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The following pairs of numbers are reciprocals.
$\begin{array}{c}\underbrace{\frac{3}{4}\text{and}\frac{4}{3}}\\ \frac{3}{4}\cdot \frac{4}{3}=1\end{array}$
$\begin{array}{c}\underbrace{\frac{7}{16}\text{and}\frac{16}{7}}\\ \frac{7}{16}\cdot \frac{16}{7}=1\end{array}$
$\begin{array}{c}\underbrace{\frac{1}{6}\text{and}\frac{6}{1}}\\ \frac{1}{6}\cdot \frac{6}{1}=1\end{array}$
Notice that we can find the reciprocal of a nonzero number in fractional form by inverting it (exchanging positions of the numerator and denominator).
Find the reciprocal of each number.
$\frac{3}{\text{10}}$
$\frac{\text{10}}{3}$
$\frac{2}{3}$
$\frac{3}{2}$
$\frac{7}{8}$
$\frac{8}{7}$
$\frac{1}{5}$
5
$2\frac{2}{7}$
$\frac{7}{\text{16}}$
$5\frac{1}{4}$
$\frac{4}{\text{21}}$
$\text{10}\frac{3}{\text{16}}$
$\frac{\text{16}}{\text{163}}$
Our concept of division is that it indicates how many times one quantity is contained in another quantity. For example, using the diagram we can see that there are 6 one-thirds in 2.
There are 6 one-thirds in 2.
Since 2 contains six $\frac{1}{3}$ 's we express this as
Using these observations, we can suggest the following method for dividing a number by a fraction.
Perform the following divisions.
$\frac{1}{3}\xf7\frac{3}{4}$ . The divisor is $\frac{3}{4}$ . Its reciprocal is $\frac{4}{3}$ . Multiply $\frac{1}{3}$ by $\frac{4}{3}$ .
$\frac{1}{3}\cdot \frac{4}{3}=\frac{1\cdot 4}{3\cdot 3}=\frac{4}{9}$
$\frac{1}{3}\xf7\frac{3}{4}=\frac{4}{9}$
$\frac{3}{8}\xf7\frac{5}{4}$ The divisor is $\frac{5}{4}$ . Its reciprocal is $\frac{4}{5}$ . Multiply $\frac{3}{8}$ by $\frac{4}{5}$ .
$\frac{3}{\underset{2}{\overline{)3}}}\cdot \frac{\stackrel{1}{\overline{)4}}}{5}=\frac{3\cdot 1}{2\cdot 5}=\frac{3}{\text{10}}$
$\frac{3}{8}\xf7\frac{5}{4}=\frac{3}{\text{10}}$
$\frac{5}{6}\xf7\frac{5}{\text{12}}$ . The divisor is $\frac{5}{\text{12}}$ . Its reciprocal is $\frac{\text{12}}{5}$ . Multiply $\frac{5}{6}$ by $\frac{\text{12}}{5}$ .
$\frac{\stackrel{1}{\overline{)5}}}{\underset{1}{\overline{)6}}}\cdot \frac{\stackrel{2}{\overline{)\text{12}}}}{\underset{1}{\overline{)5}}}=\frac{1\cdot 2}{1\cdot 1}=\frac{2}{1}=2$
$\frac{5}{6}\xf7\frac{5}{12}=2$
$2\frac{2}{9}\xf73\frac{1}{3}$ . Convert each mixed number to an improper fraction.
$2\frac{2}{9}=\frac{9\cdot 2+2}{9}=\frac{\text{20}}{9}$ .
$3\frac{1}{3}=\frac{3\cdot 3+1}{3}=\frac{10}{3}$ .
$\frac{\text{20}}{9}\xf7\frac{\text{10}}{3}$ The divisor is $\frac{\text{10}}{3}$ . Its reciprocal is $\frac{3}{\text{10}}$ . Multiply $\frac{\text{20}}{9}$ by $\frac{3}{\text{10}}$ .
$\frac{\stackrel{2}{\overline{)20}}}{\underset{3}{\overline{)9}}}\cdot \frac{\stackrel{1}{\overline{)3}}}{\underset{1}{\overline{)10}}}=\frac{2\cdot 1}{3\cdot 1}=\frac{2}{3}$
$2\frac{2}{9}\xf73\frac{1}{3}=\frac{2}{3}$
$\frac{\text{12}}{\text{11}}\xf78$ . First conveniently write 8 as $\frac{8}{1}$ .
