Question

Write the decimal as a fraction or mixed number.

$\mathbf{2}\mathbf{.}\overline{\mathbf{6}}$

Short answer

$\mathbf{2}\frac{\mathbf{2}}{\mathbf{3}}$ is the fraction number of decimal $\mathbf{2}\mathbf{.}\overline{\mathbf{6}}$.

Step-by-step solution

Step 1. Given information

Consider, the decimal $2.\overline{6}$ provided in question

Step 2. Calculation 

To evaluate fraction number of decimal $2.\overline{6}$, it is needed to remove the bar as here one digit is repeating so multiply the number by 10

Therefore, the fraction number obtained as follows:

$\begin{array}{c}x=2.\overline{6}\\ =\mathrm{2.66...}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}equation(1)\hspace{0.17em}}\end{array}$

By multiplying 10 each sides.

$\text{10}x=26.66... \text{equation(2)}$

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

$\begin{array}{c}\text{10}x-x=\mathrm{26.66...}-\mathrm{2.66...}\\ 9x=24\\ x=\frac{24}{9}\end{array}$

Simplify the fraction into mixed fraction and in decimal as follows:

$\begin{array}{c}\frac{24}{9}=\frac{8}{3}\\ =2.\overline{6}\\ =2\frac{2}{3}\end{array}$

Hence,

$2\frac{2}{3}$ is the mixed number of decimal $2.\overline{6}$