Contributed by:

1 Find the reciprocal of a number.

2 Divide two fractions.

3 Divide two mixed numbers.

4 Divide fractions, mixed numbers, and whole numbers.

5 Solve application problems that require division of fractions or mixed numbers.

2 Divide two fractions.

3 Divide two mixed numbers.

4 Divide fractions, mixed numbers, and whole numbers.

5 Solve application problems that require division of fractions or mixed numbers.

1.
2.4 Dividing Fractions and Mixed Numbers

Learning Objective(s)

1 Find the reciprocal of a number.

2 Divide two fractions.

3 Divide two mixed numbers.

4 Divide fractions, mixed numbers, and whole numbers.

5 Solve application problems that require division of fractions or mixed numbers.

There are times when you need to use division to solve a problem. For example, if

painting one coat of paint on the walls of a room requires 3 quarts of paint and there are

6 quarts of paint, how many coats of paint can you paint on the walls? You divide 6 by 3

for an answer of 2 coats. There will also be times when you need to divide by a fraction.

1

Suppose painting a closet with one coat only required quart of paint. How many coats

2

could be painted with the 6 quarts of paint? To find the answer, you need to divide 6 by

1

the fraction, .

2

Reciprocals Objective 1

If the [product] of two numbers is 1, the two numbers are [reciprocals] of each other.

Here are some examples:

Original number Reciprocal Product

3 4 3 4 3 ⋅ 4 12

⋅ = = = 1

4 3 4 3 4 ⋅ 3 12

1 2 1 2 1⋅ 2 2

⋅ = = =1

2 1 2 1 2 ⋅1 2

3 1 3 1 3 ⋅1 3

3= ⋅ = = =1

1 3 1 3 1⋅ 3 3

1 7 3 7 3 7 ⋅ 3 21

2 = ⋅ = = = 1

3 3 7 3 7 3 ⋅ 7 21

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Learning Objective(s)

1 Find the reciprocal of a number.

2 Divide two fractions.

3 Divide two mixed numbers.

4 Divide fractions, mixed numbers, and whole numbers.

5 Solve application problems that require division of fractions or mixed numbers.

There are times when you need to use division to solve a problem. For example, if

painting one coat of paint on the walls of a room requires 3 quarts of paint and there are

6 quarts of paint, how many coats of paint can you paint on the walls? You divide 6 by 3

for an answer of 2 coats. There will also be times when you need to divide by a fraction.

1

Suppose painting a closet with one coat only required quart of paint. How many coats

2

could be painted with the 6 quarts of paint? To find the answer, you need to divide 6 by

1

the fraction, .

2

Reciprocals Objective 1

If the [product] of two numbers is 1, the two numbers are [reciprocals] of each other.

Here are some examples:

Original number Reciprocal Product

3 4 3 4 3 ⋅ 4 12

⋅ = = = 1

4 3 4 3 4 ⋅ 3 12

1 2 1 2 1⋅ 2 2

⋅ = = =1

2 1 2 1 2 ⋅1 2

3 1 3 1 3 ⋅1 3

3= ⋅ = = =1

1 3 1 3 1⋅ 3 3

1 7 3 7 3 7 ⋅ 3 21

2 = ⋅ = = = 1

3 3 7 3 7 3 ⋅ 7 21

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2.
In each case, the original number, when multiplied by its reciprocal, equals 1.

To create two numbers that multiply together to give an answer of one, the numerator of

one is the denominator of the other. You sometimes say one number is the “flip” of the

2 5

other number: flip to get the reciprocal . In order to find the reciprocal of a mixed

5 2

number, write it first as an improper fraction so that it can be “flipped.”

Example

Problem 1

Find the reciprocal of 5 .

4

1 21 1

5 = Rewrite 5 as an improper

4 4 4

fraction. The numerator is

4 • 5 + 1 = 21.

4 Find the reciprocal by

Answer interchanging (“flipping”) the

21

numerator and denominator.

Self Check A

2

What is the reciprocal of 3 ?

