Chapter 11.2, Problem 41WE

a.

To determine

To express: The given fractions as the repeating decimals.

Answer to Problem 41WE

The given fractions as the repeating decimals are $0.\overline{1},0.\overline{5},\text{and 0}\text{.<mover accent="true"><mo>8</mo><mo>¯</mo></mover>}$ .

Explanation of Solution

Given information:

The given fractions are $\frac{1}{9},\frac{5}{9},\text{and }\frac{8}{9}$ .

The given fractions as the repeating decimals is expressed as,

  $\begin{array}{c}\frac{1}{9}=\mathrm{0.1111...}\\ =0.\overline{1}\end{array}$

The given fractions as the repeating decimals is expressed as,

  $\begin{array}{c}\frac{5}{9}=\mathrm{0.5555...}\\ =0.\overline{5}\end{array}$

The given fractions as the repeating decimals is expressed as,

  $\begin{array}{c}\frac{8}{9}=\mathrm{0.8888...}\\ =0.\overline{8}\end{array}$

Thus, the given fractions as the repeating decimals are $0.\overline{1},0.\overline{5},\text{and 0}\text{.<mover accent="true"><mo>8</mo><mo>¯</mo></mover>}$ .

b.

To determine

To express: The given fractions as the repeating decimals.

Answer to Problem 41WE

The given fractions as the repeating decimals are $0.\overline{037},0.\overline{185},\text{and }0.\overline{296}$ .

Explanation of Solution

Given information:

The given fractions are $\frac{1}{27},\frac{5}{27},\text{and }\frac{8}{27}$ .

The given fractions as the repeating decimals is expressed as,

  $\begin{array}{c}\frac{1}{27}=0.037037\\ =0.\overline{037}\end{array}$

The given fractions as the repeating decimals is expressed as,

  $\begin{array}{c}\frac{5}{27}=\mathrm{0.185185...}\\ =0.\overline{185}\end{array}$

The given fractions as the repeating decimals is expressed as,

  $\begin{array}{c}\frac{8}{27}=\mathrm{0.296296...}\\ =0.\overline{296}\end{array}$

Thus, the given fractions as the repeating decimals are $0.\overline{037},0.\overline{185},\text{and }0.\overline{296}$ .

c.

To determine

To find: The relation between the given numbers.

Explanation of Solution

Given information:

The given fractions are $\frac{1}{9},\frac{5}{9},\text{and }\frac{8}{9}$ and in part b , the given fractions are $\frac{1}{27},\frac{5}{27},\text{and }\frac{8}{27}$ .

The relation between the given numbers is expressed as,

  $\frac{1}{3}\left(0.\overline{1}\right)=0.\overline{037}$

The relation between the given numbers is expressed as,

  $\frac{1}{3}\left(0.\overline{5}\right)=0.\overline{185}$

The relation between the given numbers is expressed as,

  $\frac{1}{3}\left(0.\overline{8}\right)=0.\overline{296}$