Topics
1. Operations with Fractions
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Grade 6

1.04 Divide fractions and whole numbers

Lesson
Practice

Divide whole numbers by fractions

When we divide by a whole number, such as 12\div4, we ask the question, "how many groups of 4 fit into 12?". It's just like thinking about, "what number fills in the blank: 4\cdot⬚=12".

3 groups of 4 apples lined horizontally.

In this case, there are 3 whole groups of 4 in 12, so the result is 3.

We can think about dividing by a fraction in a similar way. The division 2\div\dfrac{1}{3} is equivalent to asking the question, "how many parts of size \dfrac{1}{3} fit into 2 wholes?"

2 circles divided into 3 equal parts

If we split two wholes up into thirds, we can see that there are 3 thirds in each whole, and so there are 2\cdot3=6 thirds in total.

From this we can see that 2\div\dfrac{1}{3}=2\cdot3=6

When dividing by a fraction, we can rewrite it as multiplication by the reciprocal. This also works for fractions that do not have a numerator of 1.

Reciprocal

The number that, when multiplied with the original number, results in 1

Example:

5 and \dfrac15

\dfrac 23 and \dfrac 32

Consider 6\div\dfrac{2}{3}. We cannot ask, "What is 6 divided into \dfrac{2}{3} groups of the same size" because that does not make sense. Instead, we can say, "How many groups of \dfrac{2}{3} are in 6?"

A number line showing 6 wholes divided into thirds and then grouping by two thirds.

To answer this question, we can begin by dividing the 6 wholes into thirds. Then, we can make groups of 2 thirds.

If we count, we can see there are 9 equal groups of \dfrac{2}{3} in 6. This shows 6\div\dfrac{2}{3}=9.

Again, instead of dividing by a fraction, we can multiply by the reciprocal of the fraction: 6\div\dfrac{2}{3}=6\cdot\dfrac{3}{2}=9

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 whole number by a fraction between 0 and 1?

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

Examples

Example 1

The number line below shows 4 wholes split into \dfrac{1}{3} sized parts.

Number line starting from 0 to 4 with 3 spaces in between the numbers. Each space represents 1 third.
a

Use the model to evaluate 4\div \dfrac{1}{3}.

Worked Solution
Create a strategy

To evaluate 4 \div \dfrac{1}{3}, we can use the number line model to count the number of one-third sized parts that make up the whole sections.

Apply the idea

There are 4 whole sections, and each section is divided into 3 parts (each representing \dfrac{1}{3}). There are 12 parts of size \dfrac{1}{3} in the diagram so 4\div \dfrac{1}{3}=12.

b

If 4 is divided into parts that are \dfrac{1}{3} of a whole each, how many parts are there in total?

Worked Solution
Create a strategy

We can rewrite the division as multiplication by the reciprocal and solve for the total number of parts. The reciprocal of \dfrac{1}{3} is \dfrac{3}{1} or 3.

Apply the idea
\displaystyle \text{Number of parts}\displaystyle =\displaystyle 4\div\dfrac{1}{3}Divide the whole number by the unit fraction
\displaystyle =\displaystyle 4\cdot 3Multiply by the reciprocal
\displaystyle =\displaystyle 12Evaluate

Example 2

Rewrite 9\div \dfrac{3}{7} using multiplication.

Worked Solution
Create a strategy

To rewrite the expression using multiplication, remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of \dfrac{3}{7} is \dfrac{7}{3}.

Apply the idea
\displaystyle 9\div \dfrac{3}{7}\displaystyle =\displaystyle 9 \cdot \dfrac{7}{3}Multiply by the reciprocal
Idea summary

Dividing a whole number by a fraction is the same as multiplying the whole number by the reciprocal of that fraction.

Divide fractions by whole numbers

When we divided a whole number by a fraction, such as 2\div \dfrac{1}{3}, we asked the question "how many parts of size \dfrac{1}{3} fit into 2 wholes?"

Dividing a fraction by a whole number is the reverse of this. Let's look at \dfrac{1}{3}\div 2 as an example:

Square divided into 3 vertical equal parts where 1 part is shaded.

We start with \dfrac{1}{3} of a whole, shown as the shaded area in the image.

Square divided into 6 equal parts where 1 part is shaded.

We then divide each of these thirds into 2 parts.

How big is the remaining shaded area? Well, there are now 6 parts of equal area and 1 of them is shaded, so this is equal to \dfrac{1}{6} of the whole.

