1.6 Decimal fractions

$$\begin{array}{l}57.9\\ \text{The\hspace{0.17em}9\hspace{0.17em}is\hspace{0.17em}in\hspace{0.17em}the}tenths\text{position}.57.9=57\frac{9}{10}.\end{array}$$

$$\begin{array}{l}6.8014\\ \text{The\hspace{0.17em}8\hspace{0.17em}is\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}}tenths\text{\hspace{0.17em}position}\text{.\hspace{0.17em}}\\ \text{The\hspace{0.17em}0\hspace{0.17em}is\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}}hundredths\text{\hspace{0.17em}position}\text{.\hspace{0.17em}}\\ \text{The\hspace{0.17em}1\hspace{0.17em}is\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}}thousandths\text{\hspace{0.17em}position}\text{.\hspace{0.17em}}\\ \text{The\hspace{0.17em}4\hspace{0.17em}is\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}ten\hspace{0.17em}}thousandths\text{\hspace{0.17em}position}\text{.\hspace{0.17em}}\\ 6.8014=6\frac{8014}{10000}.\end{array}$$

$$\begin{array}{l}9.183+2.140\\ \begin{array}{rrr}\downarrow \hfill & \hfill & \text{The\hspace{0.17em}decimal\hspace{0.17em}points\hspace{0.17em}are\hspace{0.17em}aligned\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}same\hspace{0.17em}column}\text{.\hspace{0.17em}}\hfill \\ \hfill \text{9}\text{.183}& \hfill & \hfill \\ \hfill \text{+}\underset{¯}{\text{\hspace{0.17em}2}\text{.140}}& \hfill & \hfill \\ \hfill \text{11}\text{.323}& \hfill & \hfill \end{array}\end{array}$$

$$\begin{array}{l}841.0056+47.016+19.058\\ \begin{array}{rrr}\hfill \downarrow & \hfill & \text{The\hspace{0.17em}decimal\hspace{0.17em}points\hspace{0.17em}are\hspace{0.17em}aligned\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}same\hspace{0.17em}column}\text{.\hspace{0.17em}}\hfill \\ \hfill 841.0056& \hfill & \hfill \\ \hfill 47.016& \hfill & \text{Place\hspace{0.17em}a\hspace{0.17em}0\hspace{0.17em}into\hspace{0.17em}the\hspace{0.17em}thousandths\hspace{0.17em}position}\text{.}\hfill \\ \hfill +\underset{¯}{19.058}& \hfill & \text{Place\hspace{0.17em}a\hspace{0.17em}0\hspace{0.17em}into\hspace{0.17em}the\hspace{0.17em}thousandths\hspace{0.17em}position}\text{.\hspace{0.17em}}\hfill \\ \hfill \downarrow & \hfill & \text{The\hspace{0.17em}decimal\hspace{0.17em}points\hspace{0.17em}are\hspace{0.17em}aligned\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}same\hspace{0.17em}column}\text{.\hspace{0.17em}}\hfill \\ \hfill 841.0056& \hfill & \hfill \\ \hfill 47.0160& \hfill & \hfill \\ \hfill +\underset{¯}{19.0580}& \hfill & \hfill \\ \hfill 907.0796& \hfill & \hfill \end{array}\end{array}$$

$$\begin{array}{l}16.01-7.053\\ \begin{array}{rrr}\hfill \downarrow & \hfill & \text{The\hspace{0.17em}decimal\hspace{0.17em}points\hspace{0.17em}are\hspace{0.17em}aligned\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}same\hspace{0.17em}column}\text{.\hspace{0.17em}}\hfill \\ \hfill 16.01& \hfill & \text{Place\hspace{0.17em}a\hspace{0.17em}0\hspace{0.17em}into\hspace{0.17em}the\hspace{0.17em}thousandths\hspace{0.17em}position}\text{.\hspace{0.17em}}\hfill \\ \hfill -\underset{¯}{7.053}& \hfill & \hfill \\ \hfill \downarrow & \hfill & \text{The\hspace{0.17em}decimal\hspace{0.17em}points\hspace{0.17em}are\hspace{0.17em}aligned\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}same\hspace{0.17em}column}\text{.\hspace{0.17em}}\hfill \\ \hfill 16.010& \hfill & \hfill \\ \hfill -\underset{¯}{7.053}& \hfill & \hfill \\ \hfill 8.957& \hfill & \hfill \end{array}\end{array}$$

