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In this pdf, we will multiply fractions and mixed numbers. There are three simple rules for multiplying fractions. First, multiply the numerators, and then the denominators of both the fractions to obtain the resultant.

1.
2.3 Multiplying Fractions and Mixed Numbers

Learning Objective(s)

1 Multiply two or more fractions.

2 Multiply a fraction by a whole number.

3 Multiply two or more mixed numbers.

4 Solve application problems that require multiplication of fractions or mixed numbers.

5 Find the area of triangles.

Just as you add, subtract, multiply, and divide when working with whole numbers, you

also use these operations when working with fractions. There are many times when it is

necessary to multiply fractions and mixed numbers. For example, this recipe will make

4 crumb piecrusts:

5 cups graham crackers 8 T. sugar

1 1

1 cups melted butter tsp. vanilla

2 4

Suppose you only want to make 2 crumb piecrusts. You can multiply all the ingredients

1

by , since only half of the number of piecrusts are needed. After learning how to

2

multiply a fraction by another fraction, a whole number or a mixed number, you should

be able to calculate the ingredients needed for 2 piecrusts.

Multiplying Fractions Objective 1

When you multiply a fraction by a fraction, you are finding a “fraction of a fraction.”

3 1 3

Suppose you have of a candy bar and you want to find of the :

4 2 4

By dividing each fourth in half, you can divide the candy bar into eighths.

3

Then, choose half of those to get .

8

In both of the above cases, to find the answer, you can multiply the numerators together

and the denominators together.

2.35

Learning Objective(s)

1 Multiply two or more fractions.

2 Multiply a fraction by a whole number.

3 Multiply two or more mixed numbers.

4 Solve application problems that require multiplication of fractions or mixed numbers.

5 Find the area of triangles.

Just as you add, subtract, multiply, and divide when working with whole numbers, you

also use these operations when working with fractions. There are many times when it is

necessary to multiply fractions and mixed numbers. For example, this recipe will make

4 crumb piecrusts:

5 cups graham crackers 8 T. sugar

1 1

1 cups melted butter tsp. vanilla

2 4

Suppose you only want to make 2 crumb piecrusts. You can multiply all the ingredients

1

by , since only half of the number of piecrusts are needed. After learning how to

2

multiply a fraction by another fraction, a whole number or a mixed number, you should

be able to calculate the ingredients needed for 2 piecrusts.

Multiplying Fractions Objective 1

When you multiply a fraction by a fraction, you are finding a “fraction of a fraction.”

3 1 3

Suppose you have of a candy bar and you want to find of the :

4 2 4

By dividing each fourth in half, you can divide the candy bar into eighths.

3

Then, choose half of those to get .

8

In both of the above cases, to find the answer, you can multiply the numerators together

and the denominators together.

2.35

2.
Multiplying Two Fractions

a c a⋅c product of the numerators

⋅ = =

b d b ⋅ d product of the denominators

3 1 3 ⋅1 3

⋅ = =

4 2 4⋅2 8

Multiplying More Than Two Fractions

a c e a⋅c ⋅e

⋅ ⋅ =

b d f b⋅d ⋅f

1 2 3 1⋅ 2 ⋅ 3 6

⋅ =

⋅ =

3 4 5 3 ⋅ 4 ⋅ 5 60

Example

Problem 2 4 Multiply.

⋅

3 5

2⋅4 Multiply the numerators and

3⋅5 multiply the denominators.

8 Simplify, if possible. This

fraction is already in lowest

15

terms.

8

Answer

15

If the resulting product needs to be simplified to lowest terms, divide the numerator and

denominator by common factors.

2.36

a c a⋅c product of the numerators

⋅ = =

b d b ⋅ d product of the denominators

3 1 3 ⋅1 3

⋅ = =

4 2 4⋅2 8

Multiplying More Than Two Fractions

a c e a⋅c ⋅e

⋅ ⋅ =

b d f b⋅d ⋅f

1 2 3 1⋅ 2 ⋅ 3 6

⋅ =

⋅ =

3 4 5 3 ⋅ 4 ⋅ 5 60

Example

Problem 2 4 Multiply.

