1.6: Operations with Fractions
Overview
- Multiplication of Fractions
- Division of Fractions
- Addition and Subtraction of Fractions
Multiplication of Fractions
To multiply two fractions, multiply the numerators together and multiply the denominators together. Reduce to lowest terms if possible.
For example, multiply \(\dfrac{3}{4} \cdot \dfrac{1}{6}\):
\(
\begin{aligned}
&\begin{aligned}\dfrac{3}{4} \cdot \dfrac{1}{6} &=\dfrac{3 \cdot 1}{4 \cdot 6} \\
&=\dfrac{3}{24} \quad \text { Now reduce. } \\
&=\dfrac{3 \cdot 1}{2 \cdot 2 \cdot 2 \cdot 3}
\end{aligned}\\
&=\dfrac{\not{3} \cdot 1}{2 \cdot 2 \cdot 2 \cdot \not 3}\\
&\text { 3 is the only common factor. }\\
&=\dfrac{1}{8}
\end{aligned}
\)
Notice that we since had to reduce, we nearly started over again with the original two fractions. If we factor first, then cancel, then multiply, we will save time and energy and still obtain the correct product.
Sample Set A
Perform the following multiplications:
\(
\begin{aligned}
&\dfrac{1}{4} \cdot \dfrac{8}{9}=\dfrac{1}{2 \cdot 2} \cdot \dfrac{2 \cdot 2 \cdot 2}{3 \cdot 3}\\&\begin{array}{l}
=\dfrac{1}{\not 2 \cdot \not 2} \cdot \dfrac{\not 2 \cdot \not 2 \cdot 2}{3 \cdot 3} \text{ 2 is a common factor} \\
=\dfrac{1}{1} \cdot \dfrac{2}{3 \cdot 3} \\
=\dfrac{1 \cdot 2}{1 \cdot 3 \cdot 3} \\
=\dfrac{2}{9}
\end{array}\\
\end{aligned}
\)
\(
\begin{aligned}
\dfrac{3}{4} \cdot \dfrac{8}{9} \cdot \dfrac{5}{12} &=\dfrac{3}{2 \cdot 2} \cdot \dfrac{2 \cdot 2 \cdot 2}{3 \cdot 3} \cdot \dfrac{5}{2 \cdot 2 \cdot 3} \\
&=\dfrac{\not{3}}{\not 2 \cdot \not 2} \cdot \dfrac{\not 2 \cdot \not 2 \cdot \not 2}{\not{3} \cdot 3} \cdot \dfrac{5}{\not 2 \cdot 2 \cdot 3} \quad 2 \text { and } 3 \text { are common factors. } \\
&=\dfrac{1 \cdot 1 \cdot 5}{3 \cdot 2 \cdot 3} \\
&=\dfrac{5}{18}
\end{aligned}
\)
Division of Fractions
Reciprocals
Two numbers whose product is 1 are reciprocals of each other. For example, since \(\dfrac{4}{5} \cdot \dfrac{5}{4}=1, \dfrac{4}{5} \text { and } \dfrac{5}{4}\) are reciprocals of each other. Some other pairs of reciprocals are listed below.
\( \dfrac{2}{7}, \dfrac{7}{2} \quad \dfrac{3}{4}, \dfrac{4}{3} \quad \dfrac{6}{1}, \dfrac{1}{6} \)
Reciprocals are used in division of fractions.
To divide a first fraction by a second fraction, multiply the first fraction by the reciprocal of the second fraction. Reduce if possible.
This method is sometimes called the “invert and multiply” method.
