The denominator of any fraction cannot have the value zero. If the denominator of a fraction is zero, the expression is not a legal fraction because it's overall value is undefined
This means
,
,
are not legal fractions. Their values are all undefined, and hence they have no meaning. Once you encounter such a fraction in a problem, stop. You cannot proceed with the problem.
A numerator is allowed to take on the value of zero in a fraction. Any legal fraction (denominator not equal to zero) with a numerator equal to zero has an overall value of zero.
This means
,
,
all have a fraction value of zero because the numerators are equal to zero.
Different Faces of the Number 1
The number 1 can take on many forms. 4 - 3 = 1 and 10 - 9 = 1 can be used as a substitution for the number 1 because they have a value of 1. When the numerator of a fraction is equivalent to the denominator of a fraction, the value of the fraction is 1. This only works when you have a legal fraction; i.e., the denominator does not equal zero. You can substitute one of these fractions for the numer 1.
Any Integer Can Be Written as a Fraction
You can express an integer as a fraction by simply dividing by 1, or you can express any integer as a fraction by simply choosing a numerator and denominator so that the overall value is equal to the integer.
To reduce a simple fraction, follow the following three steps:
- 1.
- Factor the numerator.
- 2.
- Factor the denominator.
- 3.
- Find the fraction mix that equals 1.
For example, reduce
.
First: Rewrite the fraction with the numerator and the denominator factored.
Note all factors in the numerator and denominator are separated by multiplication signs.
Second: Find the fraction that equals 1.
can be written
which in turn can be written
which in turn can be written
.
Third: We have just illustrated that
. Although the left side of the equal sign does not look identical to the right side of the equal sign, both fractions are equivalent because they have the same value. Check it with your calculator.
and
. This proves that the fraction
can be reduced to the equivalent fraction
.
A compound fraction is sometimes called a mixed number. Recall that 
, 
, and 
are examples of compound fractions.
- To manipulate compound fractions, just convert them to simple fractions and follow rules 1 through 23 for simple fractions.Example 1: Convert to a simple fraction.
Answer. can be written 
. Write 3 as the fraction 
. Now can be written 
and
- To convert a simple fraction to a compound fraction, the numerator must be larger than the denominator. Separate the whole number first.Example 2: Convert
to a compound fraction.
Answer. Rewrite as 
. We have
- To add or subtract two compound fractions, convert the fractions to simple fractions and follow the steps you use to add or subtract two simple fractions.Example 3: Calculate
.
Solution We have
- To multiply or divide two compound fractions, convert the fractions to simple fractions and follow the steps you use to multiply or divide two simple fractions.Example 4: Calculate
.
Solution. We have
For this review, we will focus on terminating fractions. Divide the fraction by 1, and then multiply the result by 1 in a form that will remove the decimal.
Example: Convert 2.3 to a fraction.
Answer. 2.3 can be written
which can be multiplied by 1 and divided by 1 without changing its value. We have
Recall that 
. To convert a percentage to a fraction, simply convert 1% to 
. To convert a percentage to a decimal, simply convert 1% to .01.
Example 1: Convert
to a fraction.
Answer. can be written
which in turn can be written
. Multiply the two fractions using the multiplication rule to get
Example 2: Convert
to a decimal.
Answer. can be written
