Question

10 years ago the age of the father was $5$ times that of the son. $20$ Years hence the age of the father will be twice that of the son. The present age of the father (in years) is

Answer 1

Hint: Here we will solve this arithmetic problem by considering a pair of linear equations with two variables and solve these linear equations using a substitution method for finding the values for the given two variables. That gives the solution for this problem.

Complete step-by-step solution:
Let consider the present age of the father and son be $x$ and $y$ respectively,
$10$ Years ago we knew that the age of the father was $5$ times that of his son. So we can express this relationship in terms of $x$ and $y$ that is,
$ \Rightarrow x - 10 = 5\left( {y - 10} \right)$
$ \Rightarrow x - 10 = 5y - 50$
$ \Rightarrow x = 5y - 50 + 10$
$ \Rightarrow x = 5y - 40$--------- (1)
This equation will come in handy later on.
Now $20$ years after age of the father will be twice that of the son,
$ \Rightarrow x + 20 = 2\left( {y + 20} \right)$
$ \Rightarrow x + 20 = 2y + 40$
$ \Rightarrow x = 2y + 40 - 20$
$ \Rightarrow x - 2y = 20$---------- (2)
So in the end we have two equations (1) and (2) in terms of $x$ and $y$, so we have two simultaneous equations and we can solve this by using substitution method,
$ \Rightarrow x = 5y - 40$--------- (1)
$ \Rightarrow x - 2y = 20$--------- (2)
In the substitution method, one variable is expressed in terms of another. This would allow the number of variables to be reduced to one, and so we can solve for that one variable. Once we get the value of that variable, we use it to solve for the other variable. Let’s see how this works,
So since we have $x = 5y - 40$, will choose to substitute every instance of $x$ with $5y - 40$, and so reduce the number of variables to just one, namely $y$. We can substitute this in equation (2),
$ \Rightarrow 5y - 40 - 2y = 20$
$ \Rightarrow 5y - 2y = 20 + 40$
$ \Rightarrow 3y = 60$
$ \Rightarrow y = \dfrac{{60}}{3}$
$ \Rightarrow y = 20$
We have found the value of $y$ to be $20$. Now, we can substitute this back into equation (1), to solve for $x$,
$ \Rightarrow x = 5\left( {20} \right) - 40$
$ \Rightarrow x = 100 - 40$
$ \Rightarrow x = 60$
We have successfully found the values of $x$ and $y$ to be $60$ and $20$ respectively.
Therefore the present age of the father and son will be $60$ and $20$ respectively.

Note: We can check if these values are correct or not by substituting these values in the given equation.
Substitute the values of $x = 60$ and $y = 20$ in any of the equations. Here we will select the equation (1),
$ \Rightarrow x = 5y - 40$
$ \Rightarrow 60 = 5\left( {20} \right) - 40$
$ \Rightarrow 60 = 100 - 40$
$ \Rightarrow 60 = 60$
Thus we conclude that our values are correct.