Multiply and divide fractions, whole numbers, and mixed numbers

Ontario Grade 8 Mathematics

Number

Ontario Grade 8 Mathematics — B2.6

Fractions can describe part of a whole or part of a group. Multiplying and dividing them helps answer questions such as “What is three-quarters of two-thirds?” or “How many one-quarter portions fit into two-thirds?” First, remember that the denominator tells how many equal parts make a whole, and the numerator tells how many of those parts are counted. A mixed number has a whole-number part and a fraction part, such as

1\frac{1}{2}

. We will use these ideas to multiply and divide fractions, whole numbers, and mixed numbers.

What you will learn

Multiply fractions by fractions and whole numbers.
Divide fractions and explain why the reciprocal is used.
Change mixed numbers to improper fractions before calculating.
Write answers in simplest form and check whether they make sense.

1. Multiplying fractions

Imagine a strip divided into four equal parts. Three parts are shaded, so the shaded amount is

\frac{3}{4}

. Now take half of that shaded amount. Each shaded part is split into two equal pieces, making eight pieces in all. Half of the three shaded parts is three of those eight pieces, or

\frac{3}{8}

.

This model shows why multiplying by a fraction can make an amount smaller. To multiply two fractions, multiply the numerators to find the number of small parts counted. Multiply the denominators to find the total number of equal small parts.

A whole number can be written as a fraction with denominator

1

. For example,

3=\frac{3}{1}

. This lets you use the same multiplication rule for whole numbers and fractions. A mixed number must first be changed to an improper fraction. An improper fraction has a numerator that is at least as large as its denominator.

\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}

Multiply the numerators together and the denominators together.
Write a whole number over $1$ when using the fraction rule.
Change mixed numbers to improper fractions before multiplying.

2. Dividing fractions

Division asks how many groups of a certain size fit into an amount. Suppose you have

\frac{2}{3}

of a strip and want portions that are each

\frac{1}{6}

of a strip. Two-thirds contains four sixths, so the answer is

4

portions.

To divide by a fraction, multiply by its reciprocal. A reciprocal is the fraction with its numerator and denominator switched. For example, the reciprocal of

\frac{2}{5}

is

\frac{5}{2}

. The divisor is the number you divide by; it is the second number in a division question.

Why does this rule fit the portion question? Each sixth is half of one-third, and there are two thirds in the amount. Counting sixths gives four. The rule gives the same result:

\frac{2}{3}\div\frac{1}{6}=\frac{2}{3}\times\frac{6}{1}=4

.

A whole-number divisor can also be written as a fraction over

1

. For division by a whole number, use its reciprocal as the multiplier. Do not switch the first fraction; switch only the divisor. Mixed numbers must be changed to improper fractions before dividing.

\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}

Keep the first number, change division to multiplication, and flip the divisor.
The reciprocal is made by switching the numerator and denominator.
Change mixed numbers to improper fractions before dividing.

3. Mixed numbers, simplest form, and sense checks

To change a mixed number to an improper fraction, multiply the whole-number part by the denominator, add the numerator, and keep the same denominator. For example,

2\frac{1}{3}=\frac{7}{3}

because

2

wholes contain

6

thirds, and one more third makes

7

thirds.

After calculating, simplify when the numerator and denominator have a common factor. A factor is a whole number that divides another whole number evenly. For example, both

6

and

8

can be divided by

2

, so

\frac{6}{8}=\frac{3}{4}

. If an improper fraction is the answer, it may also be written as a mixed number.

Check the size of the answer. Multiplying a positive amount by a fraction less than

1

should make it smaller. Dividing by a fraction less than

1

can make the answer larger because several small portions may fit into the amount. These checks can help catch an error, but they do not replace the calculation.

w\frac{n}{d}=\frac{wd+n}{d}

For a mixed number, multiply the whole number by the denominator, then add the numerator.
Simplify by dividing the numerator and denominator by the same factor.
Use the meaning of the question to judge whether the answer is reasonable.

4. Try these, then check your thinking

Use the rules in order: change any mixed numbers, choose multiplication or division, and simplify the result. For division, identify the divisor before taking its reciprocal.

Practice without looking at the examples: calculate

\frac{5}{6}\times\frac{3}{10}

,

4\div\frac{2}{3}

, and

1\frac{1}{4}\times\frac{2}{5}

. Write each answer in simplest form. For the division question, ask how many two-thirds groups fit into four wholes.

When you finish, compare your work with the quick check. If an answer differs, review whether you changed the correct mixed number, flipped only the divisor, and simplified correctly.

Keep multiplication and division steps in a clear order.
Use the question’s meaning to check the size of an answer.

Worked example

Find a fraction of a fraction

Calculate

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

and give the answer in simplest form.

