Introduction

Fraction multiplication occurs regularly in real-life situations.

For example, suppose that a bag contains \dfrac{3}{5}\,\text{kg} of flour. If Nancy used \dfrac{2}{5} of the bag to make a pie, how much flour did she use to make the pie?

To find out how much flour Nancy used, we need to multiply \dfrac{3}{5} by \dfrac{2}{5}.

To multiply these two fractions, we multiply the numerators, and we multiply the denominators: \dfrac{3}{5} \times \dfrac{2}{5} = \dfrac{3\times 2}{5\times 5} = \dfrac{6}{25}

Therefore, Nancy used \dfrac{6}{25} kilograms of flour.

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Example: Multiplying Two Proper Fractions

Huck gave \dfrac{1}{3} of a bag of marbles to his friend Tom. If Tom gave \dfrac{6}{7} of his share to Jim, what fraction of the original bag did Jim receive?

EXPLANATION

To find the fraction of the original bag Jim received, we need to multiply \dfrac{1}{3} by \dfrac{6}{7}.

To do that, we multiply the numerators, and we multiply the denominators: \dfrac{1}{3} \times \dfrac{6}{7} = \dfrac {1 \times 6} {3 \times 7} = \dfrac{6}{21}

We can simplify this fraction by dividing the numerator and denominator by 3\mathbin{:} \dfrac{6}{21} = \dfrac{6 \div 3}{21 \div 3} = \dfrac{2}{7}

So, Jim received \dfrac{2}{7} of the initial bag of marbles.

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Practice: Multiplying Two Proper Fractions

Sam drank $\dfrac{1}{5}$ of the contents of a bottle containing $\dfrac 3 4$ gallons of milk. How many gallons of milk did Sam drink?

a
 $\dfrac{15}{4}$ gallons
b
 $\dfrac{4}{20}$ gallons
c
 $\dfrac{3}{20}$ gallons
d
 $\dfrac{1}{20}$ gallons
e
 $\dfrac{4}{9}$ gallons
Practice: Multiplying Two Proper Fractions

In Mrs. Morgan's class, $\dfrac 35$ of the students are girls, and $\dfrac 12$ of the girls like to play soccer. What fraction of the students in the class are girls who like to play soccer?

a
 $\dfrac{2}{15}$
b
 $\dfrac{5}{6}$
c
 $\dfrac{3}{10}$
d
 $\dfrac{6}{5}$
e
 $\dfrac{3}{7}$
Example: Multiplying an Improper Fraction by a Proper Fraction

Jacob drank \dfrac{1}{4} of the contents of a pitcher that had been filled with \dfrac{8}{3} cups of coffee. How many cups of coffee did Jacob drink?

EXPLANATION

To find out the amount of coffee Jacob drank, we need to calculate \dfrac{1}{4} \times \dfrac{8}{3}.

To do that, we multiply the numerators, and we multiply the denominators: \dfrac{1}{4} \times \dfrac{8}{3} = \dfrac{1 \times 8}{4 \times 3} = \dfrac{8}{12}

Next, we simplify: \dfrac{8}{12} = \dfrac{8 \div 4}{12\div 4} =\dfrac{2}{3}

So, Jacob drank \dfrac{2}{3} cups of coffee.

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Practice: Multiplying an Improper Fraction by a Proper Fraction

A bag contained $\dfrac{9}{7}\,\textrm{kg}$ of soil. Annie poured $\dfrac{1}{3}$ of the bag into a pot. How many kilograms of soil did Annie pour into the pot?

a
 $\dfrac{9}{14}\,\textrm{kg}$
b
 $\dfrac{6}{7}\,\textrm{kg}$
c
 $\dfrac{3}{7}\,\textrm{kg}$
d
 $\dfrac{3}{14}\,\textrm{kg}$
e
 $\dfrac{9}{10}\,\textrm{kg}$
Practice: Multiplying an Improper Fraction by a Proper Fraction

Daryl drank $\dfrac 15$ of a can containing $\dfrac54\,\textrm{L}$ of juice. How many liters of juice did Daryl drink?

a
 $\dfrac 14\,\textrm{L}$
b
 $\dfrac 25\,\textrm{L}$
c
 $\dfrac 45\,\textrm{L}$
d
 $\dfrac 54\,\textrm{L}$
e
 $\dfrac 43\,\textrm{L}$
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