5

Dividing a Fraction or a Mixed Number by a Whole Number Objective 3, 4

When you divide by a whole number, you multiply by the reciprocal of the divisor. In the

painting example where you need 3 quarts of paint for a coat and have 6 quarts of paint,

you can find the total number of coats that can be painted by dividing 6 by 3, 6 ÷3 = 2.

1

You can also multiply 6 by the reciprocal of 3, which is , so the multiplication problem

3

6 1 6

becomes ⋅ = = 2.

1 3 3

3

The same idea will work when the divisor is a fraction. If you have of a candy bar and

4

1 1 3

need to divide it among 5 people, each person gets of the available candy: of is

5 5 4

1 3 3 3

⋅ = , so each person gets of a whole candy bar.

5 4 20 20

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To create two numbers that multiply together to give an answer of one, the numerator of

one is the denominator of the other. You sometimes say one number is the “flip” of the

2 5

other number: flip to get the reciprocal . In order to find the reciprocal of a mixed

5 2

number, write it first as an improper fraction so that it can be “flipped.”

Example

Problem 1

Find the reciprocal of 5 .

4

1 21 1

5 = Rewrite 5 as an improper

4 4 4

fraction. The numerator is

4 • 5 + 1 = 21.

4 Find the reciprocal by

Answer interchanging (“flipping”) the

21

numerator and denominator.

Self Check A

2

What is the reciprocal of 3 ?

5

Dividing a Fraction or a Mixed Number by a Whole Number Objective 3, 4

When you divide by a whole number, you multiply by the reciprocal of the divisor. In the

painting example where you need 3 quarts of paint for a coat and have 6 quarts of paint,

you can find the total number of coats that can be painted by dividing 6 by 3, 6 ÷3 = 2.

1

You can also multiply 6 by the reciprocal of 3, which is , so the multiplication problem

3

6 1 6

becomes ⋅ = = 2.

1 3 3

3

The same idea will work when the divisor is a fraction. If you have of a candy bar and

4

1 1 3

need to divide it among 5 people, each person gets of the available candy: of is

5 5 4

1 3 3 3

⋅ = , so each person gets of a whole candy bar.

5 4 20 20

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3.
If you have a recipe that needs to be divided in half, you can divide each ingredient by 2,

1

or you can multiply each ingredient by to find the new amount.

2

Similarly, with a mixed number, you can either divide by the whole number or you can

1

multiply by the reciprocal. Suppose you have 1 pizzas that you want to divide evenly

2

among 6 people.

1

Dividing by 6 is the same as multiplying by the reciprocal of 6, which is . Cut the

6

available pizza into six equal-sized pieces.

1

Each person gets one piece, so each person gets of a pizza.

4

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1

or you can multiply each ingredient by to find the new amount.

2

Similarly, with a mixed number, you can either divide by the whole number or you can

1

multiply by the reciprocal. Suppose you have 1 pizzas that you want to divide evenly

2

among 6 people.

1

Dividing by 6 is the same as multiplying by the reciprocal of 6, which is . Cut the

6

available pizza into six equal-sized pieces.

1

Each person gets one piece, so each person gets of a pizza.

4

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4.
Dividing a fraction by a whole number is the same as multiplying by the reciprocal, so

you can always use multiplication of fractions to solve such division problems.

Example

Problem 2

Find 2 ÷ 4 . Write your answer as a mixed number

3

with any fraction part in lowest terms.

2 8 2

2 = Rewrite 2 as an improper fraction.

3 3 3

The numerator is 2 • 3 + 2. The

denominator is still 3.

8 8 1 4

÷4 = ⋅ Dividing by 4 or is the same as

3 3 4 1

multiplying by the reciprocal of 4,

1

which is .

4

8 ⋅1 8 Multiply numerators and multiply

= denominators.

3 ⋅ 4 12

Simplify to lowest terms by dividing

2 numerator and denominator by the

common factor 4.

3

Answer 2 2

2 ÷4 =

3 3

Self Check B

3

Find 4 ÷ 2 Simplify the answer and write as a mixed number.

5

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you can always use multiplication of fractions to solve such division problems.

Example

Problem 2

Find 2 ÷ 4 . Write your answer as a mixed number

3

with any fraction part in lowest terms.

