2.2 Add and Subtract Fractions
Learning Objectives
By the end of this section, you will be able to:
- Add or subtract fractions with a common denominator
- Add or subtract fractions with different denominators
- Use the order of operations to simplify complex fractions
- Evaluate variable expressions with fractions
Add or Subtract Fractions with a Common Denominator
When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.
Fraction Addition and Subtraction
If are numbers where , then
To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
EXAMPLE 1
Find the sum: .
| Add the numerators and place the sum over the common denominator. |
TRY IT 1.1
Find the sum: .
Answer
TRY IT 1.2
Find the sum: .
Answer
EXAMPLE 2
Find the difference: .
| Subtract the numerators and place the difference over the common denominator. | |
| Simplify. | |
| Simplify. Remember, . |
TRY IT 2.1
Find the difference: .
Answer
TRY IT 2.2
Find the difference: .
Answer
EXAMPLE 3
Simplify: .
| Subtract the numerators and place the difference over the common denominator. | |
| Rewrite with the sign in front of the fraction. |
TRY IT 3.1
Find the difference: .
Answer
TRY IT 3.2
Find the difference: .
Answer
Now we will do an example that has both addition and subtraction.
EXAMPLE 4
Simplify: .
| Add and subtract fractions—do they have a common denominator? Yes. | |
| Add and subtract the numerators and place the difference over the common denominator. | |
| Simplify left to right. | |
| Simplify. |
TRY IT 4.1
Simplify: .
Answer
TRY IT 4.2
Simplify: .
Answer
Add or Subtract Fractions with Different Denominators
As we have seen, to add or subtract fractions, their denominators must be the same. The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.
Least Common Denominator
The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.
After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!
EXAMPLE 5
Add: .
| Step 1. Do they have a common denominator? No-rewrite each fraction with the LCD (least common denominator). |
Find the LCD of 12, 18.
Change into equivalent fractions with the LCD, 36. |
|
| Step 2. Add or subtract the fractions. | Add. | |
| Step 3. Simplify, if possible. | Because 31 is a prime number, it has no factors in common with 36. The answer is simplified. |
TRY IT 5.1
Add: .
Answer
TRY IT 5.2
Add: .
Answer
HOW TO: Add or Subtract Fractions
- Do they have a common denominator?
- Yes—go to step 2.
- No—r