$\frac{\text{12}}{\text{11}}\xf7\frac{8}{1}$ The divisor is $\frac{8}{1}$ . Its reciprocal is $\frac{1}{8}$ . Multiply $\frac{\text{12}}{\text{11}}$ by $\frac{1}{8}$ .
$\frac{\stackrel{3}{\overline{)12}}}{\text{11}}\cdot \frac{1}{\underset{2}{\overline{)8}}}=\frac{3\cdot 1}{\text{11}\cdot 2}=\frac{3}{\text{22}}$
$\frac{\text{12}}{\text{11}}\xf78=\frac{3}{\text{22}}$
$\frac{7}{8}\xf7\frac{\text{21}}{\text{20}}\cdot \frac{3}{\text{35}}$ . The divisor is $\frac{\text{21}}{\text{20}}$ . Its reciprocal is $\frac{\text{20}}{\text{21}}$ .
$\frac{\stackrel{1}{\overline{)7}}}{\underset{2}{\overline{)8}}}\cdot \frac{\stackrel{\stackrel{1}{\overline{)5}}}{\overline{)\text{20}}}}{\underset{\underset{1}{\overline{)3}}}{\overline{)\text{21}}}}\frac{\stackrel{1}{\overline{)3}}}{\underset{7}{\overline{)\text{35}}}}=\frac{1\cdot 1\cdot 1}{2\cdot 1\cdot 7}=\frac{1}{\text{14}}$
$\frac{7}{8}\xf7\frac{\text{21}}{\text{20}}\cdot \frac{3}{\text{25}}=\frac{1}{\text{14}}$
How many $2\frac{3}{8}$ -inch-wide packages can be placed in a box 19 inches wide?
The problem is to determine how many two and three eighths are contained in 19, that is, what is $\text{19}\xf72\frac{3}{8}$ ?
$2\frac{3}{8}=\frac{\text{19}}{8}$ Convert the divisor $2\frac{3}{8}$ to an improper fraction.
$\text{19}=\frac{\text{19}}{1}$ Write the dividend 19 as $\frac{\text{19}}{1}$ .
$\frac{\text{19}}{1}\xf7\frac{\text{19}}{8}$ The divisor is $\frac{\text{19}}{8}$ . Its reciprocal is $\frac{8}{\text{19}}$ .
$\frac{\stackrel{1}{\overline{)\text{19}}}}{1}\cdot \frac{8}{\underset{1}{\overline{)\text{19}}}}=\frac{1\cdot 8}{1\cdot 1}=\frac{8}{1}=8$
Thus, 8 packages will fit into the box.
Perform the following divisions.
$\frac{1}{2}\xf7\frac{9}{8}$
$\frac{4}{9}$
$\frac{3}{8}\xf7\frac{9}{\text{24}}$
1
$\frac{7}{\text{15}}\xf7\frac{\text{14}}{\text{15}}$
$\frac{1}{2}$
$8\xf7\frac{8}{\text{15}}$
15
$6\frac{1}{4}\xf7\frac{5}{\text{12}}$
15
$3\frac{1}{3}\xf71\frac{2}{3}$
2
$\frac{5}{6}\xf7\frac{2}{3}\cdot \frac{8}{\text{25}}$
$\frac{2}{5}$
A container will hold 106 ounces of grape juice. How many $6\frac{5}{8}$ -ounce glasses of grape juice can be served from this container?
16 glasses
Determine each of the following quotients and then write a rule for this type of division.
$1\xf7\frac{2}{3}$
$\frac{3}{2}$
$1\xf7\frac{3}{8}$
$\frac{8}{3}$
$1\xf7\frac{3}{4}$
$\frac{4}{3}$
$1\xf7\frac{5}{2}$
$\frac{2}{5}$
When dividing 1 by a fraction, the quotient is the
is the reciprocal of the fraction.
For the following problems, find the reciprocal of each number.
