Question

Solve the following problems:

The numerator of a fraction is $4$ less than its denominator. If both numerator and denominator are divided by $4$, the fraction becomes $\frac{2}{3}$. Find the fraction.

Answer 1

Let the denominator of the given fraction be $\mathrm{x}$

So, the numerator is $(\mathrm{x}-4)$

According to the given condition,

$$\begin{gathered} \Rightarrow \frac{(\mathrm{x}-4)÷4}{\mathrm{x}÷4}=\frac{2}{3} \\ \Rightarrow \frac{(\mathrm{x}-4)\times 4}{\mathrm{x}\times 4}=\frac{2}{3}[\mathrm{Rule}\mathrm{of}\mathrm{identity}] \\ \Rightarrow \frac{(4\mathrm{x}-16)}{4\mathrm{x}}=\frac{2}{3} \\ \Rightarrow 3(4\mathrm{x}-16)=4\mathrm{x}\times 2 \\ \Rightarrow 12\mathrm{x}-48=8\mathrm{x}[\mathrm{Distributive}\mathrm{property}] \\ \Rightarrow 12\mathrm{x}-8\mathrm{x}=48 \\ \Rightarrow 4\mathrm{x}=48 \\ \Rightarrow \mathrm{x}=\frac{48}{4} \\ \therefore \mathrm{x}=12 \end{gathered}$$

So, the denominator of the given fraction $12$ and

the numerator $=(\mathrm{x}-4)=12-4=8$

Hence, the numerator and denominator of the given fraction are $8$ and $12$ respectively.