Chapter 5.1, Problem 17E

To determine

To find: fraction or mixed number of decimal $2.\overline{6}$

Answer to Problem 17E

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

Explanation of Solution

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

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

  $\begin{array}{l}\text{let}x=2.\overline{6}=\mathrm{2.66...}\text{equation(1)}\\ \text{by}\text{multiplying}\text{10}\text{each}\text{sides}\\ \Rightarrow \text{10}x=\mathrm{26.66...}\text{equation(2)}\\ \text{subtract}\text{equation(1)}\text{from}\text{equation(2)}\\ \Rightarrow \text{10}x-x=\mathrm{26.66...}-\mathrm{2.66...}\\ \Rightarrow 9x=24\\ \Rightarrow x=\frac{24}{9}\\ \Rightarrow x=\frac{8}{3}=2\frac{2}{3}\\ \Rightarrow x=2.\overline{6}=2\frac{2}{3}\end{array}$

Hence,

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