⋅

3 5

2⋅4 Multiply the numerators and

3⋅5 multiply the denominators.

8 Simplify, if possible. This

fraction is already in lowest

15

terms.

8

Answer

15

If the resulting product needs to be simplified to lowest terms, divide the numerator and

denominator by common factors.

2.36

3.
Example

Problem 2 1 Multiply. Simplify the

⋅ answer.

3 4

2 ⋅1 Multiply the numerators and

multiply the denominators.

3⋅4

2 Simplify, if possible.

12

2÷2 Simplify by dividing the

numerator and denominator by

12 ÷ 2

the common factor 2.

2 1 1

Answer ⋅ =

3 4 6

You can also simplify the problem before multiplying, by dividing common factors.

Example

Problem 2 1 Multiply. Simplify the

⋅ answer.

3 4

2 ⋅ 1 1⋅ 2 Reorder the numerators so that

= you can see a fraction that has

3⋅4 3⋅4

a common factor.

Simplify.

1⋅ 1 2 2÷2 1

= =

3⋅2 4 4÷2 2

2 1 1

Answer ⋅ =

3 4 6

You do not have to use the “simplify first” shortcut, but it could make your work easier

because it keeps the numbers in the numerator and denominator smaller while you are

working with them.

Self Check A

3 1

⋅ Multiply. Simplify the answer.

4 3

2.37

Problem 2 1 Multiply. Simplify the

⋅ answer.

3 4

2 ⋅1 Multiply the numerators and

multiply the denominators.

3⋅4

2 Simplify, if possible.

12

2÷2 Simplify by dividing the

numerator and denominator by

12 ÷ 2

the common factor 2.

2 1 1

Answer ⋅ =

3 4 6

You can also simplify the problem before multiplying, by dividing common factors.

Example

Problem 2 1 Multiply. Simplify the

⋅ answer.

3 4

2 ⋅ 1 1⋅ 2 Reorder the numerators so that

= you can see a fraction that has

3⋅4 3⋅4

a common factor.

Simplify.

1⋅ 1 2 2÷2 1

= =

3⋅2 4 4÷2 2

2 1 1

Answer ⋅ =

3 4 6

You do not have to use the “simplify first” shortcut, but it could make your work easier

because it keeps the numbers in the numerator and denominator smaller while you are

working with them.

Self Check A

3 1

⋅ Multiply. Simplify the answer.

4 3

2.37

4.
Multiplying a Fraction by a Whole Number Objective 2

When working with both fractions and whole numbers, it is useful to write the whole

number as an improper fraction (a fraction where the numerator is greater than or

equal to the denominator). All whole numbers can be written with a “1” in the

2 5 100

denominator. For example: 2 = , 5 = , and 100 = . Remember that the

1 1 1

denominator tells how many parts there are in one whole, and the numerator tells how

many parts you have.

Multiplying a Fraction and a Whole Number

b a b

a⋅ = ⋅

c 1 c

2 4 2 8

4⋅ = ⋅ =

3 1 3 3

Often when multiplying a whole number and a fraction the resulting product will be an

improper fraction. It is often desirable to write improper fractions as a mixed number for

the final answer. You can simplify the fraction before or after rewriting as a mixed

number. See the examples below.

Example

Problem 3 Multiply. Simplify the answer

7⋅ and write as a mixed number.

5

7 3 Rewrite 7 as the improper fraction

⋅ 7

1 5 .

1

7 ⋅ 3 21 Multiply the numerators and

= multiply the denominators.

1⋅ 5 5

1 Rewrite as a mixed number.

4 21 ÷ 5 =4 with a remainder of 1 .

5

3 1

Answer 7⋅ =4

5 5

2.38

When working with both fractions and whole numbers, it is useful to write the whole

number as an improper fraction (a fraction where the numerator is greater than or

equal to the denominator). All whole numbers can be written with a “1” in the

2 5 100

denominator. For example: 2 = , 5 = , and 100 = . Remember that the

1 1 1

denominator tells how many parts there are in one whole, and the numerator tells how

many parts you have.

