Chapter 11, Problem 24CLR

To determine

To express: The $0.\overline{5}$ and $0.\overline{15}$ as fractions and find their sum.

Answer to Problem 24CLR

The $0.\overline{5}$ can be expressed in fraction as $\frac{5}{9}$ , $0.\overline{15}$ can be expressed in fraction as $\frac{5}{33}$ ,

The sum of both the numbers is $\frac{70}{99}$.

Explanation of Solution

Given information:

The numbers are $0.\overline{5}$ and $0.\overline{15}$ .

Calculation:

Consider the number $0.\overline{5}$ .

  $x= \mathrm{0.5555555...}$ ….. (1)

Multiply equation (1) by $10$ .

  $10x= \mathrm{5.5555...}$ …. (2)

Subtract (1) from (2).

  $\begin{array}{c}9x=5\\ x=\frac{5}{9}\end{array}$ .

Consider the number $0.\overline{15}$ .

  $x= \mathrm{0.15151515...}$ ….. (3)

Multiply equation (3) by $100$ .

  $100x=\mathrm{15.15151515...}$ …. (4)

Subtract (1) from (2).

  $\begin{array}{c}99x=15\\ x=\frac{15}{99}\\ x=\frac{5}{33}\end{array}$

Add both the fractions.

  $\begin{array}{c}\text{sum}=\frac{5}{9}+\frac{5}{33}\\ =\frac{55+15}{99}\\ =\frac{70}{99}\end{array}$

Therefore, the $0.\overline{5}$ can be expressed in fraction as $\frac{5}{9}$ , $0.\overline{15}$ can be expressed in fraction as $\frac{5}{33}$ , the sum of both the numbers is $\frac{70}{99}$.