Question

A father is 25 years older than his son. After 8 years the ratio of their ages will be 13:8. Find their present ages.

Answer 1

Hint: We will start by letting the age of father and son as x and y respectively. Then we will use the fact that the father is 25 years older than his son to form an equation. Then we will use the fact given in the question that after 8 years the ratio of their ages will be 13:8. Now finally, we will solve these two equations to find the value of their present age.

Complete step-by-step solution -
Now, we let the ages of father and son as x and y years.
Now, we have been given that the father is 25 years older than son. So, this means,
$x=25+y......\left( 1 \right)$
Also, we have been given that the ratio of their after 8 years will be 13:8. So, we have,
$\dfrac{x+8}{y+8}=\dfrac{13}{8}$
Now, on cross – multiplying we have,
$\begin{align}
  & 8\left( x+8 \right)=13\left( y+8 \right) \\
 & 8x+64=13y+104 \\
 & 8x-13y=104-64 \\
 & 8x-13y=40............\left( 2 \right) \\
\end{align}$
Now, we substitute x from (1) and (2). So, we have,
$\begin{align}
  & 8\left( y+25 \right)-13y=40 \\
 & 8y+25\times 8-13y=40 \\
 & 8y+200-13y=40 \\
 & 200-5y=40 \\
 & 200-40=5y \\
 & 160=5y \\
 & 32=y \\
\end{align}$
Now, we substitute y = 32 in (1). So, we have,
$\begin{align}
  & x=25+32 \\
 & =57years \\
\end{align}$
Hence, the present age of the father and son is 57 and 32 years.

Note: It is important to note that we have used the elimination method of solving the system of equations in two variables. Also, we have to note that to solve the questions we have first assumed the ages of father and son and then formed an equation as per the data given in the question.