Question

One year ago, a man was 8 times as old as his son. Now his age is equal to the square of his son’s age. Find their present ages.

Answer 1

Hint: Assign variables for the present ages of man and his son. Write down the equations involving these two variables and solve them to find their present ages.

Complete step-by-step answer:
Let the present age of the man be x and the present age of the man’s son be y.
We have two variables and we need at least two equations to solve them.
It is given that one year ago, the man was 8 times as old as his son.
The age of the man one year ago is x – 1 and the age of his son one year ago is y – 1.
Then, we have the equation as follows:
$\Rightarrow$ \[x - 1 = 8(y - 1)\]
Simplifying by multiplying inside the bracket, we have:
$\Rightarrow$ \[x - 1 = 8y - 8\]
Now, solving for x in terms of y, we get:
$\Rightarrow$ \[x = 8y - 7............(1)\]
It is given that now the man’s age is equal to the square of his son’s age. Hence, we have the equation as follows:
$\Rightarrow$ \[x = {y^2}..........(2)\]
Hence, we have two equations in two variables, hence, we can solve them.
Substituting equation (1) in equation (2), we have:
$\Rightarrow$ \[8y - 7 = {y^2}..........(2)\]
We now take all terms to the right-hand side of the equation.
$\Rightarrow$ \[{y^2} - 8y + 7 = 0\]
We can solve this quadratic equation by splitting the middle term – 8y into – 7y and – y.
$\Rightarrow$ \[{y^2} - 7y - y + 7 = 0\]
Taking common terms and simplifying, we have:
$\Rightarrow$ \[y(y - 7) - 1(y - 7) = 0\]
$\Rightarrow$ \[(y - 1)(y - 7) = 0\]
$\Rightarrow$ \[y = 1,y = 7\]
We got two values of y, we find x and then check if the answer agrees with the scenario.
For y = 1, from equation (2), we get:
$\Rightarrow$ \[x = {1^2}\]
$\Rightarrow$ \[x = 1\]
Son should be born after the man, hence, their ages can’t be the same, this solution is not correct.
For y = 7, we find the x and then check if the answer agrees with the scenario.
$\Rightarrow$ \[x = {7^2}\]
$\Rightarrow$ \[x = 49\]
The value of x is 49.
Hence, the present age of the son is 49 and the present age of his son is 7.

Note: Note that a year ago the age of the man and the son should be one year less than the actual age. Many students miss this fact and hence, obtain a wrong answer.