Question

A father is three times as old as his son. In 12 years time, he will be twice as old as his son. Find the present ages of father and the son.

Answer 1

Hint: Here, we will first assume the present age of the father and the present age of the son to be any variable. Then we will form two linear equations in two variables using the given conditions. Then we will solve both the equations to get the required value of one of the variables. Using this we will find the value of another variable and hence we will get the required present value of the father and the son.

Complete step by step solution:
Let the present age of the father be \[x\] and the present age of the son be \[y\].
It is given that the age of the father is three times as that of the age of the son. So we can write it as:
\[x = 3y\] ………. \[\left( 1 \right)\]
12 years later, the age of the father will become \[x + 12\] and the age of the son will become \[y + 12\].
It is also given that the age of the father will be equal to 2 times the age of the son after 12 years.
So we can write it as:-
\[x + 12 = 2\left( {y + 12} \right)\]
Using the distributive property of multiplication, we get
\[ \Rightarrow x + 12 = 2y + 24\]
Now, we will subtract 12 from both sides. Therefore, we get
\[\begin{array}{l} \Rightarrow x + 12 - 12 = 2y + 24 - 12\\ \Rightarrow x = 2y + 12\end{array}\]
Now, we will substitute the value of \[x\] from equation \[\left( 1 \right)\] in the above equation.
\[ \Rightarrow 3y = 2y + 12\]
Subtracting the term \[2y\] from both sides, we get
\[\begin{array}{l} \Rightarrow 3y - 2y = 2y - 2y + 12\\ \Rightarrow y = 12\end{array}\]
Now, we will substitute the value of \[y\] in equation \[\left( 1 \right)\]. So, we get
\[x = 3 \times 12 = 36\]

Hence, the present age of father is equal to 36 years and the present age of the son is equal to 12 years.

Note:
We need to keep in mind that if we are forming the equation after certain years then the present age of the father and the son will get added by the same number of years to avoid mistakes. Here the age after 12 years is given and not before 12 years, so we will add the number of years and not subtract it. So, we need to be careful while forming an equation. A linear equation in two variables is defined as an equation which has two distinct variables and has a highest degree of 1.