Reciprocals can help you divideby fractions. Two numbers arereciprocals if their product is 1.
Additional Example 1A: Finding Reciprocals
Find the reciprocal.
1
9
__
1
9
__
• = 1
Think: of what number is 1?
1
9
__
1
9
__
1
9
__
• 9 = 1
of is 1.
1
9
__
9
1
__
The reciprocal of is 9.
1
9
__
Additional Example 1B: Finding Reciprocals
Find the reciprocal.
2
3
__
2
3
__
• = 1
Think: of what number is 1?
2
3
__
3
2
__
2
3
__
• = = 1
6
6
__
of is 1.
2
3
__
3
2
__
The reciprocal of is .
2
3
__
3
2
__
1
5
__
16
5
__
• = 1
Write 3 as .
1
5
__
5
16
__
16
5
__
• = = 1
80
80
__
of is 1.
16
5
__
5
16
__
The reciprocal of is .
16
5
__
5
16
__
16
5
__
Additional Example 1C: Finding Reciprocals
Find the reciprocal.
3
Check It Out: Example 1A
Find the reciprocal.
1
4
__
1
4
__
• = 1
Think: of what number is 1?
1
4
__
1
4
__
• 4 = 1
of is 1.
1
4
__
4
1
__
The reciprocal of is 4.
1
4
__
Check It Out: Example 1B
Find the reciprocal.
4
5
__
4
5
__
• = 1
Think: of what number is 1?
4
5
__
5
4
__
4
5
__
• = = 1
20
20
__
of is 1.
4
5
__
5
4
__
The reciprocal of is .
4
5
__
5
4
__
1
8
__
33
8
__
• = 1
Write 4 as .
1
8
__
8
33
__
33
8
__
• = = 1
264264
___
of is 1.
33
8
__
8
33
__
The reciprocal of is .
33
8
__
8
33
__
33
8
__
Check It Out: Example 1C
Find the reciprocal.
4
Look at the relationship between the
fractions and . If you switch the
numerator and denominator of a fraction,
you will find its reciprocal. Dividing by a
number is the same as multiplying by its
reciprocal.
3
4
__
4
3
__
24 ÷ 4 = 6
24 • = 6
1
4
__
Additional Example 2A: Using Reciprocals toDivide Fractions and Mixed Numbers
Divide. Write each answer in simplest form.
÷ 7
8
7
__
8
7
__
÷ 7 = •
Rewrite as multiplication using the
reciprocal of 7, .
1
7
__
Multiply by the reciprocal.
8
7
__
1
7
__
8 • 1
7 • 7
____
=
The answer is in simplest form.
=
8
49
__
When you divide by a proper fraction, the quotientwill be greater than the dividend. For example,since there are 8 halves in 4, 4 ½ = 8.
Caution!
Additional Example 2B: Using Reciprocals toDivide Fractions and Mixed Numbers
÷
5
6
__
5
6
__
÷ = •
Rewrite as multiplication using the
reciprocal of , .
3
2
__
Simplify before multiplying.
5
6
__
3
2
__
5 • 3
6 • 2
____
=
You can write the answer as amixed number.
=
5
4
__
2
3
__
2
3
__
2
3
__
1
2
Multiply.
= 1
1
4
__
Additional Example 2C: Using Reciprocals toDivide Fractions and Mixed Numbers
2 ÷ 1
3
4
__
Write the mixed numbers as
improper fractions. 2 =
and 1 =
13
12
__
Rewrite as multiplication.
11 • 12
4 • 13
______
=
You can write the answer as amixed number.
=
33
13
__
1
12
__
3
4
__
1
3
Simplify before multiplying.
= 2
7
13
__
2 ÷ 1 = ÷
3
4
__
1
12
__
11
4
__
13
12
__
11
4
__
1
12
__
11
4
__
12
13
__
•
Multiply.
=
Check It Out: Example 2A
Divide. Write each answer in simplest form.
÷ 3
2
3
__
2
3
__
÷ 3 = •
Rewrite as multiplication using the
reciprocal of 3, .
1
3
__
Multiply by the reciprocal.
2
3
__
1
3
__
2 • 1
3 • 3
____
=
The answer is in simplest form.
=
2
9
__
Check It Out: Example 2B
÷
7
10
__
7
10
__
÷ = •
Rewrite as multiplication using the
reciprocal of , .
5
1
__
Simplify before multiplying.
7
10
__
5
1
__
7 • 5
10 • 1
____
=
You can write the answer as amixed number.
=
7
2
__
1
5
__
1
5
__
1
5
__
1
2
Multiply.
= 3
1
2
__
Check It Out: Example 2C
3 ÷ 1
2
3
__
Write the mixed numbers as
improper fractions. 3 =
and 1 = .
10
9
__
Rewrite as multiplication.
11 • 9
3 • 10
______
=
You can write the answer as amixed number.
=
33
10
__
1
9
__
2
3
__
1
3
Simplify before multiplying.
= 3
3
10
__
3 ÷ 1 = ÷
2
3
__
1
9
__
11
3
__
10
9
__
11
3
__
1
9
__
11
3
__
9
10
__
•
Multiply.
=