Download 4.3 Dividing Fractions Example 1: Write the reciprocal of the

Math 40
Prealgebra
Section 4.3 – Dividing Fractions
4.3 Dividing Fractions
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
1
1
1
1
The reciprocal of the fraction a is b since a  b  a  b  1 1  1  1
b
a
b a
1 1 1
b a
Reciprocals are sometimes called multiplicative inverses.
Example 1: Write the reciprocal of the following numbers.
a)
5
3
d)  2
b) 1
c) 6
Number
Reciprocal
8
17
Solution:
5
3
1
8
think of  6 as 
6
6
1
Reason
1
1
1
1
1
1
1
1
5 3
5 3
1 1 1
 

 1
3 5
1 1 1
3 5
3
5
1 8
1 8
1 1 1
 

 1
8 1
1 1 1
8 1
8
or 8
1
1
1
1
1
6 1
1 1 1
 6  1 

 1
      
1 1 1
1 6
 1  6 
1

6
1
2

17
1
2  17
1 1 1
 2   17 

 1

 

1 1 1
17  2
 17   2 
17

2
1
1
Notice that you “flip” or “invert” a fraction to find its reciprocal.
Caution 1: The number 0 does not have a reciprocal since the product of zero and a number is always zero (and
never 1). In order for two numbers to be reciprocals of each other, their product must be 1.
0   any number   0  1
Caution 2: Reciprocals are not equal. Be careful when writing your answers! For example, in part (a), the
reciprocal of 5 is 3 . We don’t write 5  3 since this is a false statement.
3
5
3
5
1
2015 Carreon
Math 40
Prealgebra
Section 4.3 – Dividing Fractions
You Try It 1: Write the reciprocal of each of the following numbers.
a)
b)  5
24
c) 1
6
13
d)  9
8
Dividing Fractions
If a, b, c, and d are numbers with (b, c, and d not zero), then:
a c
a d
  
b d
b c
In other words, we change division to multiplying by the reciprocal of the divisor.
Then use methods of section 4.2 to multiply the fractions.
Example 2: Find the quotient.
5  12 
 
8  35 
Solution:
Figure out the sign of the quotient first! The quotient of two numbers with different signs is negative.
Write the negative sign at the beginning of your work.
Rewrite division as multiplying by the reciprocal of the divisor.
Use methods of 4.2 to multiply:
5  12 
5 12
    
8  35 
8 35
5 12
8  35
5 2 23

2 2 257
5  2  2 3

2  2 2 5 7
1 1 1  3

1 1  2 1  7
3

14

2
2015 Carreon
Math 40
Prealgebra
Section 4.3 – Dividing Fractions
You Try It 2: Find the quotient.
a)
24 12

15 25
b)
48  72 
 
42  56 
Example 3: Find the quotient.
a) 12 
1
3
3
4
b)    8
Solution:
a) Figure out the sign of the quotient first! The quotient of two numbers with different signs is negative.
Write the negative sign at the beginning of your work.
Rewrite division as multiplying by the reciprocal of the divisor.
Rewrite 12 in fraction form as
12
.
1
Use methods of 4.2 to multiply:
1
3
12   12 
3
1
12 3
 
1 1
12  3

1 1
36

1
 36
3
2015 Carreon
Math 40
Prealgebra
Section 4.3 – Dividing Fractions
b) Figure out the sign of the quotient first! The quotient of two numbers with the same sign is positive.
Rewrite 8 in fraction form as
8
.
1
Rewrite division as multiplying by the reciprocal of the divisor.
Use methods of 4.2 to multiply:
3
3 8
   8     
4
4 1
3 1
 
4 8
3 1

4 8
3

32
You Try It 3: Find the quotient.
 1
 3
5
8
b)   6
a) 9    
Example 4: Find the quotient.
a) 8a 
12c
a
b)
3x 2 8 x5

12 y 3 24 y 3
Solution:
a) Figure out the sign of the quotient first! The quotient of two numbers with the same sign is positive.
Rewrite division as multiplying by the reciprocal of the divisor.
Rewrite 8a in fraction form as
Use methods of 4.2 to multiply:
8a
.
1
8a 
12c
a
 8a 
a
12c
8a a
 
