# Fractions

## Multiply and Divide Fractions


### Simplify Fractions

In working with equivalent fractions, you saw that there are many ways to write fractions that have the same value, or represent the same part of the whole. How do you know which one to use? Often, we’ll use the fraction that is in simplified form.
A fraction is considered simplified if there are no common factors, other than  in the numerator and denominator. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.
For example,
1.  is simplified because there are no common factors of  and 
2.  is not simplified because  is a common factor of  and 
The process of simplifying a fraction is often called reducing the fraction. In the previous section, we used the Equivalent Fractions Property to find equivalent fractions. We can also use the Equivalent Fractions Property in reverse to simplify fractions. We rewrite the property to show both forms together.
Notice that  is a common factor in the numerator and denominator. Anytime we have a common factor in the numerator and denominator, it can be removed.
To simplify a negative fraction, we use the same process as in . Remember to keep the negative sign.
After simplifying a fraction, it is always important to check the result to make sure that the numerator and denominator do not have any more factors in common. Remember, the definition of a simplified fraction: a fraction is considered simplified if there are no common factors in the numerator and denominator. 
When we simplify an improper fraction, there is no need to change it to a mixed number.
Sometimes it may not be easy to find common factors of the numerator and denominator. A good idea, then, is to factor the numerator and the denominator into prime numbers. (You may want to use the factor tree method to identify the prime factors.) Then divide out the common factors using the Equivalent Fractions Property.
We can also simplify fractions containing variables. If a variable is a common factor in the numerator and denominator, we remove it just as we do with an integer factor.

### Multiply Fractions

A model may help you understand multiplication of fractions. We will use fraction tiles to model  To multiply  and  think  of 
Start with fraction tiles for three-fourths. To find one-half of three-fourths, we need to divide them into two equal groups. Since we cannot divide the three  tiles evenly into two parts, we exchange them for smaller tiles.
We see  is equivalent to  Taking half of the six  tiles gives us three  tiles, which is 
Therefore,
Look at the result we got from the model in . We found that  Do you notice that we could have gotten the same answer by multiplying the numerators and multiplying the denominators?
This leads to the definition of fraction multiplication. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.
When multiplying fractions, the properties of positive and negative numbers still apply. It is a good idea to determine the sign of the product as the first step. In Example 4.26 we will multiply two negatives, so the product will be positive.
When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer,  can be written as  So,  for example.

### Find Reciprocals

The fractions  and  are related to each other in a special way. So are  and  Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be 
Such pairs of numbers are called reciprocals.
To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator.
To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign. 
To find the reciprocal, keep the same sign and invert the fraction. The number zero does not have a reciprocal. Why? A number and its reciprocal multiply to  Is there any number  so that  No. So, the number  does not have a reciprocal.
In a previous chapter, we worked with opposites and absolute values.  compares opposites, absolute values, and reciprocals.

### Divide Fractions

Why is  We previously modeled this with counters. How many groups of  counters can be made from a group of  counters?
There are  groups of  counters. In other words, there are four s in  So, 
What about dividing fractions? Suppose we want to find the quotient:  We need to figure out how many s there are in  We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown in . Notice, there are three  tiles in  so 
Let’s use money to model  in another way. We often read  as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in . So we can think of  as, “How many quarters are there in two dollars?” One dollar is  quarters, so  dollars would be  quarters. So again, 
Using fraction tiles, we showed that  Notice that  also. How are  and  related? They are reciprocals. This leads us to the procedure for fraction division.
We need to say  and  to be sure we don’t divide by zero.

### Key Concepts

1. Equivalent Fractions Property
2. Simplify a fraction.
3. Fraction Multiplication
4. Reciprocal
5. Fraction Division

### Practice Makes Perfect

Simplify Fractions 
In the following exercises, simplify each fraction. Do not convert any improper fractions to mixed numbers.
Multiply Fractions 
In the following exercises, use a diagram to model.
In the following exercises, multiply, and write the answer in simplified form.
Find Reciprocals 
In the following exercises, find the reciprocal.
Divide Fractions 
In the following exercises, model each fraction division.
In the following exercises, divide, and write the answer in simplified form.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?
