1. Operations with 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\cdotâ¬š=12".

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?"

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.

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?"

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

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.

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

- Explain how the model is showing the division.
- How does the model relate to the final fraction?
- What is the relationship between the numbers in the original division expression and the numbers in the final fraction?
- 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.

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

a

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

Worked Solution

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

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

Worked Solution

Idea summary

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

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:

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 |

- What is the method you used to divide the fractions by whole numbers?
- Evaluate each example using a model. How does this compare to the method you developed?
- Did you get the same result using both your method and a model? Why or why not?
- 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}

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}.

a

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

A

B

C

Worked Solution

b

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

Worked Solution

Evaluate the following:

a

\dfrac 43 \div 5

Worked Solution

b

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

Worked Solution

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}}