$6.5\times 4.3$



$6.5\times 4.3=27.95$

$23.4\times 1.96$



$23.4\times 1.96=45.864$

$32.66÷7.1$



$\begin{array}{l}32.66÷7.1=4.6\\ \begin{array}{lll}Check:\hfill & \hfill & 32.66÷7.1=4.6\text{if}4.6\times 7.1=32.66\hfill \\ \hfill 4.6& \hfill & \hfill \\ \hfill \underset{¯}{7.1}& \hfill & \hfill \\ \hfill 4.6& \hfill & \hfill \\ \underset{¯}{322}\hfill & \hfill & \hfill \\ 32.66\hfill & \hfill & \text{True}\hfill \end{array}\end{array}$

Check by multiplying $2.1$ and $\mathrm{0.513.}$ This will show that we have obtained the correct result.

$12÷0.00032$

$$\begin{array}{l}0.6\\ 0.\underset{¯}{6}\to \text{tenths\hspace{0.17em}position}\\ \begin{array}{lll}\text{Reading:}\hfill & \hfill & \text{six\hspace{0.17em}tenths}\to \frac{6}{10}\hfill \\ \text{Reduce:}\hfill & \hfill & 0.6=\frac{6}{10}=\frac{3}{5}\hfill \end{array}\end{array}$$

$$\begin{array}{l}21.903\\ 21.90\underset{¯}{3}\to \text{thousandths\hspace{0.17em}position}\\ \begin{array}{ccc}\text{Reading:}& & \text{twenty-one\hspace{0.17em}and\hspace{0.17em}nine\hspace{0.17em}hundred\hspace{0.17em}three\hspace{0.17em}thousandths}\to 21\frac{903}{1000}\end{array}\end{array}$$

$\frac{3}{4}$



$\frac{3}{4}=0.75$

$\frac{1}{5}$



$\frac{1}{5}=0.2$

$\frac{5}{6}$



$\begin{array}{llll}\frac{5}{6}\hfill & =\hfill & \mathrm{0.833...}\hfill & \begin{array}{l}\\ \text{We\hspace{0.17em}are\hspace{0.17em}to\hspace{0.17em}round\hspace{0.17em}to\hspace{0.17em}2\hspace{0.17em}decimal\hspace{0.17em}places}.\end{array}\hfill \\ \frac{5}{6}\hfill & =\hfill & 0.83\text{\hspace{0.17em}to\hspace{0.17em}2\hspace{0.17em}decimal\hspace{0.17em}places}\text{.}\hfill & \hfill \end{array}$

$\begin{array}{l}5\frac{1}{8}\\ \text{Note\hspace{0.17em}that\hspace{0.17em}}5\frac{1}{8}=5+\frac{1}{8}.\end{array}$



$\begin{array}{l}\frac{1}{8}=.125\\ \text{Thus,\hspace{0.17em}}5\frac{1}{8}=5+\frac{1}{8}=5+.125=\mathrm{5.125.}\end{array}$

$0.16\frac{1}{4}$

This is a complex decimal. The “6” is in the hundredths position. The number $0.16\frac{1}{4}$ is read as “sixteen and one-fourth hundredths.”

$\begin{array}{lll}0.16\frac{1}{4}=\frac{16\frac{1}{4}}{100}=\frac{\frac{16·4+1}{4}}{100}\hfill & =\hfill & \frac{\frac{65}{4}}{\frac{100}{1}}\hfill \\ \hfill & =\hfill & \frac{\stackrel{13}{\cancel{65}}}{4}·\frac{1}{\underset{20}{\cancel{100}}}=\frac{13\times 1}{4\times 20}=\frac{13}{80}\hfill \end{array}$

Now, convert $\frac{13}{80}$ to a decimal.



$0.16\frac{1}{4}=\mathrm{0.1625.}$