Exploration

We know that 2 \div \dfrac{1}{3} = 2\cdot \dfrac{3}{1} and using a model we saw that \dfrac{1}{3} \div 2= \dfrac{1}{6}. Try to come up with a method that can be used to divide fractions by whole numbers and test it on the following examples:

\text{a. } \dfrac{1}{2}\div 4
\text{b. }\dfrac{3}{4} \div 3
\text{c. }\dfrac{5}{7}\div 5
\text{d. }\dfrac{8}{9} \div 2
  1. What is the method you used to divide the fractions by whole numbers?
  2. Evaluate each example using a model. How does this compare to the method you developed?
  3. Did you get the same result using both your method and a model? Why or why not?
  4. Can your method be applied to all fractions and whole numbers? Why or why not?

We can think about dividng by a whole number using multiplication, in a similar way to dividing by a fraction, but creating the reciprocal for a whole number looks a bit different. First, we have to remember that every whole number has a denominator of 1:

\begin{aligned} \dfrac13\div2&=\dfrac13\div\dfrac21=\dfrac13 \cdot\dfrac{1}{2}\\ &=\dfrac16 \end{aligned}

Examples

Example 3

Let's use the image below to help us find the value of \dfrac{1}{3}\div 4. This number line shows the number 1 split into 3 parts of size \dfrac{1}{3}.

0\frac{1}{3}\frac{2}{3}1
a

Which image shows that each third has been divided into 4 parts?

A
0\frac{1}{3}\frac{2}{3}1
B
0\frac{1}{3}\frac{2}{3}1
C
0\frac{1}{3}\frac{2}{3}1
Worked Solution
Create a strategy

Count the spaces between the thirds on each number line.

Apply the idea

Option A has 2 spaces between the thirds which means that each third is divided into 2 parts, not 4.

Option B has 4 spaces between the thirds which means that each third is divided into 4 parts.

Option C has 3 spaces between the thirds which means that each third is divided into 3 parts, not 4.

The answer is option B.

b

What is the size of the part created when \dfrac{1}{3} is divided by 4?

Worked Solution
Create a strategy

We can also count the number of spaces between 0 and 1 on the number line.

Apply the idea
0\frac{1}{3}\frac{2}{3}1

We can see that there are 12 spaces, so each space (or part) has a size of \dfrac{1}{12}.

Reflect and check

We can also divide the unit fraction by the whole number.

\displaystyle \text{Size}\displaystyle =\displaystyle \dfrac{1}{3}\div4Divide the unit fraction by the whole number
\displaystyle =\displaystyle \dfrac{1}{3\cdot4}Multiply the denominator by the whole number
\displaystyle =\displaystyle \dfrac{1}{12}Evaluate

Example 4

Evaluate the following:

a

\dfrac 43 \div 5

Worked Solution
Create a strategy

Rewrite the whole number as a fraction. Then multiply the first fraction by the reciprocal of the second fraction.

Apply the idea
\displaystyle \dfrac 43 \div 5\displaystyle =\displaystyle \dfrac 43 \div \dfrac 51Rewrite the whole number as fraction
\displaystyle =\displaystyle \dfrac 43 \cdot \dfrac 15Multiply by the reciprocal of \dfrac 51
\displaystyle =\displaystyle \dfrac 4{15} Multiply the numerators and denominators
b

1 \dfrac{1}{3} \div 4

Worked Solution
Create a strategy

Convert the mixed number into an improper and the whole number into a fraction. Then rewrite as multiplication by the reciprocal.

Apply the idea

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

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

Then convert the whole number 4 into a fraction.

\displaystyle 4\displaystyle =\displaystyle \dfrac{4}{1}Rewrite the whole number

Rewrite the division as multiplication by the reciprocal.

\displaystyle 1 \dfrac{1}{3} \div 4\displaystyle =\displaystyle \dfrac{4}{3} \div \dfrac{4}{1}Divide improper fractions
\displaystyle =\displaystyle \dfrac{4}{3} \cdot \dfrac{1}{4}Multiply by the reciprocal
\displaystyle =\displaystyle \dfrac{4 \cdot 1}{3 \cdot 4}Multiply the numerators and denominators
\displaystyle =\displaystyle \dfrac{4}{12}Evaluate
\displaystyle =\displaystyle \dfrac{1}{3}Simplify
Idea summary

Dividing a fraction by a whole number is the same as multiplying the fraction by the reciprocal of the whole number.

The reciprocal of a whole number is: \dfrac{1}{\text{Whole Number}}

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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