Future Ratio $\implies 4 : 2$ Now, the change from past to future for both is $4 - 3 = 1$ unit, and $2 - 1 = 1$ unit. This 1 unit increase corresponds to the time gap of 20 years. Man's age 10 years ago = $3 \times 20 = 60$. Present age = $60 + 10 = 70$. Son's age 10 years ago = $1 \times 20 = 20$. Present age = $20 + 10 = 30$. Present Ratio = $70 : 30 = 7 : 3$. ### Common Pitfall A frequent mistake is finding the ages for "10 years ago" (60 and 20) and carelessly choosing a ratio based on those past ages (3:1), or forgetting to add only 10 years to reach the present age, incorrectly adding 20 instead. Always double-check what timeline the final question is asking for. ### Final Answer **Therefore, the correct answer is 7 : 3.**