Question

Eighteen years ago, a father was three times as old as his son. Now the father is only twice as old as his son. Then the sum of the present ages of the son and the father is?
(a) 54
(b) 72
(c) 105
(d) 108

Answer 1

Hint: Assume that the present age of the father as x years and that of the son as y years. Now, use the information given in the question to form two linear equations in x and y. To form the first equation, subtract 18 years from the ages of both the father and son and equate the age of the father with three times the age of the son. To form the second equation, equate the present age of the father with twice the present age of the son. Solve the two equations to find the values of x and y and take the sum of these values to get the answer.

Complete step by step answer:
Here we have been provided with the information regarding the ages of a father and his son eighteen years ago and in the present. We have to find the sum of their present ages.
Now, let us assume the present age of the father is x years and the present age of the son is y years. In the first condition we have been given that eighteen years ago the age of the father was three times the age of the son, so mathematically we have,
\[\begin{align}
  & \Rightarrow \left( x-18 \right)=3\left( y-18 \right) \\
 & \Rightarrow x-18=3y-54 \\
 & \Rightarrow x-3y=-36........\left( i \right) \\
\end{align}\]
Now, in the second condition it is given that the present age of the father is two times the present age of the son, so mathematically we have,
\[\begin{align}
  & \Rightarrow x=2y \\
 & \Rightarrow x-2y=0........\left( ii \right) \\
\end{align}\]
Subtracting equation (ii) from equation (i) we get,
\[\begin{align}
  & \Rightarrow \left( x-3y \right)-\left( x-2y \right)=-36-0 \\
 & \Rightarrow -y=-36 \\
 & \Rightarrow y=36 \\
\end{align}\]
Substituting the obtained value of y in equation (ii) we get,
$\begin{align}
  & \Rightarrow x=2\left( 36 \right) \\
 & \Rightarrow x=72 \\
\end{align}$
Therefore, the sum of present ages of the father and the son will be x + y = 72 + 36 = 108.

Hence, option (d) is the correct answer.

Note: Do not get confused in the words of the questions as sometimes they might be confusing. For the past times subtract the provided ages from the present ages while for the future times add the provided ages in the present ages. Do not try to solve the question using only one variable as it might be confusing in certain situations.