2 8 2

2 = Rewrite 2 as an improper fraction.

3 3 3

The numerator is 2 • 3 + 2. The

denominator is still 3.

8 8 1 4

÷4 = ⋅ Dividing by 4 or is the same as

3 3 4 1

multiplying by the reciprocal of 4,

1

which is .

4

8 ⋅1 8 Multiply numerators and multiply

= denominators.

3 ⋅ 4 12

Simplify to lowest terms by dividing

2 numerator and denominator by the

common factor 4.

3

Answer 2 2

2 ÷4 =

3 3

Self Check B

3

Find 4 ÷ 2 Simplify the answer and write as a mixed number.

5

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5.
Dividing by a Fraction Objective 2

Sometimes you need to solve a problem that requires dividing by a fraction. Suppose

1

you have a pizza that is already cut into 4 slices. How many slices are there?

2

1

There are 8 slices. You can see that dividing 4 by gives the same result as

2

multiplying 4 by 2.

What would happen if you needed to divide each slice into thirds?

You would have 12 slices, which is the same as multiplying 4 by 3.

Dividing with Fractions

1. Find the reciprocal of the number that follows the division symbol.

2. Multiply the first number (the one before the division symbol) by the reciprocal of the

second number (the one after the division symbol).

Examples:

2 3 2 1 2 3

6÷ = 6 ⋅ and ÷ = ⋅

3 2 5 3 5 1

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Sometimes you need to solve a problem that requires dividing by a fraction. Suppose

1

you have a pizza that is already cut into 4 slices. How many slices are there?

2

1

There are 8 slices. You can see that dividing 4 by gives the same result as

2

multiplying 4 by 2.

What would happen if you needed to divide each slice into thirds?

You would have 12 slices, which is the same as multiplying 4 by 3.

Dividing with Fractions

1. Find the reciprocal of the number that follows the division symbol.

2. Multiply the first number (the one before the division symbol) by the reciprocal of the

second number (the one after the division symbol).

Examples:

2 3 2 1 2 3

6÷ = 6 ⋅ and ÷ = ⋅

3 2 5 3 5 1

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6.
Any easy way to remember how to divide fractions is the phrase “keep, change, flip”.

This means to KEEP the first number, CHANGE the division sign to multiplication, and

then FLIP (use the reciprocal) of the second number.

Example

Problem 2 1 Divide.

÷

3 6

2 6 Multiply by the reciprocal:

⋅ 2 1

3 1 Keep , change ÷ to •, and flip .

3 6

2 ⋅ 6 12 Multiply numerators and multiply

= denominators.

3 ⋅1 3

12 Simplify.

=4

3

2 1

Answer ÷ =4

3 6

Example

Problem 3 2 Divide.

÷

5 3

3 3 Multiply by the reciprocal:

⋅ 3 2

5 2 Keep , change ÷ to •, and flip .

5 3

3⋅3 9 Multiply numerators and multiply

= denominators.

5 ⋅ 2 10

3 2 9

Answer ÷ =

5 3 10

When solving a division problem by multiplying by the reciprocal, remember to write all

whole numbers and mixed numbers as improper fractions. The final answer should be

simplified and written as a mixed number.

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This means to KEEP the first number, CHANGE the division sign to multiplication, and

then FLIP (use the reciprocal) of the second number.

Example

Problem 2 1 Divide.

÷

3 6

2 6 Multiply by the reciprocal:

⋅ 2 1

3 1 Keep , change ÷ to •, and flip .

3 6

2 ⋅ 6 12 Multiply numerators and multiply

= denominators.

3 ⋅1 3

12 Simplify.

=4

3

2 1

Answer ÷ =4

3 6

Example

Problem 3 2 Divide.

÷

5 3

3 3 Multiply by the reciprocal:

⋅ 3 2

5 2 Keep , change ÷ to •, and flip .

5 3

3⋅3 9 Multiply numerators and multiply

= denominators.

5 ⋅ 2 10

3 2 9

Answer ÷ =

5 3 10

When solving a division problem by multiplying by the reciprocal, remember to write all

whole numbers and mixed numbers as improper fractions. The final answer should be

simplified and written as a mixed number.