$\frac{4}{5}$
$\frac{5}{4}$ or $1\frac{1}{4}$
$\frac{8}{\text{11}}$
$\frac{2}{9}$
$\frac{9}{2}$ or $4\frac{1}{2}$
$\frac{1}{5}$
$3\frac{1}{4}$
$\frac{4}{\text{13}}$
$8\frac{1}{4}$
$3\frac{2}{7}$
$\frac{7}{\text{23}}$
$5\frac{3}{4}$
1
1
4
For the following problems, find each value.
$\frac{3}{8}\xf7\frac{3}{5}$
$\frac{5}{8}$
$\frac{5}{9}\xf7\frac{5}{6}$
$\frac{9}{\text{16}}\xf7\frac{\text{15}}{8}$
$\frac{3}{\text{10}}$
$\frac{4}{9}\xf7\frac{6}{\text{15}}$
$\frac{\text{25}}{\text{49}}\xf7\frac{4}{9}$
$\frac{\text{225}}{\text{196}}$ or $1\frac{\text{29}}{\text{196}}$
$\frac{\text{15}}{4}\xf7\frac{\text{27}}{8}$
$\frac{\text{24}}{\text{75}}\xf7\frac{8}{\text{15}}$
$\frac{3}{5}$
$\frac{5}{7}\xf70$
$\frac{7}{8}\xf7\frac{7}{8}$
1
$0\xf7\frac{3}{5}$
$\frac{4}{\text{11}}\xf7\frac{4}{\text{11}}$
1
$\frac{2}{3}\xf7\frac{2}{3}$
$\frac{7}{\text{10}}\xf7\frac{\text{10}}{7}$
$\frac{\text{49}}{\text{100}}$
$\frac{3}{4}\xf76$
$\frac{9}{5}\xf73$
$\frac{3}{5}$
$4\frac{1}{6}\xf73\frac{1}{3}$
$7\frac{1}{7}\xf78\frac{1}{3}$
$\frac{6}{7}$
$1\frac{1}{2}\xf71\frac{1}{5}$
$3\frac{2}{5}\xf7\frac{6}{\text{25}}$
$\frac{\text{85}}{6}$ or $\text{14}\frac{1}{6}$
$5\frac{1}{6}\xf7\frac{\text{31}}{6}$
$\frac{\text{35}}{6}\xf73\frac{3}{4}$
$\frac{\text{28}}{\text{18}}=\frac{\text{14}}{9}$ or $1\frac{5}{9}$
$5\frac{1}{9}\xf7\frac{1}{\text{18}}$
$8\frac{3}{4}\xf7\frac{7}{8}$
10
$\frac{\text{12}}{8}\xf71\frac{1}{2}$
$3\frac{1}{8}\xf7\frac{\text{15}}{\text{16}}$
$\frac{\text{10}}{3}$ or $3\frac{1}{3}$
$\text{11}\frac{\text{11}}{\text{12}}\xf79\frac{5}{8}$
$2\frac{2}{9}\xf7\text{11}\frac{2}{3}$
$\frac{4}{\text{21}}$
$\frac{\text{16}}{3}\xf76\frac{2}{5}$
$4\frac{3}{\text{25}}\xf72\frac{\text{56}}{\text{75}}$
$\frac{3}{2}$ or $1\frac{1}{2}$
$\frac{1}{\text{1000}}\xf7\frac{1}{\text{100}}$
$\frac{3}{8}\xf7\frac{9}{\text{16}}\cdot \frac{6}{5}$
$\frac{4}{5}$
$\frac{3}{\text{16}}\cdot \frac{9}{8}\cdot \frac{6}{5}$
$\frac{4}{\text{15}}\xf7\frac{2}{\text{25}}\cdot \frac{9}{\text{10}}$
3
$\frac{\text{21}}{\text{30}}\cdot 1\frac{1}{4}\xf7\frac{9}{\text{10}}$
$8\frac{1}{3}\cdot \frac{\text{36}}{\text{75}}\xf74$
1
( [link] ) What is the value of 5 in the number 504,216?
( [link] ) Find the product of 2,010 and 160.
321,600
( [link] ) Use the numbers 8 and 5 to illustrate the commutative property of multiplication.
( [link] ) Find the least common multiple of 6, 16, and 72.
144
( [link] ) Find $\frac{8}{9}$ of $6\frac{3}{4}$ .
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