Divide whole numbers by proper fractions

Ontario Grade 6 Mathematics

Number

Dividing whole numbers by proper fractions

Division can ask how many groups of a certain size fit into an amount. A proper fraction is a fraction with a numerator smaller than its denominator, such as

\frac{2}{5}

. The denominator tells how many equal parts make one whole. The numerator tells how many of those parts are in the fraction. In this lesson, you will use equal parts to count groups, including when the last group is not complete.

What you will learn

Explain division by a proper fraction as finding how many groups of that size fit into a whole-number amount.
Use equal parts and grouping to solve a division problem.
Check a result by seeing whether the groups make the starting amount.

1. Begin with equal parts

Imagine four whole strips of paper. Cut each strip into three equal parts. Each small part is one-third of a strip. There are three such parts in each strip, so the four strips contain twelve one-third parts.

This model answers the question, “How many groups of one-third fit into four wholes?” Each small part is one group. The answer is twelve groups. The key idea is to make parts that match the denominator, then count them in groups the size named by the numerator.

For a fraction such as

\frac{2}{5}

, each whole can be split into five equal fifths. A group of two-fifths contains two of those fifths. If some fifths remain after making full groups, compare the leftover with the size of one group.

4\div\frac{1}{3}=12

The denominator tells how many equal parts make one whole.
The numerator tells how many equal parts make one group.
Count parts in groups of the stated size.

2. Use a grouping model when the group has more than one part

Consider three wholes and groups of

\frac{2}{5}

. Split each whole into five equal fifths. The three wholes now contain fifteen fifths. Make groups using two fifths at a time.

Seven full groups use fourteen fifths. One fifth remains. Since a full group needs two fifths, the remaining one fifth is half of a full group. So the amount makes seven and one-half groups.

This way of working stays close to the meaning of the question: how many groups fit? First rename the whole-number amount in parts of the denominator’s size. Then group those parts by the numerator. You can describe a leftover as a fraction of another group.

15\div 2=7\frac{1}{2}

Rename each whole as equal parts that match the denominator.
Make groups containing the number of parts named by the numerator.
A leftover may make part of another group.

3. Read, model, and check

Use a steady plan. First, say what the starting amount is and what size group you are counting. Next, split each whole into equal parts that match the denominator. Then count those parts in groups the size of the numerator.

When a model is hard to draw, you can keep track with a calculation. For example, two wholes contain eight quarters. To count groups of three-quarters, group the eight quarters in threes. There are two full groups, with two quarters left. Those two quarters are two-thirds of a group of three quarters.

A check connects the answer back to the question. Put the groups together. The complete groups and any part of a group should use exactly the starting amount. This helps catch a counting error or a leftover that has been described incorrectly.

number of groups\timesgroup size=starting amount

Keep the starting amount and group size clear.
Use equal parts to make the grouping visible.
Check that the groups together match the starting amount.

Worked example

One-third groups in four wholes

How many groups of

\frac{1}{3}

fit into

4

wholes?

Four wholes split into thirds

Each whole has three equal parts. The four wholes contain twelve one-third parts in total.

Split each whole
Each whole contains three equal one-third parts. Four wholes contain four sets of three parts.
$4\times 3=12$
Count the groups
Each small part is one group of one-third. There are twelve small parts altogether.
$4\div\frac{1}{3}=12$

Answer: Twelve groups of

\frac{1}{3}

fit into

4

wholes.

Check: Twelve one-third parts make twelve thirds, which is four wholes.

Worked example

Groups of two-fifths in three wholes

How many groups of

\frac{2}{5}

fit into

3

wholes?

Three wholes split into fifths

The three wholes contain fifteen fifths. Groups of two fifths make seven full groups and half of another.

Count fifths
Each whole contains five fifths. Three wholes contain three sets of five fifths.
$3\times 5=15$
Make groups of two fifths
Group the fifteen fifths in pairs. Seven full groups use fourteen fifths. One fifth remains. Since a full group needs two fifths, the remaining fifth is half of another group.
$15\div 2=7\frac{1}{2}$
Check the grouping
Seven and one-half groups each use two fifths per group. Together, they use fifteen fifths, or three wholes.
$7\frac{1}{2}\times\frac{2}{5}=3$

Answer: There are

7\frac{1}{2}

groups of

\frac{2}{5}

in

3

wholes.

Check: Seven full groups use fourteen fifths, and the half group uses one fifth. Together, they use all fifteen fifths.

Worked example

Groups of three-quarters in two wholes

How many groups of

\frac{3}{4}

fit into

2

wholes?

Two wholes split into quarters

The two wholes contain eight quarters. Groups of three quarters make two full groups and two-thirds of another.

Count quarters
Each whole contains four quarters. Two wholes contain eight quarters. Make groups of three quarters.
$2\times 4=8$
Count full and partial groups
Two full groups use six quarters, leaving two quarters. A group needs three quarters, so two quarters make two-thirds of one group.
$8\div 3=2\frac{2}{3}$
Check the result
Two and two-thirds groups of three-quarters use eight quarters altogether, which is two wholes.
$2\frac{2}{3}\times\frac{3}{4}=2$

Answer: There are

2\frac{2}{3}

groups of

\frac{3}{4}

in

2

wholes.

Check: The two full groups use six quarters. The remaining two quarters are two-thirds of a three-quarter group. All eight quarters are accounted for.

Common mistakes and how to avoid them

Counting each small part as a full group when the fraction names more than one part.

Correction: Check the numerator. For

\frac{2}{5}

, a full group contains two fifths, not one fifth.

Treating a leftover part as a complete group.

Correction: Compare the leftover with the number of parts in one group. One fifth is half of a group of two fifths.

Thinking the answer must be smaller than the starting whole number.

Correction: The group size is less than one whole, so more than one group can fit in a whole.

Leaving out the meaning of a fractional group in the answer.

Correction: State how many full groups there are and what fraction of another group remains.

Lesson summary

Division by a proper fraction asks how many groups of that size fit in a whole-number amount.
Split each whole into equal parts named by the denominator.
Count those parts in groups named by the numerator.
Describe any leftover as part of another group, and check that all groups match the starting amount.

Check your understanding

Question 1

How many groups of

\frac{1}{4}

fit into

3

wholes?

$\frac{3}{4}$
$7$
$12$
$\frac{1}{12}$

Show answer and explanation

12

Each whole contains four fourths. Three wholes contain twelve fourths, so twelve groups fit.

Question 2

How many groups of

\frac{2}{3}

fit into

2

wholes?

$\frac{4}{3}$
$3$
$\frac{2}{5}$
$\frac{3}{4}$

Show answer and explanation

3

Two wholes contain six thirds. Grouping the six thirds in pairs makes three groups of two-thirds.

Question 3

You have two wholes and are counting groups of

\frac{3}{5}

. How many fifths are in one full group?

$2$
$3$
$5$
$8$

Show answer and explanation

3

The numerator tells how many fifths make the group. A group of three-fifths contains three fifths.

Key terms

Proper fraction: A fraction whose numerator is smaller than its denominator, such as $\frac{2}{5}$ .
Numerator: The top number in a fraction. It tells how many equal parts are counted.
Denominator: The bottom number in a fraction. It tells how many equal parts make one whole.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation B2.10. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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B2.10 · Divide whole numbers by proper fractions

Ontario Grade 6 Mathematics

Dividing whole numbers by proper fractions

What you will learn

1. Begin with equal parts

2. Use a grouping model when the group has more than one part

3. Read, model, and check

One-third groups in four wholes

Groups of two-fifths in three wholes

Groups of three-quarters in two wholes

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 6 Math

About this lesson and its review