The present age of son is half of the present age of his mother. Ten years ago, his mother's age was thrice the age of her son. What is the present age of the son?

### Concept & Formula Define the present ages using a primary variable and a multiplier, then project those variables backwards in time by subtracting the time elapsed to establish a solvable algebraic equation. ### Step-by-Step Solution * **Given:** * Present: Son = $\frac{1}{2} \times$ Mother (which means Mother = $2 \times$ Son) * $10$ years ago: Mother = $3 \times$ Son * **Calculation / Deduction:** * Let the present age of the son be $x$. * Then, the present age of the mother is $2x$. * Exactly $10$ years ago, their ages were: * Son = $x - 10$ * Mother = $2x - 10$ * According to the condition $10$ years ago, the mother was thrice the son: * $$2x - 10 = 3(x - 10)$$ * $$2x - 10 = 3x - 30$$ * $$30 - 10 = 3x - 2x$$ * $$x = 20$$ * Since $x$ directly represents the present age of the son, he is $20$ years old. ### Exam Strategy & Shortcut Apply the Ratio Cross-Multiplication technique: Present Ratio (Mother : Son) = $2 : 1$ Past Ratio ($-10$ years) = $3 : 1$ Difference in cross-products = $(3 \times 1) - (2 \times 1) = 1$. Difference in past ratio parts = $3 - 1 = 2$. Multiply the ratio parts difference by the time gap ($10$): $2 \times 10 = 20$. Value of $1$ ratio unit = $20$. The Son's present ratio part is $1$, so his age is directly $20$ years. ### Common Pitfall A very common mistake is applying the past adjustment ($-10$) only to the mother's age while leaving the son as $x$ in the equation setup. Both individuals age backwards together. ### Final Answer **Therefore, the correct answer is 20 years.**