Multiplying a Fraction and a Whole Number

b a b

a⋅ = ⋅

c 1 c

2 4 2 8

4⋅ = ⋅ =

3 1 3 3

Often when multiplying a whole number and a fraction the resulting product will be an

improper fraction. It is often desirable to write improper fractions as a mixed number for

the final answer. You can simplify the fraction before or after rewriting as a mixed

number. See the examples below.

Example

Problem 3 Multiply. Simplify the answer

7⋅ and write as a mixed number.

5

7 3 Rewrite 7 as the improper fraction

⋅ 7

1 5 .

1

7 ⋅ 3 21 Multiply the numerators and

= multiply the denominators.

1⋅ 5 5

1 Rewrite as a mixed number.

4 21 ÷ 5 =4 with a remainder of 1 .

5

3 1

Answer 7⋅ =4

5 5

2.38

5.
Example

Problem 3 Multiply. Simplify the

4⋅ answer and write as a

4

mixed number.

4 3 Rewrite 4 as the improper fraction

⋅ 4

1 4 .

1

4⋅3 Multiply the numerators and

1⋅ 4 multiply the denominators.

12 Simplify.

=3

4

3

Answer 4⋅ =3

4

Self Check B

5

3⋅ Multiply. Simplify the answer and write it as a mixed number.

6

Multiplying Mixed Numbers Objective 3

If you want to multiply two mixed numbers, or a fraction and a mixed number, you can

again rewrite any mixed number as an improper fraction.

So, to multiply two mixed numbers, rewrite each as an improper fraction and then

multiply as usual. Multiply numerators and multiply denominators and simplify. And, as

before, when simplifying, if the answer comes out as an improper fraction, then convert

the answer to a mixed number.

Example

Problem 1 1 Multiply. Simplify the answer and write

2 ⋅4 as a mixed number.

5 2

1 11 1

2 = Change 2 to an improper fraction.

5 5 5

5 • 2 + 1 = 11, and the denominator is 5.

1 9 1

4 = Change 4 to an improper fraction.

2 2 2

2 • 4 + 1 = 9, and the denominator is 2.

2.39

Problem 3 Multiply. Simplify the

4⋅ answer and write as a

4

mixed number.

4 3 Rewrite 4 as the improper fraction

⋅ 4

1 4 .

1

4⋅3 Multiply the numerators and

1⋅ 4 multiply the denominators.

12 Simplify.

=3

4

3

Answer 4⋅ =3

4

Self Check B

5

3⋅ Multiply. Simplify the answer and write it as a mixed number.

6

Multiplying Mixed Numbers Objective 3

If you want to multiply two mixed numbers, or a fraction and a mixed number, you can

again rewrite any mixed number as an improper fraction.

So, to multiply two mixed numbers, rewrite each as an improper fraction and then

multiply as usual. Multiply numerators and multiply denominators and simplify. And, as

before, when simplifying, if the answer comes out as an improper fraction, then convert

the answer to a mixed number.

Example

Problem 1 1 Multiply. Simplify the answer and write

2 ⋅4 as a mixed number.

5 2

1 11 1

2 = Change 2 to an improper fraction.

5 5 5

5 • 2 + 1 = 11, and the denominator is 5.

1 9 1

4 = Change 4 to an improper fraction.

2 2 2

2 • 4 + 1 = 9, and the denominator is 2.

2.39

6.
11 9 Rewrite the multiplication problem, using

⋅ the improper fractions.

5 2

11⋅ 9 99 Multiply numerators and multiply

= denominators.

5 ⋅ 2 10

99 9 Write as a mixed number.

=9 99 ÷ 10 =

9 with a remainder of 9.

10 10

1 1 9

Answer 2 ⋅4 =

9

5 2 10

Example

Problem 1 1 Multiply. Simplify the answer and

⋅3 write as a mixed number.

2 3

1 10 1

3 = Change 3 to an improper

3 3 3

fraction. 3 • 3 + 1 = 10, and the

denominator is 3.

1 10 Rewrite the multiplication problem,

⋅ using the improper fraction in place

2 3 of the mixed number.

1⋅ 10 10 Multiply numerators and multiply

= denominators.

