Rules for Division
 Like signs; positive
 Unlike signs; negative
Multiplicative Inverse Property
Remember that when you divide fractions, you change the division problem to a multiplication problem and multiply by the reciprocal or multiplicative inverse.
Numbers that are reciprocals or multiplicative inverses are numbers whose product is one.
Examples
Reasoning

2/3 and 3/2
2/3(3/2) = 1

5/4 and 4/5
5/4(4/5) = 1

5 and 1/5
5/1(1/5) = 1

Change an integer into a fraction by putting it over one 
Multiplicative Inverse Property:
For every nonzero number a, there is a 1/a so
a(1/a) = 1 or 1/a(a) = 1
Examples
Reasoning

25/5 = 5
25/5 = 5 
Like signs; positive 
24/8 = 3
24/8 = 3 
Unlike signs; negative 
Remember to use your rules to find the sign of your answer first.
Suggestion: In a division of fractions problem, if one of the numbers is not a fraction, make it a fraction by putting it over one.
Examples
Reasoning

2/3 / 4 
Make 4 a fraction by putting it over one 
2/3 / 4/1 
Multiply by the reciprocal 1/4 
12/3 / 1/42 
Unlike signs; negative
Reduce 2 into 4 twice 
1/6 
Multiply 1/3(1/2) = 1/6 
When multiplying fractions, you multiply numerators and numerators and denominators and denominators.
Reasoning

6 / 3/8 
Make the 6 a fraction by putting it over one 
6/1 / 3/8 
Multiply by the reciprocal 8/3 
26/1 / 8/13 
Like signs; positive
Reduce 3 into 6 twice 
16 
Multiply (2/1)(8/1) = 16/1 = 16 
Complex Fractions
There are even more complex fractions where the numerator, the denominator, or both are fractions. These fractions are in fact called complex fractions.
Examples
To divide complex fractions, rewrite as a division of fractions problem with the becoming the division sign.
Examples
Reasoning


Rewrite as a division of fractions problem. 
2/3 / 4/1 
Place the 4 over one
Multiply by the reciprocal 1/4 
12/3 / 1/42 
Like signs; positive
Reduce 2 into 4 twice 
1/6 
Multiply 1/3(1/2) = 1/6 
Reasoning


Rewrite as a division of fractions problem 
5/1 / 10/3 
Place the five over one
Multiply by the reciprocal 3/10 
15/1 / 3/102

Unlike signs; negative
Reduce 5 into 10 twice 
3/2

Multiply 1/1(3/2) = 3/2 
Reasoning


Rewrite as a division of fractions problem 
2/3 / 4/5 
Multiply by the reciprocal 5/4 
12/3 / 5/42 
Like signs; positive
Reduce 2 into 4 twice 
5/6 
Multiply 1/3(5/2) = 5/6 
Not only can you use the distributive property for multiplication, but you can also use it for division.
Examples
Reasoning


Since a fraction bar is a grouping symbol, this problem tells us to divide both the 3x and the 6 by 3 
x + 2 
3x/3 = x the 3s will cancel
6/3 = 2 
Reasoning


Divide both 
2x + 3 
10x/5 = 2x Unlike signs; negative
and
15/5 = 3 Like signs; positive 
Simplify
 15/3
 30/6
 56/8
 63/9
 Hint: To divide complex fractions, rewrite as a division of fractions problem with
becoming the division sign.
 Hint: To divide complex fractions, rewrite as a division of fractions problem with
becoming the division sign.
5x  25
5
7x + 42
7
12x + 18
3
 12 / 3
 24 / 6
Answers to Practice Problems
 15/3 = 5
 30/6 = 5
 56/8 = 7
 63/9 = 7
 2/1 / 4/5
12/1 / 5/42
5/2
 7/8 / 6/1
7/8 / 1/6
7/48
 2/3 / 4/9
12/31 / 39/42
3/2
5x  25 = x  5
5
7x + 42 = x  6
7
 12x + 18 = 4x + 6
3

12 / 3 = 4
 24 / 6 = 4