8.5 Decimals: converting a decimal to a fraction

Reading: six tenths→ $\frac{6}{10}$ .

Reduce: $\frac{3}{5}$ .

Reading: nine hundred three thousands→ $\frac{\text{903}}{\text{1000}}$ .

Reading: eighteen and sixty-one hundredths→ $\text{18}\frac{\text{61}}{\text{100}}$ .

Reading: five hundred eight and five ten thousandths→ $\text{508}\frac{5}{\text{10},\text{000}}$ .

Reduce: $\text{508}\frac{1}{\mathrm{2,}\text{000}}$ .

$0\text{.}\text{11}\frac{2}{3}$

The $\frac{2}{3}$ appears to occur in the thousands position, but it is referring to $\frac{2}{3}$ of a hundredth. So, we read $0\text{.}\text{11}\frac{2}{3}$ as "eleven and two-thirds hundredths."

$\begin{array}{ccc}0.11\frac{2}{3}=\frac{\text{11}\frac{2}{3}}{\text{100}}& =& \frac{\frac{\text{11}\cdot 3+2}{3}}{\text{100}}\\ & =& \frac{\frac{\text{35}}{3}}{\frac{\text{100}}{1}}\hfill \\ & =& \frac{\text{35}}{3}÷\frac{\text{100}}{1}\hfill \\ & =& \frac{\stackrel{7}{\cancel{35}}}{3}\cdot \frac{1}{\underset{\text{20}}{\cancel{100}}}\hfill \\ & =& \frac{7}{\text{60}}\hfill \end{array}$

$4\text{.}\text{006}\frac{1}{4}$

Note that $4\text{.}\text{006}\frac{1}{4}=4+\text{.}\text{006}\frac{1}{4}$

$\begin{array}{ccc}\hfill 4+.006\frac{1}{4}& =& 4+\frac{6\frac{1}{4}}{\text{1000}}\hfill \\ & =& 4+\frac{\frac{\text{25}}{4}}{\frac{\text{1000}}{1}}\hfill \\ & =& 4+\frac{\stackrel{1}{\cancel{25}}}{4}\cdot \frac{1}{\underset{40}{\cancel{1000}}}\hfill \\ & =& 4+\frac{1\cdot 1}{4\cdot \text{40}}\hfill \\ & =& 4+\frac{1}{\text{160}}\hfill \\ & =& 4\frac{1}{\text{160}}\hfill \end{array}$