Question

Write the decimal as a fraction or mixed number.

$\mathbf{0}\mathbf{.}\overline{\mathbf{78}}$

Short answer

$\frac{\mathbf{78}}{\mathbf{99}}$ is the fraction number of decimal $\mathbf{0}\mathbf{.}\overline{\mathbf{78}}$

Step-by-step solution

Step 1. Given information

Consider, the decimal $0.\overline{78}$ provided in question

Step 2. Calculation 

To evaluate fraction number of decimal $0.\overline{78}$, it is needed to remove the bar as here two digits are repeating so multiply the number by 100

Therefore, the fraction number obtained as

$\begin{array}{c}x=0.\overline{78}\\ =\mathrm{0.7878...}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}equation}\left(1\right)\end{array}$

By multiplying 100 each sides

$\text{100}x=78.7878... \text{equation}\left(2\right)$

Subtract equation (1) from the equation (2).

$\begin{array}{c}\text{100}x-x=\mathrm{78.7878...}-\mathrm{.7878...}\\ 99x=78\\ x=0.\overline{78}\\ =\frac{78}{99}\end{array}$

Hence,

$\frac{78}{99}$is the fraction number of decimal $0.\overline{78}$