Grade 6

2.04 Multiply and divide mixed numbers

Introduction

Previously, we worked on multiplying and dividing fractions. Now, we'll apply the same rules to multiply and divide mixed numbers.

Ideas

Multiply and divide mixed numbers

Multiply and divide mixed numbers

Recall that a mixed number is a number consisting of an integer and a fraction. A mixed number can also be expressed as an improper fraction or a fraction where the numerator is larger than the denominator.

Watch the video below, where we discuss how to multiply and divide mixed numbers.

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In what ways are the steps the same as multiplying and dividing other fractions?
In what ways are the steps different?

Notice that we can use the same steps for multiplying and dividing fractions that we previously saw, with one added step. That is, we can convert mixed numbers to improper fractions first.

Examples

Example 1

Evaluate 5\dfrac47\times4\dfrac23.

Worked Solution

Create a strategy

Convert mixed numbers into improper form, then multiply.

Apply the idea

First, convert the mixed fraction 5\dfrac47 into its improper form.

\displaystyle 5\dfrac47	\displaystyle =	\displaystyle \dfrac{7\times5+4}{7}	Rewrite the mixed fraction
	\displaystyle =	\displaystyle \dfrac{35+4}{7}	Perform the multiplication
	\displaystyle =	\displaystyle \dfrac{39}{7}	Evaluate

Then convert the mixed fraction 4\dfrac23 into its improper form.

\displaystyle 4\dfrac23	\displaystyle =	\displaystyle \dfrac{3\times4+2}{3}	Rewrite the mixed fraction
	\displaystyle =	\displaystyle \dfrac{12+2}{3}	Perform the multiplication
	\displaystyle =	\displaystyle \dfrac{14}{3}	Evaluate

Evaluate the expression using their improper fraction forms.

\displaystyle 5\dfrac47\times4\dfrac23	\displaystyle =	\displaystyle \dfrac{39}{7} \times \dfrac{14}{3}	Multiply improper fractions
	\displaystyle =	\displaystyle \dfrac{39\times14}{7\times3}	Multiply numerators and denominators
	\displaystyle =	\displaystyle \dfrac{546}{21}	Evaluate

Reflect and check

We can give the answer as an improper fraction, or simplify \dfrac{546}{21} by dividing 546 by 21 resulting in 26.

Example 2

Evaluate 2\dfrac3{20} \div 2\dfrac{7}{10}.

Worked Solution

Create a strategy

Convert mixed numbers into improper form, then multiply the first fraction by the reciprocal of the second fraction.

Apply the idea

First, convert the mixed fraction 2\dfrac{3}{20} into its improper form.

\displaystyle 2\dfrac3{20}	\displaystyle =	\displaystyle \dfrac{2\times20+3}{20}	Rewrite the mixed fraction
	\displaystyle =	\displaystyle \dfrac{40+3}{20}	Perform the multiplication
	\displaystyle =	\displaystyle \dfrac{43}{20}	Evaluate

Then convert the mixed fraction 2\dfrac{7}{10} into its improper form.

\displaystyle 2\dfrac{7}{10}	\displaystyle =	\displaystyle \dfrac{2\times10+7}{10}	Rewrite the mixed fraction
	\displaystyle =	\displaystyle \dfrac{20+7}{10}	Perform the multiplication
	\displaystyle =	\displaystyle \dfrac{27}{10}	Evaluate

Evaluate the expression using their improper fraction forms.

\displaystyle 2\dfrac3{20} \div 2\dfrac{7}{10}	\displaystyle =	\displaystyle \dfrac{43}{20} \div \dfrac{27}{10}	Multply by the reciplrocal
	\displaystyle =	\displaystyle \dfrac{43}{20} \times \dfrac{10}{27}	Multiply numerators and denominators
	\displaystyle =	\displaystyle \dfrac{430}{540}	Evaluate
	\displaystyle =	\displaystyle \dfrac{43}{54}	Simplify

Idea summary

To multiply or divide mixed numbers, first convert them to improper fractions, then perform the operation.

Outcomes

6.NS.A.1

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g. By using visual fraction models and equations to represent the problem.