Question

Six years before, the age of the mother was equal to the square of her son’s age. Three years hence, her age will be thrice the age of her son then find the present ages of the mother and son?

Answer 1

Hint: We can assume the age of the mother to be \[x \] and the age of the son to be \[y \] . Then we can form an equation using the conditions that are given in the question. We can consider the ages at 3 years later and from an equation with the given relation, then we can eliminate one variable by substitution.

Complete step-by-step answer:
Let \[x \] be the present age of the mother and \[y \] be the present age of the son.
Given, six years before, the age of the mother was equal to the square of her son’s age,
We know before six years the age of the mother will be \[x - 6 \] and the son’s age will be \[y - 6 \] .
So we can write,
  \[x - 6 = {(y - 6)^2} \] ---(1)
There are two unknowns, using the given condition that after three years mother’s age will be thrice the age of her son. Using this we can convert it into one variable.
 \[x + 3 = 3(y + 3) \]
 \[ \Rightarrow x + 3 = 3y + 9 \]
 \[ \Rightarrow x = 3y + 6 \]
 \[ \Rightarrow x = 3y + 6 \] ---(2)
Now we have the value of \[x \] , substituting this in equation (1)
We get,
 \[ \Rightarrow 3y + 6 - 6 = {(y - 6)^2} \]
 \[ \Rightarrow 3y = {(y - 6)^2} \]
Applying formula \[{(a - b)^2} = {a^2} + {b^2} - 2ab \] .
 \[ \Rightarrow 3y = {y^2} - 12y + 36 \]
 \[ \Rightarrow {y^2} - 12y - 3y + 36 = 0 \]
 \[ \Rightarrow y(y - 12) - 3(y - 12) = 0 \]
 \[ \Rightarrow (y - 3)(y - 12) = 0 \]
 \[ \Rightarrow y = 3 \] , \[y = 12 \]
When \[y = 3 \] , the age before six years will be negative, \[3 - 9 = - 3 \]
So we take \[ \Rightarrow y = 12 \]
Substituting in equation (2), we get,
 \[ \Rightarrow x = 3 \times 12 + 6 \]
 \[ \Rightarrow x = 36 + 6 \]
 \[ \Rightarrow x = 42 \]
Thus, we obtained the present age of the mother is \[42 \] and of her son is \[12 \] .
So, the correct answer is “age of the mother is \[42 \] and of her son is \[12 \]”.

Note: In the problem is based on the age, all you need to remember is we add if we find “after” and subtract if we find” before”. Most of the students get confused with this. So remember it well. We cannot solve an equation with a variable. So we converted it into only one variable. Age will not take negative value. Hence we did not take \[y = 3 \] .