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Fractions are one way we can represent parts of whole numbers. Decimal fractions are another way of representing parts of whole numbers.
A decimal fraction uses a
decimal point to separate whole parts and fractional parts. Whole parts are written to the
left of the decimal point and fractional parts are written to the
right of the decimal point. Just as each digit in a whole number has a particular value, so do the digits in decimal positions.
The following numbers are decimal fractions.
$$\begin{array}{l}57.9\\ \text{The\hspace{0.17em}9\hspace{0.17em}is\hspace{0.17em}in\hspace{0.17em}the}\text{\hspace{0.17em}}tenths\text{\hspace{0.17em}}\text{position}.\text{\hspace{0.17em}}57.9=57\frac{9}{10}.\end{array}$$
$$\begin{array}{l}6.8014\text{\hspace{0.17em}}\\ \text{The\hspace{0.17em}8\hspace{0.17em}is\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}}tenths\text{\hspace{0.17em}position}\text{.\hspace{0.17em}}\\ \text{The\hspace{0.17em}0\hspace{0.17em}is\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}}hundredths\text{\hspace{0.17em}position}\text{.\hspace{0.17em}}\\ \text{The\hspace{0.17em}1\hspace{0.17em}is\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}}thousandths\text{\hspace{0.17em}position}\text{.\hspace{0.17em}}\\ \text{The\hspace{0.17em}4\hspace{0.17em}is\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}ten\hspace{0.17em}}thousandths\text{\hspace{0.17em}position}\text{.\hspace{0.17em}}\\ 6.8014=6\frac{8014}{10000}.\end{array}$$
Find each sum or difference.
$$\begin{array}{l}9.183+2.140\\ \begin{array}{rrr}\text{\hspace{0.17em}}\downarrow \hfill & \hfill & \text{The\hspace{0.17em}decimal\hspace{0.17em}points\hspace{0.17em}are\hspace{0.17em}aligned\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}same\hspace{0.17em}column}\text{.\hspace{0.17em}}\hfill \\ \hfill \text{9}\text{.183}& \hfill & \hfill \\ \hfill \text{+}\underset{\xaf}{\text{\hspace{0.17em}2}\text{.140}}& \hfill & \hfill \\ \hfill \text{11}\text{.323}& \hfill & \hfill \end{array}\end{array}$$
$$\begin{array}{l}841.0056\text{\hspace{0.17em}}+\text{\hspace{0.17em}}47.016\text{\hspace{0.17em}}+\text{\hspace{0.17em}}19.058\text{\hspace{0.17em}}\\ \begin{array}{rrr}\hfill \downarrow & \hfill & \text{The\hspace{0.17em}decimal\hspace{0.17em}points\hspace{0.17em}are\hspace{0.17em}aligned\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}same\hspace{0.17em}column}\text{.\hspace{0.17em}}\hfill \\ \hfill 841.0056& \hfill & \hfill \\ \hfill 47.016& \hfill & \text{Place\hspace{0.17em}a\hspace{0.17em}0\hspace{0.17em}into\hspace{0.17em}the\hspace{0.17em}thousandths\hspace{0.17em}position}\text{.}\hfill \\ \hfill +\text{\hspace{0.17em}}\underset{\xaf}{19.058}& \hfill & \text{Place\hspace{0.17em}a\hspace{0.17em}0\hspace{0.17em}into\hspace{0.17em}the\hspace{0.17em}thousandths\hspace{0.17em}position}\text{.\hspace{0.17em}}\hfill \\ \hfill \downarrow & \hfill & \text{The\hspace{0.17em}decimal\hspace{0.17em}points\hspace{0.17em}are\hspace{0.17em}aligned\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}same\hspace{0.17em}column}\text{.\hspace{0.17em}}\hfill \\ \hfill 841.0056& \hfill & \hfill \\ \hfill 47.0160& \hfill & \hfill \\ \hfill +\text{\hspace{0.17em}}\underset{\xaf}{19.0580}& \hfill & \hfill \\ \hfill 907.0796& \hfill & \hfill \end{array}\end{array}$$
$$\begin{array}{l}16.01\text{\hspace{0.17em}}-\text{\hspace{0.17em}}7.053\\ \begin{array}{rrr}\hfill \downarrow & \hfill & \text{The\hspace{0.17em}decimal\hspace{0.17em}points\hspace{0.17em}are\hspace{0.17em}aligned\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}same\hspace{0.17em}column}\text{.\hspace{0.17em}}\hfill \\ \hfill 16.01& \hfill & \text{Place\hspace{0.17em}a\hspace{0.17em}0\hspace{0.17em}into\hspace{0.17em}the\hspace{0.17em}thousandths\hspace{0.17em}position}\text{.\hspace{0.17em}}\hfill \\ \hfill -\text{\hspace{0.17em}}\underset{\xaf}{7.053}& \hfill & \hfill \\ \hfill \downarrow & \hfill & \text{The\hspace{0.17em}decimal\hspace{0.17em}points\hspace{0.17em}are\hspace{0.17em}aligned\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}same\hspace{0.17em}column}\text{.\hspace{0.17em}}\hfill \\ \hfill 16.010& \hfill & \hfill \\ \hfill -\text{\hspace{0.17em}}\underset{\xaf}{7.053}& \hfill & \hfill \\ \hfill 8.957& \hfill & \hfill \end{array}\end{array}$$
Find the following products.