2⋅3 6

10 4 Rewrite as a mixed number.

=1 10 ÷ 6 = 1 with a remainder of 4.

6 6

2 Simplify the fractional part to lowest

1 terms by dividing the numerator and

3 denominator by the common factor

2.

1 1 2

Answer ⋅3 =

1

2 3 3

As you saw earlier, sometimes it’s helpful to look for common factors in the numerator

and denominator before you simplify the products.

2.40

⋅ the improper fractions.

5 2

11⋅ 9 99 Multiply numerators and multiply

= denominators.

5 ⋅ 2 10

99 9 Write as a mixed number.

=9 99 ÷ 10 =

9 with a remainder of 9.

10 10

1 1 9

Answer 2 ⋅4 =

9

5 2 10

Example

Problem 1 1 Multiply. Simplify the answer and

⋅3 write as a mixed number.

2 3

1 10 1

3 = Change 3 to an improper

3 3 3

fraction. 3 • 3 + 1 = 10, and the

denominator is 3.

1 10 Rewrite the multiplication problem,

⋅ using the improper fraction in place

2 3 of the mixed number.

1⋅ 10 10 Multiply numerators and multiply

= denominators.

2⋅3 6

10 4 Rewrite as a mixed number.

=1 10 ÷ 6 = 1 with a remainder of 4.

6 6

2 Simplify the fractional part to lowest

1 terms by dividing the numerator and

3 denominator by the common factor

2.

1 1 2

Answer ⋅3 =

1

2 3 3

As you saw earlier, sometimes it’s helpful to look for common factors in the numerator

and denominator before you simplify the products.

2.40

7.
Example

Problem 3 1 Multiply. Simplify the answer and

1 ⋅2 write as a mixed number.

5 4

3 8 3

1 = Change 1 to an improper fraction. 5 •

5 5 5

1 + 3 = 8, and the denominator is 5.

1 9 1

2 = Change 2 to an improper fraction. 4 •

4 4 4

2 + 1 = 9, and the denominator is 4.

8 9 Rewrite the multiplication problem using

⋅ the improper fractions.

5 4

8⋅9 9⋅8 Reorder the numerators so that you can

=

5⋅4 5⋅4 see a fraction that has a common factor.

9⋅8 9⋅2 8 8÷4 2

= Simplify. = =

5 ⋅ 4 5 ⋅1 4 4÷4 1

18 Multiply.

5

18 3 Write as a mixed fraction.

=3

5 5

3 1 3

Answer 1 ⋅2 = 3

5 4 5

In the last example, the same answer would be found if you multiplied numerators and

multiplied denominators without removing the common factor. However, you would get

, and then you would need to simplify more to get your final answer.

Self Check C

3 1

1 ⋅ 3 Multiply. Simplify the answer and write as a mixed number.

5 3

2.41

Problem 3 1 Multiply. Simplify the answer and

1 ⋅2 write as a mixed number.

5 4

3 8 3

1 = Change 1 to an improper fraction. 5 •

5 5 5

1 + 3 = 8, and the denominator is 5.

1 9 1

2 = Change 2 to an improper fraction. 4 •

4 4 4

2 + 1 = 9, and the denominator is 4.

8 9 Rewrite the multiplication problem using

⋅ the improper fractions.

5 4

8⋅9 9⋅8 Reorder the numerators so that you can

=

5⋅4 5⋅4 see a fraction that has a common factor.

9⋅8 9⋅2 8 8÷4 2

= Simplify. = =

5 ⋅ 4 5 ⋅1 4 4÷4 1

18 Multiply.

5

18 3 Write as a mixed fraction.

=3

5 5

3 1 3

Answer 1 ⋅2 = 3

5 4 5

In the last example, the same answer would be found if you multiplied numerators and

multiplied denominators without removing the common factor. However, you would get

, and then you would need to simplify more to get your final answer.

Self Check C

3 1

1 ⋅ 3 Multiply. Simplify the answer and write as a mixed number.

