In this section, we will learn how to (1) simplify fractions and (2) multiply and divide fractions.
Fractions are rational numbers. They look like $\dfrac{p}{q}$ where numerator p and denominator q are integers, and denominator q can never be 0.
When operating with fractions, your final answer should be simplified.
Simplifying Fractions :
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Cancel out the common factors of the numerator and the denominator.
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A fraction is simplified when the numerator and the denominator have no common factors.
Example: Simplify the following fractions.
a) $\dfrac{15}{35}=\dfrac{3\cdot 5}{7\cdot 5}=\dfrac{3}{7}$ (cancel out 5)
b) $-\dfrac{48}{18}=-\dfrac{8\cdot 6}{3\cdot 6}=-\dfrac{8}{3}$ (cancel out 6)
Multiplying Fractions :
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Multiply the numerators and the denominators of the two fractions.
$\dfrac{a}{b}\cdot \dfrac{c}{d} = \dfrac{a\cdot c}{b\cdot d}$
2. Simply the result.
Example: Multiply the following fractions.
a) $\dfrac{14}{3}\cdot\dfrac{9}{7}=\dfrac{14\cdot 9}{3\cdot 7} = \dfrac{2\cdot 7\cdot 3\cdot 3}{3\cdot 7} = \dfrac{2\cdot 3}{1}=6$
Note: when multiplying fractions, it is easier to cancel out the common factors before multiplying out the numerators and denominators.
b) $\dfrac{3}{11}\cdot\dfrac{5}{6}=\dfrac{3\cdot 5}{11\cdot 6} = \dfrac{3\cdot 5}{11\cdot 2\cdot 3} = \dfrac{5}{11\cdot 2}=\dfrac{5}{22}$
The reciprocal of a fraction $\dfrac{p}{q}$ is the fraction formed by switching the numerator and denominator, namely $\dfrac{q}{p}$.
For example:
the reciprocal of $\dfrac{3}{5}$ is $\dfrac{5}{3}$;
the reciprocal of $\dfrac{-2}{7}$ is $\dfrac{7}{-2}=\dfrac{-7}{2}=-\dfrac{7}{2}$;
the reciprocal of $\dfrac{1}{8}$ is $\dfrac{8}{1}=8$;
the reciprocal of $4=\dfrac{4}{1}$ is $\dfrac{1}{4}$.
Dividing Fractions :
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Dividing fractions is to multiply the fraction by the reciprocal of the second fraction.
$\dfrac{a}{b}\div \dfrac{c}{d} =\dfrac{a}{b}\cdot \dfrac{d}{c} = \dfrac{a\cdot d}{b\cdot c}$
2. Simply the result.
Example: Divide the following fractions.
a) $\dfrac{8}{3}\div\dfrac{4}{5}=\dfrac{8}{3}\cdot\dfrac{5}{ 4}= \dfrac{8\cdot 5}{3\cdot 4} = \dfrac{2\cdot 4\cdot 5}{3\cdot 4} = \dfrac{2\cdot 5}{3}=\dfrac{10}{3}$
b) $\dfrac{5}{12}\div-\dfrac{3}{16}=\dfrac{5}{12}\cdot-\dfrac{16}{ 3}= -\dfrac{5\cdot 16}{12\cdot 3} = -\dfrac{5\cdot 4\cdot 4}{3\cdot 4\cdot 3} =-\dfrac{5\cdot 4}{3\cdot 3}=-\dfrac{20}{9}$
c) $13\div-\dfrac{1}{4}=13\cdot (-4)= -52$
Practice Problems:
1. Simplify the following fractions:
(a) $\dfrac{32}{36}$
(b) $-\dfrac{45}{15}$
2. Multiply the following fractions:
(a) $\dfrac{6}{10}\cdot\dfrac{5}{15}$
(b) $-\dfrac{7}{11}\cdot\dfrac{33}{14}$
3. Divide the following fractions:
(a) $\dfrac{2}{5}\div\dfrac{6}{7}$
(b) $-\dfrac{5}{9}\div\dfrac{7}{12}$
Answer Key: (a). (b). (c). (d). (e).
For more detailed explanation, please read: Arithmetic|Algebra Chapter 2.
