Question

Express as a fraction in the simplest form $2.2\overline{4}$

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{x}=2.24444.....................(\mathrm{equation}1) \\ \Rightarrow 10\mathrm{x}=22.4444......................\left(\mathrm{equation}2\right)\left(\mathrm{multiply}\mathrm{by}10\mathrm{in}\mathrm{equation}1\right) \\ \Rightarrow 100\mathrm{x}=224.4444.....................(\mathrm{equation}3)(\mathrm{multiply}\mathrm{by}10\mathrm{in}\mathrm{equation}2) \\ \mathrm{Substracting}\mathrm{equation}\left(2\right)\mathrm{from}\mathrm{equation}\left(3\right), \\ \Rightarrow 100\mathrm{x}-10\mathrm{x}=224.4444......-22.4444..... \\ \Rightarrow 90\mathrm{x}=202 \\ \Rightarrow \mathrm{x}=\frac{202}{90}\left(\mathrm{Divide}\mathrm{by}2\mathrm{in}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\right) \\ \Rightarrow \mathrm{x}\mathbf{=}\frac{\mathbf{101}}{\mathbf{45}}\mathbf{}\mathbf{}\mathbf{} \end{gathered}$$

Final answer:

Hence, the required fraction form of $2.2\overline{4}$$\mathbf{=}\frac{\mathbf{101}}{\mathbf{45}}$