The ratio of a man's age and his son's age is 7 : 3 and the product of their ages is 756. The ratio of their ages after 6 years will be

### Concept & Formula When dealing with the product of variables given in a ratio, you must introduce a common multiplier ($x$). Multiplying two ratio parts results in an $x^2$ term, not a linear $x$. $$ (ax) \times (bx) = abx^2 $$ ### Step-by-Step Solution * **Given:** Present ratio of Man to Son = $7 : 3$. Product of present ages = 756. Time shift = $+6$ years. * **Calculation:** Let the present age of the man be $7x$ and the son be $3x$. * Set up the product equation: $(7x)(3x) = 756$ * Multiply the terms: $21x^2 = 756$ * Isolate $x^2$: $x^2 = \frac{756}{21} = 36$ * Solve for the multiplier: $x = 6$ * Find their exact present ages: Man = $7 \times 6 = 42$ years. Son = $3 \times 6 = 18$ years. * Find their ages after 6 years: Man = $42 + 6 = 48$ years. Son = $18 + 6 = 24$ years. * Determine the new ratio: $48 : 24 = 2 : 1$. ### Exam Strategy & Shortcut **Direct Ratio Scaling:** Once you see $21x^2 = 756$, recognize that 756 is close to $20 \times 36 = 720$. This quickly points to $x^2 = 36$ and $x = 6$. The rest is simple arithmetic. Notice that adding 6 years is exactly equivalent to adding "1 unit" of the multiplier $x$ to both sides. So the new ratio is simply $(7+1) : (3+1) = 8 : 4 = 2 : 1$. This skips finding their actual ages entirely! ### Common Pitfall A fatal mistake is treating the product like a sum and forgetting to square the variable. Students often write $21x = 756$, leading to $x = 36$, resulting in absurdly high ages and guaranteed failure on the problem. ### Final Answer Therefore, the correct answer is **2 : 1**.