The ages of Shakti and Kanti are in the ratio of 8 : 7 respectively. After 10 years, the ratio of their ages will be 13 : 12. What is the difference between their ages ?

Let Shakti's age be 8$$x$$ years,
Then Kanti's age = 7$$x$$ years
$$\eqalign{ & \therefore \frac{{8x + 10}}{{7x + 10}} = \frac{{13}}{{12}} \cr & \Rightarrow 12\left( {8x + 10} \right) = 13\left( {7x + 10} \right) \cr & \Rightarrow 96x + 120 = 91x + 130 \cr & \Rightarrow 5x = 10 \cr & \Rightarrow x = 2 \cr} $$
Difference between their ages
$$\eqalign{ & = \left( {8x - 7x} \right) \cr & = {\text{ }}x\,{\text{years}} \cr & {\text{ = 2}}\,{\text{years}} \cr} $$