Question

Find the age of a child whose age is two-third of his elder brother, after $5$ years he will be four-fifths.

Answer 1

Step 1: Form equations from the given information

Let the age of child be $x$

Let the age of elder brother be $y$

Given, age of child is two-third of his elder brother. So, $x=\frac{2}{3}y$

Also, it is given that after $5$ years, child's age will be four-fifths of the elder brother's age, i.e.,

$(x+5)=\frac{4}{5}(y+5)$

Step 2: Solve the equations

Replace $y=\frac{3}{2}x$ in $(x+5)=\frac{4}{5}(y+5)$, we get

$$\begin{gathered} \Rightarrow x+5=\frac{4}{5}\left(\frac{3}{2}x+5\right) \\ \Rightarrow 5x+25=6x+20 \\ \Rightarrow 6x-5x=25-20 \\ \Rightarrow x=5 \end{gathered}$$

Therefore, the age of the child is $5$ years.