Foundations
Fractions
Learning Objectives
By the end of this section, you will be able to:
- Simplify fractions
- Multiply and divide fractions
- Add and subtract fractions
- Use the order of operations to simplify fractions
- Evaluate variable expressions with fractions
A more thorough introduction to the topics covered in this section can be found in the Elementary Algebra chapter, Foundations.
Simplify Fractions
A fraction is a way to represent parts of a whole. The fraction represents two of three equal parts. See (Figure). In the fraction the 2 is called the numerator and the 3 is called the denominator. The line is called the fraction bar.
A fraction is written where and
a is the numerator and b is the denominator.
A fraction represents parts of a whole. The denominator is the number of equal parts the whole has been divided into, and the numerator indicates how many parts are included.
Fractions that have the same value are equivalent fractions. The Equivalent Fractions
Property allows us to find equivalent fractions and also simplify fractions.
If a, b, and c are numbers where
then and
A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.
For example,
is simplified because there are no common factors of 2 and
is not simplified because 5 is a common factor of 10 and
We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.
Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the Equivalent Fractions Property.
Simplify:
Simplify:
Simplify:
We now summarize the steps you should follow to simplify fractions.
- Rewrite the numerator and denominator to show the common factors.
If needed, factor the numerator and denominator into prime numbers first.
- Simplify using the Equivalent Fractions Property by dividing out common factors.
- Multiply any remaining factors.
Multiply and Divide Fractions
Many people find multiplying and dividing fractions easier than adding and subtracting fractions.
To multiply fractions, we multiply the numerators and multiply the denominators.
If a, b, c, and d are numbers where and 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 (Figure), we will multiply negative and a positive, so the product will be negative.
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,
Multiply:
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