5 3

2.41

8.
Solving Problems by Multiplying Fractions and Mixed Numbers Objective 4

Now that you know how to multiply a fraction by another fraction, by a whole number, or

by a mixed number, you can use this knowledge to solve problems that involve

multiplication and fractional amounts. For example, you can now calculate the

ingredients needed for the 2 crumb piecrusts.

Example

Problem 5 cups graham crackers 8 T. sugar The recipe at left makes 4

1 1 piecrusts. Find the ingredients

1 cups melted butter tsp. vanilla needed to make only 2 piecrusts.

2 4

Since the recipe is for 4 piecrusts,

you can multiply each of the

1

ingredients by to find the

2

measurements for just 2 piecrusts.

1 5 1 5 5 cups graham crackers: Since the

5⋅ = ⋅ = result is an improper fraction,

2 1 2 2

5

rewrite as the improper fraction

1 2

2 cups of graham crackers are needed. 1

2 2 .

2

1 8 1 8 8 T. sugar: This is another

8⋅ = ⋅ = =4 example of a whole number

2 1 2 2

multiplied by a fraction.

4 T. sugar is needed.

1

1 cups melted butter: You need

2

to multiply a mixed number by a

1

fraction. So, first rewrite 1 as the

2

3

improper fraction : 2 • 1 + 1, and

2

3 1 3 the denominator is 2. Then, rewrite

⋅ =

2 2 4 the multiplication problem, using

the improper fraction in place of the

3 mixed number. Multiply.

cup melted butter is needed.

4

2.42

Now that you know how to multiply a fraction by another fraction, by a whole number, or

by a mixed number, you can use this knowledge to solve problems that involve

multiplication and fractional amounts. For example, you can now calculate the

ingredients needed for the 2 crumb piecrusts.

Example

Problem 5 cups graham crackers 8 T. sugar The recipe at left makes 4

1 1 piecrusts. Find the ingredients

1 cups melted butter tsp. vanilla needed to make only 2 piecrusts.

2 4

Since the recipe is for 4 piecrusts,

you can multiply each of the

1

ingredients by to find the

2

measurements for just 2 piecrusts.

1 5 1 5 5 cups graham crackers: Since the

5⋅ = ⋅ = result is an improper fraction,

2 1 2 2

5

rewrite as the improper fraction

1 2

2 cups of graham crackers are needed. 1

2 2 .

2

1 8 1 8 8 T. sugar: This is another

8⋅ = ⋅ = =4 example of a whole number

2 1 2 2

multiplied by a fraction.

4 T. sugar is needed.

1

1 cups melted butter: You need

2

to multiply a mixed number by a

1

fraction. So, first rewrite 1 as the

2

3

improper fraction : 2 • 1 + 1, and

2

3 1 3 the denominator is 2. Then, rewrite

⋅ =

2 2 4 the multiplication problem, using

the improper fraction in place of the

3 mixed number. Multiply.

cup melted butter is needed.

4

2.42

9.
1 1 1 1

⋅ = tsp. vanilla: Here, you multiply a

4 2 8 4

fraction by a fraction.

1

tsp. vanilla is needed.

8

Answer The ingredients needed for 2 pie crusts

1

are: 2 cups graham crackers

2

4 T. sugar

3

cup melted butter

4

1

tsp. vanilla

8

Often, a problem indicates that multiplication by a fraction is needed by using phrases

3

like “half of,” “a third of,” or “ of.”

4

Example

Problem 1

The cost of a vacation is $4,500 and you are required to pay of

5

that amount when you reserve the trip. How much will you have to

pay when you reserve the trip?

1 1

4,500 ⋅ You need to find of 4,500. “Of”

5 5

tells you to multiply.

4,500 1 Change 4,500 to an improper

⋅ fraction by rewriting it with 1 as

1 5

the denominator.

4,500 Divide.

5

900 Simplify.

Answer You will need to pay $900 when you reserve the trip.

2.43

⋅ = tsp. vanilla: Here, you multiply a

4 2 8 4

fraction by a fraction.

1

tsp. vanilla is needed.

8

Answer The ingredients needed for 2 pie crusts

1

are: 2 cups graham crackers

2

4 T. sugar

3

cup melted butter

4

1

tsp. vanilla

8

Often, a problem indicates that multiplication by a fraction is needed by using phrases

3

like “half of,” “a third of,” or “ of.”