1 12c
8a  a

1 12c
222a a

1 2  2  3  c
4
2015 Carreon
Math 40
Prealgebra
Section 4.3 – Dividing Fractions
2  2 2aa
1 2  2  3  c
1 1  2  a  a

1 1 1  3  c
2a 2

3c

b) Figure out the sign of the quotient first! The quotient of two numbers with the same sign is positive.
Rewrite division as multiplying by the reciprocal of the divisor.
3x 2 8 x5
3x 2 24 y 3




12 y 3 24 y 3
12 y 3 8 x5
Use methods of 4.2 to multiply:
3 x 2  24 y 3

12 y 3  8 x5
3 x  x  2  2  2 3 y  y  y

2  2 3 y  y  y  2  2  2  x  x  x  x  x

3  x  x  2  2  2 3 y  y  y
2  2  3  y  y  y  2 22 x  x  x x x
1 1 1 1 1 1  3 1 1 1
1 1 1 1 1 1 1  2  2 1 1  x  x  x
3
 3
4x

You Try It 4: Find the quotient.
15a 2 35a 5

a)
6c 2 24c 3
7 y2
 28 x 2
b)
x
5
2015 Carreon
Math 40
Prealgebra
Section 4.3 – Dividing Fractions
Example 5: Lester made 10 casseroles for a party. How many guests can be fed if each guest will be served
1
of a casserole?
8
Solution:
The problem is asking how many 1 ’s there are in 10. This is a key phase for division.
8
1
8
10 1
 
1 8
8
 10 
1
Translate the words of the problem to the following mathematical expression: 10 
Rewrite 10 in fraction form as
10
.
1
Rewrite division as multiplying by the reciprocal of the divisor.
Use methods of 4.2 to multiply:
10  8
1 1
80

1
 80

State the answer to the problem.
80 guests can be fed from the casseroles.
You Try It 5: Christian is making scarves to give to a local shelter. A fabric store donated a 12 yard bolt of
fabric for the project. How many scarves can Christian make if each scarf requires 3 yard of
4
fabric?
6
2015 Carreon
Math 40
Prealgebra
Section 4.3 – Dividing Fractions
Example 6: Desiree has a truck that can carry 2 ton of gravel. Find the number of trips needed to deliver 6
3
tons of gravel.
Solution:
The problem is asking how many 2 ’s there are in 6. This is a key phase for division.
3
2
3
6 2
 
1 3
6 3
 
1 2
63

1 2
2 33

1 2
2 3 3

1 2
1 3  3

1 1
9

1
9
Translate the words of the problem to the following mathematical expression: 6 
Rewrite 6 in fraction form as
6
.
1
Rewrite division as multiplying by the reciprocal of the divisor.
Use methods of 4.2 to multiply:
State the answer to the problem.
9 trips are needed to deliver 6 tons of gravel.
You Try It 6: How many times can a 2 quart spray bottle be filled from an 18-quart bottle of window cleaner?
3
7
2015 Carreon
Math 40
Prealgebra
Section 4.3 – Dividing Fractions
Example 7: A production process applies several layers of a clear plastic coat to outdoor furniture to help
protect it from the weather. If each protective coat is 3 inch thick, how many applications will
32
be needed to build up 3 inch of clear finish?
8
Solution:
The problem is asking how many 3 ’s there are in 3 . This is a key phase for division.
32
8
3 3

8 32
3 32
 
8 3
3  32

83
3 2  2  2  2  2

2 2 23
3  2  2  2 22

2223
1 1 1 1  2  2

1 1 1 1
4

1
4
Translate the words of the problem to the following mathematical expression:
Rewrite division as multiplying by the reciprocal of the divisor.
Use methods of 4.2 to multiply:
State the answer to the problem.
4 applications are needed .
You Try It 7: A production process applies several layers of a clear plastic coat to outdoor furniture to help
protect it from the weather. If each protective coat is 3 inch thick, how many applications
16
will be needed to build up 3 inch of clear finish?
8
8
2015 Carreon