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.
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 |
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. |
|
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 |
Answers to Practice Problems
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 |