Topics
1. Operations with Fractions
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United States of AmericaVirginia
Grade 6

1.05 Divide fractions and mixed numbers

Lesson
Practice

Divide fractions and mixed numbers

We've divided fractions and whole numbers, but what does it look like to divide a fraction by another fraction?

Let's look at \dfrac{8}{9} \div \dfrac{2}{9}.

This image shows a circle split into 9 parts. 8 sectors are shaded

Let's start by asking, "How many groups of \dfrac{2}{9} make up \dfrac{8}{9}.

First we can draw a diagram of \dfrac{8}{9}.

This image shows a circle split into 9 parts. 8 sectors are shaded. Ask your teacher for more information.

Then we can split it into groups of \dfrac{2}{9}.

We can see that there are 4 equal groups of \dfrac {2}{9} in \dfrac{8}{9}. So \dfrac {8}{9} \div \dfrac {2}{9} = 4.

We can also apply the method of rewriting division as multiplication by the reciprocal like we did when dividing with whole numbers.

\displaystyle \dfrac {8}{9} \div \dfrac {2}{9}\displaystyle =\displaystyle \dfrac {8}{9} \cdot \dfrac {9}{2}Multiply \dfrac{8}{9} by the reciprocal of \dfrac{2}{9}
\displaystyle =\displaystyle \dfrac {8\cdot 9}{9\cdot 2} Multiply the numerators together and denominators together
\displaystyle =\displaystyle \dfrac {8}{2} Evaluate the multiplication
\displaystyle =\displaystyle 4 Divide

Both methods give us the same result.

We can apply the same process to dividing with mixed numbers, because remember a mixed number is just a different form of a fraction. We just have to convert the mixed number into an improper fraction first.

Exploration

Use the dropdown boxes to create a division expression.

Press the 'Start animation' button to see a model of the division.

Continue pressing 'Show next step' until the animation is complete.

Loading interactive...

Observe the animation for several different division expressions then answer the following questions:

  1. Explain how the model is showing the division.
  2. How does the model relate to the final fraction?
  3. What is the relationship between the numbers in the original division expression and the numbers in the final fraction?
  4. What do you notice about dividing a fraction or mixed number by a fraction between 0 and 1?

When a fraction or mixed number is divided by a fraction between 0 and 1, the result is larger than the original fraction or mixed number.

Examples

Example 1

Evaluate each expression.

a

\dfrac{1}{8}\div\dfrac{1}{5}

Worked Solution
Create a strategy

Divide fractions by multiplying the 1st fraction by the reciprocal of the 2nd.

Apply the idea
\displaystyle \dfrac18\div\dfrac15\displaystyle =\displaystyle \dfrac{1}{8} \cdot \dfrac{5}{1}Multiply by the reciprocal
\displaystyle =\displaystyle \dfrac{1\cdot5}{8\cdot1}Multiply numerators and denominators
\displaystyle =\displaystyle \dfrac{5}{8}Evaluate
Reflect and check

We can use an area model to verify the answer:

A rectangle divided into 8 equal parts and 1 part shaded divided by another rectangle divided into 5 equal parts and 1 part shade.
Area model representing 1/8 divided by 1/5. First row with 8 squares are shaded and first column with 5 squares are shaded.

Overlap the models and count the number of shaded parts in the row and column.

The row has 5 shaded parts and the column has 8 shaded parts. So, the area model shows \dfrac{5}{8}.

b

\dfrac{4}{5}\div\dfrac{11}{10}

Worked Solution
Create a strategy

Divide fractions by multiplying the 1st fraction by the reciprocal of the 2nd.

Apply the idea
\displaystyle \dfrac{4}{5}\div\dfrac{11}{10}\displaystyle =\displaystyle \dfrac{4}{5} \cdot \dfrac{10}{11}Multiply by the reciprocal
\displaystyle =\displaystyle \dfrac{4\cdot 10}{5\cdot 11}Multiply numerators and denominators
\displaystyle =\displaystyle \dfrac{40}{55}Evaluate
\displaystyle =\displaystyle \dfrac{8}{11}Simplify

Example 2

Evaluate and write your answer in its simplest form.

a

2 \dfrac {3}{5} \div 2 \dfrac {7}{10},

Worked Solution
Create a strategy

Rewrite the mixed numbers as improper fractions, then divide the fractions.

