Question

Express each repeating decimal as a fraction.

$\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{5}\mathbf{.}\overline{23}$

$\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{1}.3\overline{7}$

$\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{2}.1\overline{35}$

Short answer

1. The fraction of repeating decimal is 518/99
2. The fraction of repeating decimal is 124/90
3. The fraction of repeating decimal is 2114/990

Step-by-step solution

. Apply the concept of eliminating repeating decimals

A repeating decimal such as is a rational number. To convert it to a ratio of two integers, we write

$\begin{array}{l}1000x=3547.\overline{47}\\ 10x=35.\overline{47}\end{array}$

Subtracting both, we have:

$\begin{array}{l}990x=3512.0\\ \Rightarrow x=\frac{3512}{990}\end{array}$

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

. Analyze repeating part

Given$\text{5.}\overline{23}$

Letrole="math" $x=\text{5.}\overline{23}\mathrm{.............}\left(1\right)$

Multiply it by 100 to get$100x=\text{523.}\overline{23}\mathrm{...........}\left(2\right)$

. Express repeating decimal as fraction

Subtract (1) from (2) to get:

$\begin{array}{l}99x=518\\ \Rightarrow x=\frac{518}{99}\\ \Rightarrow \text{5.}\overline{23}=\frac{518}{99}\end{array}$

. Apply the concept of eliminating repeating decimals

A repeating decimal such as$x=3.5\overline{47}$is a rational number. To convert it to a ratio of two integers, we write

$\begin{array}{l}1000x=3547.\overline{47}\\ 10x=35.\overline{47}\end{array}$

Subtracting both, we have:

role="math" $\begin{array}{l}990x=3512.0\\ \Rightarrow x=\frac{3512}{990}\end{array}$

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

. Analyze repeating part

Given$\text{1.3}\overline{7}$

Let$x=\text{1.3}\overline{7}$

Multiply it by 10 to get$10x=\text{13.}\overline{7}\mathrm{..............}\left(1\right)$

Multiply it by 100 to get$100x=\text{137.}\overline{7}\mathrm{...........}\left(2\right)$

. Express repeating decimal as fraction

Subtract (1) from (2) to get:

$\begin{array}{l}90x=124\\ \Rightarrow x=\frac{124}{90}\\ \Rightarrow \text{1.3}\overline{7}=\frac{124}{90}\end{array}$

. Apply the concept of eliminating repeating decimals

A repeating decimal such as$x=3.5\overline{47}$is a rational number. To convert it to a ratio of two integers, we write

$\begin{array}{l}1000x=3547.\overline{47}\\ 10x=35.\overline{47}\end{array}$

Subtracting both, we have:

$\begin{array}{l}990x=3512.0\\ \Rightarrow x=\frac{3512}{990}\end{array}$

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

. Analyze repeating part

Given$\text{2.1}\overline{35}$

Let$x=\text{2.1}\overline{35}$

Multiply it by 10 to get$10x=\text{21.}\overline{35}\mathrm{..............}\left(1\right)$

Multiply it by 1000 to get$1000x=\text{2135.}\overline{35}\mathrm{...........}\left(2\right)$

. Express repeating decimal as fraction

Subtract (1) from (2) to get:

$\begin{array}{l}990x=2114\\ \Rightarrow x=\frac{2114}{990}\\ \Rightarrow \text{2.1}\overline{35}=\frac{2114}{990}\end{array}$