Sample Set B
Perform the following divisions:
\(\dfrac{1}{3} \div \dfrac{3}{4} . \quad \text{ The divisor is } \dfrac{3}{4} . \text{ Its reciprocal is } \dfrac{4}{3}\)
\(
\begin{aligned}
\dfrac{1}{3} \div \dfrac{3}{4} &=\dfrac{1}{3} \cdot \dfrac{4}{3} \\
&=\dfrac{1 \cdot 4}{3 \cdot 3} \\
&=\dfrac{4}{9}
\end{aligned}
\)
\(
\begin{aligned}
&\dfrac{3}{8} \div \dfrac{5}{4} . \quad \text { The divisor is } \dfrac{5}{4} . \text { Its reciprocal is } \dfrac{4}{5}\\
&\begin{aligned}
\dfrac{3}{8} \div \dfrac{5}{4} &=\dfrac{3}{8} \cdot \dfrac{4}{5} \\
&=\dfrac{3}{\not 2 \cdot \not 2 \cdot 2} \cdot \dfrac{\not 2 \cdot \not 2}{5}
\end{aligned}\\
&=\dfrac{3 \cdot 1}{2 \cdot 5}\\
&=\dfrac{3}{10}
\end{aligned}
\)
\(
\begin{aligned}
&\dfrac{5}{6} \div \dfrac{5}{12} . \quad \text { The divisor is } \dfrac{5}{12} . \text { Its reciprocal is } \dfrac{12}{5}\\
&\begin{aligned}
\dfrac{5}{6} \div \dfrac{5}{12} &=\dfrac{5}{6} \cdot \dfrac{12}{5} \\
&=\dfrac{5}{2 \cdot 3} \cdot \dfrac{2 \cdot 2 \cdot 3}{5} \\
&=\dfrac{\not 5}{\not 2 \cdot \not{3}} \cdot \dfrac{\not 2 \cdot 2 \cdot \not{3}}{\not 5} \\
&=\dfrac{1 \cdot 2}{1} \\
&=2
\end{aligned}
\end{aligned}
\)
Addition and Subtraction of Fractions
To add (or subtract) two or more fractions that have the same denominators, add (or subtract) the numerators and place the resulting sum over the common denominator. Reduce if possible.
Add or subtract only the numerators. Do not add or subtract the denominators!
Sample Set C
Find the following sums.
\(
\begin{aligned}
&\dfrac{3}{7}+\dfrac{2}{7} \text { . The denominators are the same. Add the numerators and place the sum over } 7 .\\
&\dfrac{3}{7}+\dfrac{2}{7}=\dfrac{3+2}{7}=\dfrac{5}{7}
\end{aligned}
\)
\(
\begin{aligned}
&\dfrac{7}{9}-\dfrac{4}{9} . \quad \text { The denominators are the same. Subtract } 4 \text { from } 7 \text { and place the difference over } 9 .\\
&\dfrac{7}{9}-\dfrac{4}{9}=\dfrac{7-4}{9}=\dfrac{3}{9}=\dfrac{1}{3}
\end{aligned}
\)
To add or subtract fractions having unlike denominators, convert each fraction to an equivalent fraction having as the denominator the least common multiple of the original denominators.
The least common multiple of the original denominators is commonly referred to as the least common denominator (LCD). See Section 3.4 for the technique of finding the least common multiple of several numbers.
Sample Set D
Find each sum or difference
The denominators are not alike. Find the LCD of 6 and 4 .
\(\begin{array}{ll}\dfrac{1}{6}+\dfrac{3}{4} . & \text { The denominators are not alike. } \\ \left\{\begin{array}{l}6=2 \cdot 3 \\ 4=2^{2}\end{array}\right. & \text { The LCD is } 2^{2} \cdot 3=4 \cdot 3=12 \text { . }\end{array}\)
Convert each of the original fractions to equivalent fractions having the common denominator 12 .
\(\dfrac{1}{6}=\dfrac{1 \cdot 2}{6 \cdot 2}=\dfrac{2}{12} \quad \dfrac{3}{4}=\dfrac{3 \cdot 3}{4 \cdot 3}=\dfrac{9}{12}\)
Now we can proceed with the addition.
\(
\begin{aligned}
\dfrac{1}{6}+\dfrac{3}{4} &=\dfrac{2}{12}+\dfrac{9}{12} \\
&=\dfrac{2+9}{12} \\
&=\dfrac{11}{12}
\end{aligned}
\)
The denominators are not alike. Find the LCD of 9 and 12 .
\(\begin{array}{ll}\dfrac{5}{9}-\dfrac{5}{12} . & \text { The denominators are not alike. } \\ \left\{\begin{array}{l}9=3^{2} \\ 12=2^{2} \cdot 3\end{array}\right. & \text { The LCD is } 2^{2} \cdot 3^{2}=4 \cdot 9=36 .\end{array}\)
Convert each of the original fractions to equivalent fractions having the common denominator 36 .
\(\dfrac{5}{9}=\dfrac{5 \cdot 4}{9 \cdot 4}=\dfrac{20}{36} \quad \dfrac{5}{12}=\dfrac{5 \cdot 3}{12 \cdot 3}=\dfrac{15}{36}\)
Now we can proceed with the subtraction.