2.53

7.
Example

Problem 1 3 Divide.

2 ÷

4 4

9 3 1

÷ Write 2 as an improper fraction.

4 4 4

9 4 Multiply by the reciprocal:

⋅ 9 3

4 3 Keep , change ÷ to •, and flip .

4 4

9 ⋅ 4 36 Multiply numerators and multiply

= denominators.

4 ⋅ 3 12

36 Simplify.

=3

12

1 3

Answer 2 ÷ =3

4 4

Example

Problem 1 1 Divide. Simplify the answer and write as

3 ÷2 a mixed number.

5 10

16 21 1 1

÷ Write 3 and 2 as improper fractions.

5 10 5 10

16 10 21

⋅ Multiply by the reciprocal of .

5 21 10

16 ⋅ 10 Multiply numerators, multiply denominators,

21⋅ 5 and regroup.

16 ⋅ 2 10 2

Simplify: = .

21⋅ 1 5 1

16 ⋅ 2 32 Multiply.

=

21⋅ 1 21

32 11 Rewrite as a mixed number.

=1

21 21

Answer 1 1 11

3 ÷2 1

=

5 10 21

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Problem 1 3 Divide.

2 ÷

4 4

9 3 1

÷ Write 2 as an improper fraction.

4 4 4

9 4 Multiply by the reciprocal:

⋅ 9 3

4 3 Keep , change ÷ to •, and flip .

4 4

9 ⋅ 4 36 Multiply numerators and multiply

= denominators.

4 ⋅ 3 12

36 Simplify.

=3

12

1 3

Answer 2 ÷ =3

4 4

Example

Problem 1 1 Divide. Simplify the answer and write as

3 ÷2 a mixed number.

5 10

16 21 1 1

÷ Write 3 and 2 as improper fractions.

5 10 5 10

16 10 21

⋅ Multiply by the reciprocal of .

5 21 10

16 ⋅ 10 Multiply numerators, multiply denominators,

21⋅ 5 and regroup.

16 ⋅ 2 10 2

Simplify: = .

21⋅ 1 5 1

16 ⋅ 2 32 Multiply.

=

21⋅ 1 21

32 11 Rewrite as a mixed number.

=1

21 21

Answer 1 1 11

3 ÷2 1

=

5 10 21

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8.
Self Check C

1 2

Find 5 ÷ . Simplify the answer and write as a mixed number.

3 3

Dividing Fractions or Mixed Numbers to Solve Problems Objective 5

Using multiplication by the reciprocal instead of division can be very useful to solve

problems that require division and fractions.

Example

Problem 3

A cook has 18 pounds of ground beef. How many quarter-

4

pound burgers can he make?

3 1 You need to find how many quarter pounds there

18 ÷ 3

4 4 are in 18 , so use division.

4

75 1 3

÷ Write 18 as an improper fraction.

4 4 4

75 4 Multiply by the reciprocal.

⋅

4 1

75 ⋅ 4 Multiply numerators and multiply denominators.

4 ⋅1

4 75 4

⋅ Regroup and simplify , which is 1.

4 1 4

Answer 75 burgers

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1 2

Find 5 ÷ . Simplify the answer and write as a mixed number.

3 3

Dividing Fractions or Mixed Numbers to Solve Problems Objective 5

Using multiplication by the reciprocal instead of division can be very useful to solve

problems that require division and fractions.

Example

Problem 3

A cook has 18 pounds of ground beef. How many quarter-

4

pound burgers can he make?

3 1 You need to find how many quarter pounds there

18 ÷ 3

4 4 are in 18 , so use division.

4

75 1 3

÷ Write 18 as an improper fraction.

4 4 4

75 4 Multiply by the reciprocal.

⋅

4 1

75 ⋅ 4 Multiply numerators and multiply denominators.

4 ⋅1

4 75 4

⋅ Regroup and simplify , which is 1.

4 1 4

Answer 75 burgers

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9.
Example

Problem 1

A child needs to take 2 tablespoons of medicine per day in 4 equal doses.

