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## Dividing Rational Numbers

### 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.

### 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

### 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.

### 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.

### 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.

### 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 6 Hint:  To divide complex fractions, rewrite as a division of fractions problem with becoming the division sign. 7 Hint:  To divide complex fractions, rewrite as a division of fractions problem with becoming the division sign. 8 5x - 25        5 9 -7x + 42        -7 10 12x + 18        3 11 -12 / -3 12 -24 / 6