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 ?

Let the man's age be 7x years ,
Then son's age = 3x years
$$\eqalign{ & \therefore {\text{7}}x \times 3x = 756 \cr & \Rightarrow 21{x^2} = 756 \cr & \Rightarrow {x^2} = 756 \cr & \Rightarrow {x^2} = 36 \cr & \Rightarrow x = 6 \cr} $$
The ratio of their ages after 6 years
$$\eqalign{ & {\text{ = }}\left( {7x + 6} \right):\left( {3x + 6} \right) \cr & = \left( {7 \times 6 + 6} \right):\left( {3 \times 6 + 6} \right) \cr & = 48:24 \cr & = 2:1 \cr} $$