Question

The combined ages of a man and his son are 60 years. Twelve years ago, the father was five times as old as his son. Find their present ages.

Answer 1

Hint: We will assume the present ages of the father and son as two unknowns. From the given information in the question, we will form two equations. These equations will be linear equations in two variables. We will solve the two equations to find the two unknowns which are the present ages of the father and the son.

Complete step-by-step answer:
Let us assume that the present age of the father and the son is $x$ years and $y$ years respectively. We are given that the combined ages of the father and the son are 60 years. So, we get the following equation,
$x+y=60....(i)$
We are also given that twelve years ago, the father was five times as old as his son. Twelve years ago, the age of the father was $x-12$ and the age of the son was$y-12$. So, we get the following equation,
$x-12=5\left( y-12 \right)....(ii)$
Now, we have two linear equations in two variables. We will use the substitution method to find the values of $x$ and$y$. From equation$(i)$, we will substitute $x=60-y$ in equation $(ii)$ in the following manner,
$60-y-12=5\left( y-12 \right)$
Simplifying the above equation, we get
$\begin{align}
  & 48-y=5y-60 \\
 & \Rightarrow 48+60=5y+y \\
 & \Rightarrow 108=6y \\
 & \therefore y=18 \\
\end{align}$
Now, substituting the value of $y$ in equation$(i)$, we get
$\begin{align}
  & x+18=60 \\
 & \Rightarrow x=60-18 \\
 & \therefore x=42 \\
\end{align}$
Hence, the present ages of the father and the son are 42 years and 18 years respectively.

Note: It is important that we form correct equations by using the information that is given to us in the question. There are various methods to solve the system of linear equations. These methods are Gauss elimination method and graphing method. The calculations should be done explicitly so that minor mistakes can be avoided.