# Fractions

## Add and Subtract Fractions with Different Denominators


### Find the Least Common Denominator

In the previous section, we explained how to add and subtract fractions with a common denominator. But how can we add and subtract fractions with unlike denominators?
Let’s think about coins again. Can you add one quarter and one dime? You could say there are two coins, but that’s not very useful. To find the total value of one quarter plus one dime, you change them to the same kind of unit—cents. One quarter equals  cents and one dime equals  cents, so the sum is  cents. See .
Similarly, when we add fractions with different denominators we have to convert them to equivalent fractions with a common denominator. With the coins, when we convert to cents, the denominator is  Since there are  cents in one dollar,  cents is  and  cents is  So we add  to get  which is  cents.
You have practiced adding and subtracting fractions with common denominators. Now let’s see what you need to do with fractions that have different denominators.
First, we will use fraction tiles to model finding the common denominator of  and 
We’ll start with one  tile and  tile. We want to find a common fraction tile that we can use to match both  and  exactly.
If we try the  pieces,  of them exactly match the  piece, but they do not exactly match the  piece.
If we try the  pieces, they do not exactly cover the  piece or the  piece.
If we try the  pieces, we see that exactly  of them cover the  piece, and exactly  of them cover the  piece.
If we were to try the  pieces, they would also work.
Even smaller tiles, such as  and  would also exactly cover the  piece and the  piece.
The denominator of the largest piece that covers both fractions is the least common denominator (LCD) of the two fractions. So, the least common denominator of  and  is 
Notice that all of the tiles that cover  and  have something in common: Their denominators are common multiples of  and  the denominators of  and  The least common multiple (LCM) of the denominators is  and so we say that  is the least common denominator (LCD) of the fractions  and 
To find the LCD of two fractions, we will find the LCM of their denominators. We follow the procedure we used earlier to find the LCM of two numbers. We only use the denominators of the fractions, not the numerators, when finding the LCD.
To find the LCD of two fractions, find the LCM of their denominators. Notice how the steps shown below are similar to the steps we took to find the LCM.

### Convert Fractions to Equivalent Fractions with the LCD

Earlier, we used fraction tiles to see that the LCD of  when  is  We saw that three  pieces exactly covered  and two  pieces exactly covered  so 
We say that  and  are equivalent fractions and also that  and  are equivalent fractions.
We can use the Equivalent Fractions Property to algebraically change a fraction to an equivalent one. Remember, two fractions are equivalent if they have the same value. The Equivalent Fractions Property is repeated below for reference.
To add or subtract fractions with different denominators, we will first have to convert each fraction to an equivalent fraction with the LCD. Let’s see how to change  and  to equivalent fractions with denominator  without using models.

### Add and Subtract Fractions with Different Denominators

Once we have converted two fractions to equivalent forms with common denominators, we can add or subtract them by adding or subtracting the numerators.
When we use the Equivalent Fractions Property, there is a quick way to find the number you need to multiply by to get the LCD. Write the factors of the denominators and the LCD just as you did to find the LCD. The “missing” factors of each denominator are the numbers you need.
The LCD,  has  factors of  and  factors of 
Twelve has two factors of  but only one of —so it is ‘missing‘ one  We multiplied the numerator and denominator of  by  to get an equivalent fraction with denominator 
Eighteen is missing one factor of —so you multiply the numerator and denominator  by  to get an equivalent fraction with denominator  We will apply this method as we subtract the fractions in the next example.
In the next example, one of the fractions has a variable in its numerator. We follow the same steps as when both numerators are numbers.

### Identify and Use Fraction Operations

By now in this chapter, you have practiced multiplying, dividing, adding, and subtracting fractions. The following table summarizes these four fraction operations. Remember: You need a common denominator to add or subtract fractions, but not to multiply or divide fractions

### Use the Order of Operations to Simplify Complex Fractions

In Multiply and Divide Mixed Numbers and Complex Fractions, we saw that a complex fraction is a fraction in which the numerator or denominator contains a fraction. We simplified complex fractions by rewriting them as division problems. For example,
Now we will look at complex fractions in which the numerator or denominator can be simplified. To follow the order of operations, we simplify the numerator and denominator separately first. Then we divide the numerator by the denominator.

### Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can also evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

### Key Concepts

1. Find the least common denominator (LCD) of two fractions.
2. Equivalent Fractions Property
3. Convert two fractions to equivalent fractions with their LCD as the common denominator.
4. Add or subtract fractions with different denominators.
5. Summary of Fraction Operations
6. Simplify complex fractions.

### Practice Makes Perfect

Find the Least Common Denominator (LCD) 
In the following exercises, find the least common denominator (LCD) for each set of fractions.
Convert Fractions to Equivalent Fractions with the LCD 
In the following exercises, convert to equivalent fractions using the LCD.
Add and Subtract Fractions with Different Denominators 
In the following exercises, add or subtract. Write the result in simplified form.
Identify and Use Fraction Operations 
In the following exercises, perform the indicated operations. Write your answers in simplified form.
Use the Order of Operations to Simplify Complex Fractions 
In the following exercises, simplify.
Mixed Practice 
In the following exercises, simplify.
In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

### Everyday Math


### Writing Exercises


### Self Check

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



ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?