2

How much medicine is in each dose?

1 You need to make 4 equal doses, so you can use division.

2 ÷4

2

5 1

÷4 Write 2 as an improper fraction.

2 2

5 1

⋅ Multiply by the reciprocal.

2 4

5 ⋅1 5 Multiply numerators and multiply denominators. Simplify, if

= possible.

2⋅4 8

Answer 5

tablespoon in each dose.

8

Self Check D

2

How many -cup salt shakers can be filled from 12 cups of salt?

5

Summary

Division is the same as multiplying by the reciprocal. When working with fractions, this is

the easiest way to divide. Whether you divide by a number or multiply by the reciprocal

of the number, the result will be the same. You can use these techniques to help you

solve problems that involve division, fractions, and/or mixed numbers.

2.4 Self Check Solutions

Self Check A

2

What is the reciprocal of 3 ?

5

5 2 17 17

. First, write 3 as an improper fraction , . The reciprocal of is found by

17 5 5 5

interchanging (“flipping”) the numerator and denominator.

2.56

Problem 1

A child needs to take 2 tablespoons of medicine per day in 4 equal doses.

2

How much medicine is in each dose?

1 You need to make 4 equal doses, so you can use division.

2 ÷4

2

5 1

÷4 Write 2 as an improper fraction.

2 2

5 1

⋅ Multiply by the reciprocal.

2 4

5 ⋅1 5 Multiply numerators and multiply denominators. Simplify, if

= possible.

2⋅4 8

Answer 5

tablespoon in each dose.

8

Self Check D

2

How many -cup salt shakers can be filled from 12 cups of salt?

5

Summary

Division is the same as multiplying by the reciprocal. When working with fractions, this is

the easiest way to divide. Whether you divide by a number or multiply by the reciprocal

of the number, the result will be the same. You can use these techniques to help you

solve problems that involve division, fractions, and/or mixed numbers.

2.4 Self Check Solutions

Self Check A

2

What is the reciprocal of 3 ?

5

5 2 17 17

. First, write 3 as an improper fraction , . The reciprocal of is found by

17 5 5 5

interchanging (“flipping”) the numerator and denominator.

2.56

10.
Self Check B

3

Find 4 ÷ 2 Simplify the answer and write as a mixed number.

5

3 23 1

Write 4 as the improper fraction . Then multiply by , the reciprocal of 2. This

5 5 2

23 3

gives the improper fraction , and the mixed number is 23 ÷ 10 = 2R3 , 2 .

10 10

Self Check C

1 2

Find 5 ÷ . Simplify the answer and write as a mixed number.

3 3

1 16 2

Write 5 as an improper fraction, . Then multiply by the reciprocal of , which is

3 3 3

3 16 3 3 16 16

, giving you ⋅ = ⋅ = =8.

2 3 2 3 2 2

Self Check D

2

How many -cup salt shakers can be filled from 12 cups of salt?

5

2 12

12 ÷ will show how many salt shakers can be filled. Write 12 as and multiply by

5 1

2 12 5 60

the reciprocal (“flip”) of , giving you ⋅ = = 30 .

5 1 2 2

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3

Find 4 ÷ 2 Simplify the answer and write as a mixed number.

5

3 23 1

Write 4 as the improper fraction . Then multiply by , the reciprocal of 2. This

5 5 2

23 3

gives the improper fraction , and the mixed number is 23 ÷ 10 = 2R3 , 2 .

10 10

Self Check C

1 2

Find 5 ÷ . Simplify the answer and write as a mixed number.

3 3

1 16 2

Write 5 as an improper fraction, . Then multiply by the reciprocal of , which is

3 3 3

3 16 3 3 16 16

, giving you ⋅ = ⋅ = =8.

2 3 2 3 2 2

Self Check D

2

How many -cup salt shakers can be filled from 12 cups of salt?

5

2 12

12 ÷ will show how many salt shakers can be filled. Write 12 as and multiply by

5 1

2 12 5 60

the reciprocal (“flip”) of , giving you ⋅ = = 30 .

5 1 2 2

2.57