4

Example

Problem 1

The cost of a vacation is $4,500 and you are required to pay of

5

that amount when you reserve the trip. How much will you have to

pay when you reserve the trip?

1 1

4,500 ⋅ You need to find of 4,500. “Of”

5 5

tells you to multiply.

4,500 1 Change 4,500 to an improper

⋅ fraction by rewriting it with 1 as

1 5

the denominator.

4,500 Divide.

5

900 Simplify.

Answer You will need to pay $900 when you reserve the trip.

2.43

10.
Example

Problem The pie chart at left represents

the fractional part of daily

activities.

Given a 24-hour day, how many

hours are spent sleeping?

Attending school? Eating? Use

the pie chart to determine your

answers.

1 1

⋅ 24 =number of hours sleeping Sleeping is of the pie, so the number of

3 3

1

hours spent sleeping is of 24.

3

1 24 Rewrite 24 as an improper fraction with a

⋅ denominator of 1.

3 1

24 Multiply numerators and multiply

=8 24

3 denominators. Simplify to 8.

8 hours sleeping 3

1 1

⋅ 24 =

number of hours spent at school Attending school is of the pie, so the

6 6

number of hours spent attending school is

1

of 24.

6

1 24 Rewrite 24 as an improper fraction with a

⋅ denominator of 1.

6 1

24 Multiply numerators and multiply

=4 24

6 denominators. Simplify to 4.

4 hours attending school 6

2.44

Problem The pie chart at left represents

the fractional part of daily

activities.

Given a 24-hour day, how many

hours are spent sleeping?

Attending school? Eating? Use

the pie chart to determine your

answers.

1 1

⋅ 24 =number of hours sleeping Sleeping is of the pie, so the number of

3 3

1

hours spent sleeping is of 24.

3

1 24 Rewrite 24 as an improper fraction with a

⋅ denominator of 1.

3 1

24 Multiply numerators and multiply

=8 24

3 denominators. Simplify to 8.

8 hours sleeping 3

1 1

⋅ 24 =

number of hours spent at school Attending school is of the pie, so the

6 6

number of hours spent attending school is

1

of 24.

6

1 24 Rewrite 24 as an improper fraction with a

⋅ denominator of 1.

6 1

24 Multiply numerators and multiply

=4 24

6 denominators. Simplify to 4.

4 hours attending school 6

2.44

11.
1 1

⋅ 24 =number of hours spent eating Eating is of the pie, so the number of

12 12

1

hours spent eating is of 24.

12

1 24 Rewrite 24 as an improper fraction with a

⋅ denominator of 1.

12 1

24 Multiply numerators and multiply

=2 24

12 denominators. Simplify to 2.

2 hours spent eating 12

Answer Hours spent:

sleeping: 8 hours

attending school: 4 hours

eating: 2 hours

Self Check D

1

Neil bought a dozen (12) eggs. He used of the eggs for breakfast. How many eggs

3

are left?

Area of Triangles Objective 5

The formula for the area of a triangle can be explained by looking at a right triangle.

Look at the image below—a rectangle with the same height and base as the original

triangle. The area of the triangle is one half of the rectangle!

2.45

⋅ 24 =number of hours spent eating Eating is of the pie, so the number of

12 12

1

hours spent eating is of 24.

12

1 24 Rewrite 24 as an improper fraction with a

⋅ denominator of 1.

12 1

24 Multiply numerators and multiply

=2 24

12 denominators. Simplify to 2.

2 hours spent eating 12

Answer Hours spent:

sleeping: 8 hours

attending school: 4 hours

eating: 2 hours

Self Check D

1

Neil bought a dozen (12) eggs. He used of the eggs for breakfast. How many eggs

3

are left?

Area of Triangles Objective 5

The formula for the area of a triangle can be explained by looking at a right triangle.

Look at the image below—a rectangle with the same height and base as the original

triangle. The area of the triangle is one half of the rectangle!

2.45

12.
Since the area of two congruent triangles is the same as the area of a rectangle, you can

1

come up with the formula Area = b • h to find the area of a triangle.

