Question

Arun is now half as old as his father. Twelve years ago, the father’s age was three times as old as Arun. Find their present ages.

Answer 1

Hint: Assume the present age of Arun be x. It is given that Arun is now half as old as his father. So, the present age of Arun’s father is 2x. Twelve years ago, the age of Arun was \[x-12\] and the age of Arun’s father was \[2x-12\] . It is also given that twelve years ago, the father’s age was three times as old as Arun. To make the age of Arun and Arun’s father equal, we need to multiply the age of Arun by 3. Now, solve the equation, \[3\left( x-12 \right)=\left( 2x-12 \right)\] and get the value of x. Use the value of x and get the present age of Arun and Arun’s father.

Complete step-by-step answer:
According to the question, it is given that Arun is now half as old as his father. Twelve years ago, the father’s age was three times as old as Arun.
First of all, let us assume that the present age of Arun be x years ……………..……..(1)
In case \[{{1}^{st}}\] , we have
Arun is half as old as his father.
From equation (1), we have the present age of Arun.
Since Arun is now half as old as his father and the age of Arun is x.
So, the present age of Arun’s father is 2x ………………………..(2)
Now, in case \[{{2}^{nd}}\] that is twelve years ago,
From equation (1) and equation (2), we have the present age of Arun and his father.
Twelve years ago, the age of Arun = \[x-12\] ……………………(3)
Twelve years ago, the age of Arun’s father = \[2x-12\] ………………………..(4)
It is given that Twelve years ago, the father’s age was three times as old as Arun.
Since the age of Arun’s father is thrice to Arun's, we have to multiply by 3 to Arun's age to make the age of Arun and Arun’s father equal.
From equation (3), we have the age of father and from equation (4), we have the age of the father.
Now, multiplying by 3 in equation (3) to make it equal to equation (4),
\[\begin{align}
  & \Rightarrow 3\left( x-12 \right)=2x-12 \\
 & \Rightarrow 3x-36=2x-12 \\
 & \Rightarrow 3x-2x=36-12 \\
\end{align}\]
\[\Rightarrow x=24\] …………………(5)
We have got the value of x which is equal to 24.
Putting the value of x in equation (1) and equation (2), we get
The present age of Arun is 24 years.
The present age of Arun’s father = \[2x=2\times 24=48\] years.
Therefore, the present age of Arun and Arun’s father is 24 years and 48 years respectively.

Note: While forming the equation for the condition “Twelve years ago, the father’s age was three times as old as Arun”, one can write it as \[\left( x-12 \right)=3\left( 2x-12 \right)\] . This is wrong because the father’s age is thrice so to make the age of Arun and Arun’s father equal, we have to multiply by 3 in the age of Arun. Hence, the correct equation is \[3\left( x-12 \right)=\left( 2x-12 \right)\] .