Apply the idea
\displaystyle 2 \dfrac 3{5} \div 2 \dfrac 7{10}\displaystyle =\displaystyle \dfrac {13}{5} \div \dfrac {27}{10}Rewrite as improper fractions
\displaystyle =\displaystyle \dfrac {13}{5} \cdot \dfrac {10}{27}Multiply by the reciprocal of \dfrac{27}{10}
\displaystyle =\displaystyle \dfrac {13}{1} \cdot \dfrac {2}{27}Simplify \dfrac{10}{5}=\dfrac {2}{1}
\displaystyle =\displaystyle \dfrac {26}{27} Multiply numerators and denominators
b

1 \dfrac{1}{3} \div \dfrac{1}{8}.

Worked Solution
Create a strategy

Convert the mixed number into an improper fraction, then rewrite as multiplication.

Apply the idea

First, convert the mixed number 1 \dfrac{1}{3} into its improper fraction form.

\displaystyle 1 \dfrac{1}{3}\displaystyle =\displaystyle \dfrac{1 \cdot 3 + 1}{3}Rewrite the mixed number
\displaystyle =\displaystyle \dfrac{3 + 1}{3}Evaluate the multiplication
\displaystyle =\displaystyle \dfrac{4}{3}Evaluate the addition

Then rewrite the division as multiplication.

\displaystyle 1 \dfrac{1}{3} \div \dfrac{1}{8}\displaystyle =\displaystyle \dfrac{4}{3} \div \dfrac{1}{8}Divide improper fractions
\displaystyle =\displaystyle \dfrac{4}{3} \cdot \dfrac{8}{1}Multiply by the reciprocal
\displaystyle =\displaystyle \dfrac{4 \cdot 8}{3 \cdot 1}Multiply the numerators and denominators
\displaystyle =\displaystyle \dfrac{32}{3}Evaluate

Example 3

A 8 \dfrac{1}{4} meter long roll of fabric is to be cut into sections of equal length 2 \dfrac{3}{4}. How many pieces of fabric will there be?

Worked Solution
Create a strategy

Rewrite the mixed numbers as improper fractions then perform division.

Apply the idea
\displaystyle 8 \dfrac {1}{4} \div 2 \dfrac {3}{4}\displaystyle =\displaystyle \dfrac {33}{4} \div \dfrac {11}{4}Rewrite as improper fractions
\displaystyle =\displaystyle \dfrac {33}{4} \cdot \dfrac {4}{11}Multiply by the reciprocal of \dfrac{11}{4}
\displaystyle =\displaystyle \dfrac {33\cdot \cancel{4}}{\cancel{4} \cdot 11} Write as a single fraction
\displaystyle =\displaystyle \dfrac {33}{11} Divide out the common factor
\displaystyle =\displaystyle 3Evaluate the division

There will be 3 pieces of fabric.

Reflect and check

To check the reasonableness of our answer, let's use estimation. We can round the length of the fabric roll to 8 meters, and the sections we're cutting to 3 meters long each. Dividing these, we get \dfrac{8}{3}, which is just less than 3 because \dfrac{9}{3}=3. This estimation shows that 3 is a reasonable answer.

Idea summary

To divide one fraction by another, multiply the first fraction by the reciprocal of the second.

Outcomes

6.CE.1

The student will estimate, demonstrate, solve, and justify solutions to problems using operations with fractions and mixed numbers, including those in context.

6.CE.1a

Demonstrate/model multiplication and division of fractions (proper or improper) and mixed numbers using multiple representations.*

6.CE.1b

Multiply and divide fractions (proper or improper) and mixed numbers that include denominators of 12 or less. Answers are expressed in simplest form.*

6.CE.1c

Investigate and explain the effect of multiplying or dividing a fraction, whole number, or mixed number by a number between zero and one.*

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