Multiply the numerators
The numerators count the small parts being taken. Multiply them to find the numerator of the product.
$3\times2=6$
Multiply the denominators
The denominators show the total number of equal parts in the product. Multiply them to find the denominator.
$4\times5=20$
Simplify
Both numbers can be divided evenly by $2$ . Dividing the numerator and denominator by the same number keeps the fraction’s value unchanged.
$\frac{6}{20}=\frac{3}{10}$

Answer:

\frac{3}{10}

Check: The answer is less than

\frac{3}{4}

because taking two-fifths of an amount makes it smaller.

Worked example

Divide a whole number by a fraction

Calculate

3\div\frac{2}{5}

.

Read the division question
The question asks how many groups of size $\frac{2}{5}$ fit into $3$ wholes. Write the whole number as a fraction over $1$ .
$3=\frac{3}{1}$
Use the reciprocal of the divisor
Keep the first fraction. Change division to multiplication and switch the numerator and denominator of the divisor.
$\frac{3}{1}\times\frac{5}{2}$
Multiply and write the result
Multiply across. The improper fraction equals a whole number and a remaining half.
$\frac{3\times5}{1\times2}=\frac{15}{2}=7\frac{1}{2}$

Answer:

7\frac{1}{2}

Check: Each group is less than one whole, so more than three groups can fit into three wholes. Seven and a half groups is reasonable.

Worked example

Multiply mixed numbers

Calculate

1\frac{1}{2}\times2\frac{2}{3}

and give the answer as a mixed number.

Change each mixed number
For the first number, one whole is two halves, then add one half. For the second, two wholes are six thirds, then add two thirds.
$1\frac{1}{2}=\frac{3}{2},\quad 2\frac{2}{3}=\frac{8}{3}$
Multiply the fractions
Multiply the numerators together and the denominators together.
$\frac{3}{2}\times\frac{8}{3}=\frac{24}{6}$
Simplify and change form
Divide the numerator and denominator by $6$ . The result is four wholes.
$\frac{24}{6}=4$

Answer:

4

Check: The factors are

1\frac{1}{2}

and

2\frac{2}{3}

. Their product should be greater than each factor, and

4

fits that check.

Common mistakes and how to avoid them

Adding the denominators when multiplying fractions.

Correction: Multiply the numerators together and multiply the denominators together.

Flipping the first fraction when dividing.

Correction: Keep the first number as it is. Take the reciprocal of the divisor only.

Using a mixed number without changing it first.

Correction: Change each mixed number to an improper fraction before calculating.

Simplifying only the numerator or only the denominator.

Correction: Divide both by the same factor so the value stays the same.

Lesson summary

To multiply fractions, multiply numerators and multiply denominators.
To divide by a fraction, multiply by the reciprocal of the divisor.
Write whole numbers over $1$ when needed.
Change mixed numbers to improper fractions before calculating, then simplify.

Check your understanding

Question 1

What is

\frac{5}{6}\times\frac{3}{10}

in simplest form?

$\frac{1}{4}$
$\frac{8}{16}$
$\frac{1}{2}$
correctIndex

Show answer and explanation

\frac{1}{4}

Multiply to get

\frac{15}{60}

, then divide the numerator and denominator by

15

to get

\frac{1}{4}

.

Question 2

What is

4\div\frac{2}{3}

?

$\frac{8}{3}$
$6$
$\frac{2}{12}$
correctIndex

Show answer and explanation

6

Keep

4

, multiply by the reciprocal

\frac{3}{2}

, and calculate

4\times\frac{3}{2}=6

.

Question 3

What is

1\frac{1}{4}\times\frac{2}{5}

in simplest form?

$\frac{1}{2}$
$\frac{1}{4}$
$\frac{3}{5}$
correctIndex

Show answer and explanation

\frac{1}{2}

Change

1\frac{1}{4}

to

\frac{5}{4}

. Then

\frac{5}{4}\times\frac{2}{5}=\frac{10}{20}=\frac{1}{2}

.

Key terms

Denominator: The bottom number in a fraction. It tells how many equal parts make one whole.
Numerator: The top number in a fraction. It tells how many parts are counted.
Improper fraction: A fraction whose numerator is equal to or greater than its denominator.
Reciprocal: A fraction made by switching the numerator and denominator of another fraction.
Simplest form: A fraction form that cannot be reduced by dividing its numerator and denominator by the same whole number greater than one.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation B2.6. 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.

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B2.6 · Multiply and divide fractions, whole numbers, and mixed numbers

Ontario Grade 8 Mathematics

Ontario Grade 8 Mathematics — B2.6

What you will learn

1. Multiplying fractions

2. Dividing fractions

3. Mixed numbers, simplest form, and sense checks

4. Try these, then check your thinking

Find a fraction of a fraction

Divide a whole number by a fraction

Multiply mixed numbers

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 8 Math

About this lesson and its review