2

When you use the formula for a triangle to find its area, it is important to identify a base

and its corresponding height, which is perpendicular to the base.

Example

Problem A triangle has a height of 4 inches and a base of 10

inches. Find the area.

1 Start with the formula for the area

A= bh of a triangle.

2

1 Substitute 10 for the base and 4 for

A= • 10 • 4 the height.

2

1 Multiply.

A= • 40

2

A = 20

Answer A = 20 in2

2.46

1

come up with the formula Area = b • h to find the area of a triangle.

2

When you use the formula for a triangle to find its area, it is important to identify a base

and its corresponding height, which is perpendicular to the base.

Example

Problem A triangle has a height of 4 inches and a base of 10

inches. Find the area.

1 Start with the formula for the area

A= bh of a triangle.

2

1 Substitute 10 for the base and 4 for

A= • 10 • 4 the height.

2

1 Multiply.

A= • 40

2

A = 20

Answer A = 20 in2

2.46

13.
You multiply two fractions by multiplying the numerators and multiplying the

denominators. Often the resulting product will not be in lowest terms, so you must also

simplify. If one or both fractions are whole numbers or mixed numbers, first rewrite each

as an improper fraction. Then multiply as usual, and simplify.

2.3 Self Check Solutions

Self Check A

3 1

⋅ Multiply. Simplify the answer.

4 3

3 ⋅1 3 3÷3 1

= , then simplify: = .

4 ⋅ 3 12 12 ÷ 3 4

Self Check B

5

3⋅ Multiply. Simplify the answer and write it as a mixed number.

6

1 15

2 . Multiplying the two numbers gives , and since 15 ÷ 6 = 2R3, the mixed number

2 6

3 1

is 2 . The fractional part simplifies to .

6 2

Self Check C

3 1

1 ⋅ 3 Multiply. Simplify the answer and write as a mixed number.

5 3

1 3 8 1 10

5 . First, rewrite each mixed number as an improper fraction: 1 = and 3 = .

3 5 5 3 3

8 10 80

Next, multiply numerators and multiply denominators: ⋅ = . Then write as a

5 3 15

80 5

mixed fraction =5 . Finally, simplify the fractional part by dividing both numerator

15 15

and denominator by the common factor 5.

Self Check D

1

Neil bought a dozen (12) eggs. He used of the eggs for breakfast. How many eggs

3

are left?

1 1 12 12

of 12 is 4 ( ⋅ = = 4 ), so he used 4 of the eggs. Because 12 – 4 = 8, there are

3 3 1 3

8 eggs left.

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denominators. Often the resulting product will not be in lowest terms, so you must also

simplify. If one or both fractions are whole numbers or mixed numbers, first rewrite each

as an improper fraction. Then multiply as usual, and simplify.

2.3 Self Check Solutions

Self Check A

3 1

⋅ Multiply. Simplify the answer.

4 3

3 ⋅1 3 3÷3 1

= , then simplify: = .

4 ⋅ 3 12 12 ÷ 3 4

Self Check B

5

3⋅ Multiply. Simplify the answer and write it as a mixed number.

6

1 15

2 . Multiplying the two numbers gives , and since 15 ÷ 6 = 2R3, the mixed number

2 6

3 1

is 2 . The fractional part simplifies to .

6 2

Self Check C

3 1

1 ⋅ 3 Multiply. Simplify the answer and write as a mixed number.

5 3

1 3 8 1 10

5 . First, rewrite each mixed number as an improper fraction: 1 = and 3 = .

3 5 5 3 3

8 10 80

Next, multiply numerators and multiply denominators: ⋅ = . Then write as a

5 3 15

80 5

mixed fraction =5 . Finally, simplify the fractional part by dividing both numerator

15 15

and denominator by the common factor 5.

Self Check D

1

Neil bought a dozen (12) eggs. He used of the eggs for breakfast. How many eggs

3

are left?

1 1 12 12

of 12 is 4 ( ⋅ = = 4 ), so he used 4 of the eggs. Because 12 – 4 = 8, there are

3 3 1 3

8 eggs left.

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