1. Operations with Real Numbers
1.3 Fractions
Learning Objectives
By the end of this section it is expected that you will be able to:
- Multiply and divide fractions
- Simplifying expressions with fraction bar
- Add or subtract fractions with a common denominator
- Add or subtract fractions with different denominators
- Use the order of operations to simplify complex fractions
Multiply Fractions
Many people find multiplying and dividing fractions easier than adding and subtracting fractions. So we will start with fraction multiplication.
We’ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let’s start with .
Now we’ll take of .
Notice that now, the whole is divided into 8 equal parts. So .
To multiply fractions, we multiply the numerators and multiply the denominators.
Fraction Multiplication
If are numbers where , then
To multiply fractions, multiply the numerators and multiply the denominators.
When multiplying fractions, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In Example 1, we will multiply negative and a positive, so the product will be negative.
EXAMPLE 1
Multiply: .
The first step is to find the sign of the product. Since the signs are the different, the product is negative.
| Determine the sign of the product; multiply. | |
| Are there any common factors in the numerator and the demoninator? No. |
TRY IT 1
Multiply: .
Show answer
When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a, can be written as . So, for example, .
EXAMPLE 2
Multiply: .
Determine the sign of the product. The signs are the same, so the product is positive.
| Write as a fraction. | |
| Multiply. | |
| Rewrite 20 to show the common factor 5 and divide it out. | |
| Simplify. |
TRY IT 2
Multiply: .
Show answer
Divide Fractions
Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, that we need some vocabulary.
The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of is .
Notice that . A number and its reciprocal multiply to 1.
To get a product of positive 1 when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.
The reciprocal of is , since .
Reciprocal
The reciprocal of is .
A number and its reciprocal multiply to one .
To divide fractions, we multiply the first fraction by the reciprocal of the second.
Fraction Division
If are numbers where , then
We need to say to be sure we don’t divide by zero!
EXAMPLE 3
Find the quotient: .
| To divide, multiply the first fraction by the reciprocal of the second. | |
| Determine the sign of the product, and then multiply.. | |
| Rewrite showing common factors. | |
| Remove common factors. | |
| Simplify. |
TRY IT 3
Find the quot