$6.5\times 4.3$
$6.5\times 4.3=27.95$
$23.4\times 1.96$
$23.4\times 1.96=45.864$
Find the following quotients.
$32.66\xf77.1$
$\begin{array}{l}32.66\xf77.1=4.6\\ \begin{array}{lll}Check:\hfill & \hfill & 32.66\xf77.1=4.6\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}4.6\times 7.1=32.66\hfill \\ \hfill \text{\hspace{0.17em}}4.6& \hfill & \hfill \\ \hfill \underset{\xaf}{\text{\hspace{0.17em}}7.1}& \hfill & \hfill \\ \hfill \text{\hspace{0.17em}}4.6& \hfill & \hfill \\ \underset{\xaf}{322\text{\hspace{0.17em}}\text{\hspace{0.17em}}}\hfill & \hfill & \hfill \\ 32.66\hfill & \hfill & \text{True}\hfill \end{array}\end{array}$
Check by multiplying
$2.1$ and
$\mathrm{0.513.}$ This will show that we have obtained the correct result.
$12\xf70.00032$
We can convert a decimal fraction to a fraction by reading it and then writing the phrase we have just read. As we read the decimal fraction, we note the place value farthest to the right. We may have to reduce the fraction.
Convert each decimal fraction to a fraction.
$$\begin{array}{l}0.6\\ 0.\underset{\xaf}{6}\to \text{tenths\hspace{0.17em}position}\\ \begin{array}{lll}\text{Reading:}\hfill & \hfill & \text{six\hspace{0.17em}tenths}\to \frac{6}{10}\hfill \\ \text{Reduce:}\hfill & \hfill & 0.6=\frac{6}{10}=\frac{3}{5}\hfill \end{array}\end{array}$$
$$\begin{array}{l}21.903\\ 21.90\underset{\xaf}{3}\to \text{thousandths\hspace{0.17em}position}\\ \begin{array}{ccc}\text{Reading:}& & \text{twenty-one\hspace{0.17em}and\hspace{0.17em}nine\hspace{0.17em}hundred\hspace{0.17em}three\hspace{0.17em}thousandths}\to 21\frac{903}{1000}\end{array}\end{array}$$
Convert the following fractions to decimals. If the division is nonterminating, round to 2 decimal places.
$\frac{3}{4}$
$\frac{3}{4}=0.75$
$\frac{1}{5}$
$\frac{1}{5}=0.2$
$\frac{5}{6}$
$\begin{array}{llll}\frac{5}{6}\hfill & =\hfill & \mathrm{0.833...}\hfill & \begin{array}{l}\\ \text{We\hspace{0.17em}are\hspace{0.17em}to\hspace{0.17em}round\hspace{0.17em}to\hspace{0.17em}2\hspace{0.17em}decimal\hspace{0.17em}places}.\text{\hspace{0.17em}}\end{array}\hfill \\ \frac{5}{6}\hfill & =\hfill & 0.83\text{\hspace{0.17em}to\hspace{0.17em}2\hspace{0.17em}decimal\hspace{0.17em}places}\text{.}\hfill & \hfill \end{array}$
$\begin{array}{l}5\frac{1}{8}\\ \text{Note\hspace{0.17em}that\hspace{0.17em}}5\frac{1}{8}=5+\frac{1}{8}.\end{array}$
$\begin{array}{l}\frac{1}{8}=.125\\ \text{Thus,\hspace{0.17em}}5\frac{1}{8}=5+\frac{1}{8}=5+.125=\mathrm{5.125.}\end{array}$
$0.16\frac{1}{4}$
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.”
$\begin{array}{lll}0.16\frac{1}{4}=\frac{16\frac{1}{4}}{100}=\frac{\frac{16\xb74+1}{4}}{100}\hfill & =\hfill & \frac{\frac{65}{4}}{\frac{100}{1}}\hfill \\ \hfill & =\hfill & \frac{\stackrel{13}{\overline{)65}}}{4}\xb7\frac{1}{\underset{20}{\overline{)100}}}=\frac{13\times 1}{4\times 20}=\frac{13}{80}\hfill \end{array}$
Now, convert
$\frac{13}{80}$ to a decimal.
$0.16\frac{1}{4}=\mathrm{0.1625.}$
For the following problems, perform each indicated operation.
$15.015-6.527$
$156.33-24.095$
$44.98+22.8-12.76$
$1.11+12.1212-13.131313$
$4.26\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3.2$
$13.632$
$2.97\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3.15$
$23.05\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}1.1$
$25.355$
$5.009\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}2.106$
$0.1\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3.24$
$0.324$
$100\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}12.008$
$1000\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}12.008$
$12,008$
$10,000\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}12.008$
$75.642\text{\hspace{0.17em}}\xf7\text{\hspace{0.17em}}18.01$
$4.2$
$51.811\text{\hspace{0.17em}}\xf7\text{\hspace{0.17em}}1.97$
$0.0000448\text{\hspace{0.17em}}\xf7\text{\hspace{0.17em}}0.014$
$0.0032$
$0.129516\text{\hspace{0.17em}}\xf7\text{\hspace{0.17em}}1004$
For the following problems, convert each decimal fraction to a fraction.
$48.1162$
For the following problems, convert each fraction to a decimal fraction. If the decimal form is nonterminating,round to 3 decimal places.
$\frac{5}{8}$
$15\text{\hspace{0.17em}}\xf7\text{\hspace{0.17em}}22$
$\frac{2}{9}$
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