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# Raja's age is one-fifth of his father's age. After 6 years, his age will be one-third his father's age. How old are they now?

Last updated date: 15th Sep 2024
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Hint: In this question, we are given a statement about ages of Raja and his father. We are also given a statement about the ages of Raja and his father after six years. Using this information, we need to calculate the present age of Raja and his father. For this, we will first suppose Raja's father's age to be x. And then present age's statement to calculate age of Raja's father in terms of x. Then we will find the age of both after 6 years by adding 6 to both of their ages. After that, we will form an equation of their ages using the statement of ages of both of them after 6 years and hence, solve it to find the value of x.

Here, we have to find the present age of Raja and his father. So, let us suppose that, present age of Raja's father is x years.
Since, Raja's age is ${{\dfrac{1}{5}}^{th}}$ of his father's age, so Raja's age will be $\dfrac{x}{5}$ years.
Now, let us find their ages 6 years later.
After 6 years, their ages will be:
Age of Raja after six years $\Rightarrow \left( \dfrac{x}{5}+6 \right)\text{years}$.
Age of Raja's father after six years $\Rightarrow \left( x+6 \right)\text{years}$.
We are given that, after 6 years, Raja's age will be one-third his father's age. Therefore, $\left( \dfrac{x}{5}+6 \right)$ will be ${{\dfrac{1}{3}}^{rd}}$ of (x+6). Hence, the equation becomes $\Rightarrow \dfrac{x}{5}+6=\dfrac{x+6}{3}$.
Taking LCM of 5 on the left side of the equation, we get:
$\Rightarrow \dfrac{x+30}{5}=\dfrac{x+6}{3}$.
Cross multiplying we get:
\begin{align} & \Rightarrow 3\left( x+30 \right)=5\left( x+6 \right) \\ & \Rightarrow 3x+90=5x+30 \\ & \Rightarrow 5x-3x=90-30 \\ & \Rightarrow 2x=60 \\ & \Rightarrow x=30 \\ \end{align}
Since, x was supposed to be the age of Raja's father, so the age of Raja's father is 30 years.
Now, age of Raja is ${{\dfrac{1}{5}}^{th}}$ the age of his father, so Raja's age will be $\dfrac{30}{5}=6\text{ years}$.

Hence, age of Raja = 6 years and age of Raja's father = 30 years.

Note: While solving this sum, students should not get confused with present age and age after 6 years. We can also simplify our calculations assuming Raja's age as x years which will result in Raja's father's age as 5x (${{\dfrac{1}{5}}^{th}}$ of 5x is x). After 6 years, their ages will be x+6 and 5x+6. So equation will be:
\begin{align} & \Rightarrow 3\left( 5x+6 \right)=5x+6 \\ & \Rightarrow 3x+18=5x+6 \\ & \Rightarrow 2x=12 \\ & \Rightarrow x=6 \\ \end{align}
Hence, required ages will be 6 years and $6\times 5=30\text{ years}$.