## Dividing Rational Numbers

#### Rules for Division

1. Like signs; positive
2. 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.

### 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

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

1. -15/3

2. -30/-6

3. 56/8

4. 63/-9

5. Hint: To divide complex fractions, rewrite as a division of fractions problem with
becoming the division sign.

6. Hint: To divide complex fractions, rewrite as a division of fractions problem with
becoming the division sign.

7.   5x - 25
5

8.   -7x + 42
-7

9.   12x + 18
3

10. -12 / -3

11. -24 / 6

1. -15/3 = -5
2. -30/-6 = 5
3. 56/8 = 7
4. 63/-9 = -7
5. -2/1 / 4/5
-12/1 / 5/42
-5/2
6. 7/8 / -6/1
7/8 / -1/6
-7/48
7. -2/3 / -4/9
-12/31 / -39/42
3/2
8.   5x - 25    =  x - 5
5

9.   -7x + 42    =  x - 6
-7

10.   12x + 18    =  4x + 6
3

11. -12 / -3 = 4
12. -24 / 6 = -4