The difference between the square of the present ages of P and Q is 25 and the average of their present ages is 12.5 years (consider age of P is more than Q). after how many years the ratio of ages of P and Q will be 15:14?

Correct option is C

​$$\text{Let the present ages of P and Q be } x \text{ and } y \text{ respectively, where } x > y. \\[5pt]\textbf{According to the question:} \\x^2 - y^2 = 25 \\(x - y)(x + y) = 25 \\[5pt]\text{The average of their present ages is } 12.5 \text{ years:} \\\frac{x + y}{2} = 12.5 \Rightarrow x + y = 25 \\[10pt]\text{Now substitute } x + y = 25 \text{ into the equation:} \\(x - y)(25) = 25 \Rightarrow x - y = 1 \\[10pt]\text{Now we have a system of equations:} \\1. x + y = 25 \\2. x - y = 1 \\[10pt]\text{Add the two equations:} \\(x + y) + (x - y) = 25 + 1 \\2x = 26 \Rightarrow x = 13 \\[5pt]\text{Substitute } x = 13 \text{ into } x + y = 25: \\13 + y = 25 \Rightarrow y = 12 \\[10pt]\text{Let the number of years be } n. \\[5pt]\text{New ages: } 13 + n \text{ and } 12 + n \\[5pt]\text{Given: } \frac{13 + n}{12 + n} = \frac{15}{14} \\[10pt]\textbf{Cross-multiplying to solve for } n: \\14(13 + n) = 15(12 + n) \\182 + 14n = 180 + 15n \\182 - 180 = 15n - 14n \\2 = n$$​​