Chapter 1.1, Problem 75E

(a)

To determine

To express: $0.\overline{7}$ repeating decimal as a fraction.

Answer to Problem 75E

  $0.\overline{7}=\frac{7}{9}$

Explanation of Solution

Given information:

  $0.\overline{7}$

Calculation:

The given repeating decimal is $0.\overline{7}$

A repeating decimal such as $x=0.777777$ ……………is a rational number

To convert it to a ration of two integers, we write

  $\begin{array}{c}\underset{\_}{\begin{array}{c}100x=77.7777\\ -10x=7.7777\end{array}}\\ 90x=70\end{array}$

Thus, $x=\frac{70}{90}=\frac{7}{9}$

The idea is to multiply $x$ by the appropriate powers of 10, and then subtract to eliminate the repeating part.

(b)

To determine

To express: $0.2\overline{8}$ repeating decimal as a fraction.

Answer to Problem 75E

  $0.2\overline{8}=\frac{16}{45}$

Explanation of Solution

Given information:

  $0.2\overline{8}$

Calculation:

The given repeating decimal is $0.2\overline{8}$

A repeating decimal such as $x=0.288888$ ……………is a rational number

To convert it to a ration of two integers, we write

  $\begin{array}{c}\underset{\_}{\begin{array}{c}100x=28.8888\\ -10x=2.8888\end{array}}\\ 90x=26\\ x=\frac{26}{90}\\ =\frac{16}{45}\end{array}$

(c)

To determine

To express: $0.\overline{57}$ repeating decimal as a fraction.

Answer to Problem 75E

  $0.\overline{57}=\frac{19}{33}$

Explanation of Solution

Given information:

  $0.\overline{57}$

Calculation:

The given repeating decimal is $0.\overline{57}$

A repeating decimal such as $x=0.575757$ ……………is a rational number

To convert it to a ration of two integers, we write

  $\begin{array}{c}\underset{\_}{\begin{array}{c}10000x=5757.57\\ -100x=57.57\end{array}}\\ 9900x=5700\\ x=\frac{57}{99}\\ =\frac{19}{33}\end{array}$