\(
\begin{aligned}
\dfrac{5}{9}-\dfrac{5}{12} &=\dfrac{20}{36}-\dfrac{15}{36} \\
&=\dfrac{20-15}{36} \\
&=\dfrac{5}{36}
\end{aligned}
\)
Exercises
For each of the following problems, perform each indicated operation
\(\dfrac{1}{3} \cdot \dfrac{4}{3}\)
- Answer
-
\(\dfrac{4}{9}\)
\(\dfrac{1}{3} \cdot \dfrac{2}{3}\)
\(\dfrac{2}{5} \cdot \dfrac{5}{6}\)
- Answer
-
\(\dfrac{1}{3}\)
\(\dfrac{1}{3} \cdot \dfrac{4}{3}\)
- Answer
-
\(\dfrac{4}{9}\)
\(\dfrac{5}{6} \cdot \dfrac{14}{15}\)
\(\dfrac{9}{16} \cdot \dfrac{20}{27}\)
- Answer
-
\(\dfrac{5}{12}\)
\(\dfrac{35}{36} \cdot \dfrac{48}{55}\)
\(\dfrac{21}{25} \cdot \dfrac{15}{14}\)
- Answer
-
\(\dfrac{9}{10}\)
\(\dfrac{76}{99} \cdot \dfrac{66}{38}\)
\(\dfrac{3}{7} \cdot \dfrac{14}{18} \cdot \dfrac{6}{2}\)
- Answer
-
1
\(\dfrac{14}{15} \cdot \dfrac{21}{28} \cdot \dfrac{45}{7}\)
\(\dfrac{5}{9} \div \dfrac{5}{6}\)
- Answer
-
\(\dfrac{2}{3}\)
\(\dfrac{9}{16} \div \dfrac{15}{8}\)
\(\dfrac{4}{9} \div \dfrac{6}{15}\)
- Answer
-
\(\dfrac{10}{9}\)
\(\dfrac{25}{49} \div \dfrac{4}{9}\)
\(\dfrac{15}{4} \div \dfrac{27}{8}\)
- Answer
-
\(\dfrac{10}{9}\)
\(\dfrac{24}{75} \div \dfrac{8}{15}\)
\(\dfrac{57}{8} \div \dfrac{7}{8}\)
- Answer
-
\(\dfrac{57}{7}\)
\(\dfrac{7}{10} \div \dfrac{10}{7}\)
\(\dfrac{3}{8} + \dfrac{2}{8}\)
- Answer
-
\(\dfrac{5}{8}\)
\(\dfrac{3}{11} + \dfrac{4}{11}\)
\(\dfrac{5}{12} + \dfrac{7}{12}\)
- Answer
-
1
\(\dfrac{11}{16} - \dfrac{2}{16}\)
\(\dfrac{15}{23} - \dfrac{2}{23}\)
- Answer
-
\(\dfrac{13}{23}\)
\(\dfrac{3}{11} + \dfrac{1}{11} + \dfrac{5}{11}\)
\(\dfrac{16}{20} + \dfrac{1}{20} + \dfrac{2}{20}\)
- Answer
-
\(\dfrac{19}{20}\)
\(\dfrac{3}{8} + \dfrac{2}{8} - \dfrac{1}{8}\)
\(\dfrac{11}{16} + \dfrac{9}{16} - \dfrac{5}{16}\)
- Answer
-
\(\dfrac{15}{16}\)
\(\dfrac{1}{2} + \dfrac{1}{6}\)
\(\dfrac{1}{8} + \dfrac{1}{2}\)
- Answer
-
\(\dfrac{5}{8}\)
\(\dfrac{3}{4} + \dfrac{1}{3}\)
\(\dfrac{5}{8} + \dfrac{2}{3}\)
- Answer
-
\(\dfrac{31}{24}\)
\(\dfrac{6}{7} - \dfrac{1}{4}\)
\(\dfrac{8}{15} - \dfrac{3}{10}\)
- Answer
-
\(\dfrac{5}{6}\)
\(\dfrac{1}{15} + \dfrac{5}{12}\)
\(\dfrac{25}{36} - \dfrac{7}{10}\)
- Answer
-
\(\dfrac{-1}{180}\)
\(\dfrac{9}{28} - \dfrac{4}{45}\)
\(\dfrac{8}{15} - \dfrac{3}{10}\)
- Answer
-
\(\dfrac{7}{30}\)
\(\dfrac{1}{16} + \dfrac{3}{4} - \dfrac{3}{8}\)
\(\dfrac{8}{3} - \dfrac{1}{4} + \dfrac{7}{36}\)
- Answer
-
\(\dfrac{47}{18}\)
\(\dfrac{3}{4} - \dfrac{3}{22} + \dfrac{5}{24}\)