Find practice problems and solutions for these concepts at Operations with Like and Unlike Fractions Practice Problems.
What's Around The Bend
 Adding Like Fractions
 Adding Unlike Fractions
 Subtracting Like Fractions
 Subtracting Unlike Fractions
 Multiplying Fractions
 Dividing Fractions
 Reciprocals
In this chapter, we'll look at how to handle the four major operations—addition, subtraction, multiplication, and division—with like and unlike fractions.
Adding Like Fractions
Remember, like fractions are fractions that have the same denominator. To add two like fractions, add the numerators of the fractions. The denominator of your answer is the same as the denominator of the two fractions that you are adding.
Example
In this example, the two fractions are like fractions, because they both have a denominator of 5. The denominator of our answer will be 5. Add the numerators: 1 + 3 = 4. The numerator of our answer is 4.
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Example
The two fractions have a denominator of 12, so the denominator of our answer will be 12. Add the numerators: 6 + 4 = 10. The numerator of our answer is 10. . Because many tests ask for answers to be put in simplest form, let's reduce this fraction. The greatest common factor of 10 and 12 is 2.
= 5 and = 6.
+ .
Example
These fractions have a denominator of 9, so the denominator of our answer will be 9. Add the numerators: 1 + 2 + 3 = 6. The numerator of our answer is 6. . The greatest common factor of 6 and 9 is 3: = 2 and = 3.
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Adding Unlike Fractions
This title is a little misleading. We NEVER add unlike fractions. Instead, we find common denominators and convert the unlike fractions into like fractions. You already know how to add those! Remember, to find a common denominator for two unlike fractions, you must find the least common multiple of the two denominators.
Example
First, we must find a common denominator for these fractions. List a few multiples for each number:
3: 3, 6, 9, 12, 15, 18, 21, 24, 27, …
8: 8, 16, 24, 32, 40, 48, 56, …
The least common multiple of 3 and 8 is 24. Convert each fraction to a number over 24: = 8, which means that the new denominator of the fraction is 8 times larger than the old denominator. Because we must always change the numerator in the same way that we change the denominator, multiply the old numerator by 8: 1 × 3 8 = 8. = . Now let's convert = 3, which means that the new denominator of is 3 times larger than the old denominator. Multiply the old numerator by 3: 2 × 3 = 6. . Now we have like fractions: . Add the numerators and keep the denominator: 8 + 6 = 14, so . Finally, let's simplify our answer. The greatest common factor of 14 and 24 is 2: = 7 and = 12.
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Example
Find a common denominator for these fractions.
5: 5, 10, 15, 20, 25, 30, 35, …
6: 6, 12, 18, 24, 30, 36, 42, …
The least common multiple of 5 and 6 is 30. Convert to a number over 30: = 6, so we must multiply the numerator of by 6: 3 × 6 = 18; . Convert in the same way: = 5. Multiply the numerator of by 5: 1 × 5 = 5. . Now we have like fractions: . Add the numerators and keep the denominator: 18 + 5 = 23, so
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Example
Find a common denominator for these fractions.
9: 9, 18, 27, 36, 45, 54, 63, …
12: 12, 24, 36, 48, 60, 72, 84, …
3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, …
The least common multiple of 9, 12, and 3 is 36. Convert each fraction to a number over 36:
4, 4 × 2 = 8. .
= 3, 3 × 5 = 15. .
= 12, 12 × 1 = 12. .
Now we have like fractions: . Add the numerators and keep the denominator: 8 + 15 + 12 = 35, so
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Subtracting Like Fractions
To add like fractions, we add the numerators and keep the denominators. You can probably guess how to subtract like fractions: Subtract the numerator of the second fraction from the numerator of the first, and keep the denominators.
Example
Both fractions have a denominator of 11, so the denominator of our answer will be 11. Subtract the numerator of the second fraction from the numerator of the first fraction: 7 – 4 = 3. The numerator of our answer is 3:
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Example
The denominator of our answer will be 8. Because 5 – 3 = 2, . The greatest common factor of 2 and 8 is 2, so
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Example
These fractions have a denominator of 12, so the denominator of our answer will be 12. 11 – 5 – 1 = 5. The numerator of our answer is 5:
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Subtracting Unlike Fractions
Just like with addition, we never subtract unlike fractions. We find common denominators, and then subtract.
Example
We begin by finding a common denominator for 3 and 5. List a few multiples for each number:
3: 3, 6, 9, 12, 15, 18, …
5: 5, 10, 15, 20, 25, …
The least common multiple of 3 and 5 is 15. Convert to a number over 15. = 5. Multiply the numerator and denominator of by 5: 2 × 5 = 10. = . Now convert to a number over 15. = 3. Multiply the numerator and denominator of by 3: 3 × 3 = 9; = . Now we have like fractions: – . Find the difference between 10 and 9 and keep the denominator: 10 – 9 = 1, so
– = – = .
Example
Find a common denominator for these fractions.
8: 8, 16, 24, 32, 40, …
12: 12, 24, 36, 48, 60, …
The least common multiple of 8 and 12 is 24. Convert to a number over 24. = 3, so we must multiply the numerator and denominator of by 3: 7 × 3 = 21; = . Convert to a number over 24. = 2, so we must multiply the numerator and denominator of by 2: 7 × 2 = 14. = . Now we have like fractions: – . Find the difference between 21 and 14 and keep the denominator: 21 – 14 = 7, so
– = – = .
Example
Find a common denominator for these fractions.
6: 6, 12, 18, 24, …
2: 2, 4, 6, 8, 10, 12, 14, …
12: 12, 24, 36, 48, …
The least common multiple of 6, 2, and 12 is 12. Convert each fraction to a number over 12:
Now we have like fractions: . Because 10 – 6 – 1 = 3, = . The greatest common factor of 3 and 12 is 3, so the numerator of our answer reduces to = 1 and the denominator reduces to
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Pace Yourself
Ask ten people what kind of pets they have (dogs, cats, fish, etc.). What fraction of them has dogs or cats? In other words, add the fraction of people that have dogs to the fraction of people that have cats. Find the fraction that represents the amount of people who have at least one pet. What other questions about pets can you answer by adding and subtracting fractions?

Multiplying Fractions
Good news—you don't need common denominators to multiply fractions! You can multiply two fractions whether they are like or unlike. We do not need common denominators because the denominator of our answer does not have to be the same as the denominators of the two fractions that we are multiplying.
To multiply two fractions, multiply the numerators, and then multiply the denominators. Multiplying fractions is actually easier than adding or subtracting them!
Example
First, multiply the numerators: 3 × 4 = 12. Then, multiply the denominators: 7 × 9 = 63. The product of and is . The greatest common factor of 12 and 63 is 3, so reduces to .
Example
Multiply the numerators: 7 × 5 = 35. Multiply the denominators: 10 × 6 = 60. The product of and is . The greatest common factor of 35 and 60 is 5, so reduces to .
Example
Before multiplying, look at the numerator of the first fraction and the denominator of the second fraction—they are the same. You can divide both by 3, or "cancel" the threes. The problem now becomes . Multiply the numerators: 1 × 2 = 2. Multiply the denominators: 11 × 1 = 11. = = .
Sometimes you can divide the numerators and denominators more than once before multiplying.
Example
We could begin with any part of this problem—we could simplify the first fraction, divide the numerator of the first fraction and the denominator of the second fraction, or divide the denominator of the first fraction and the numerator of the second fraction.
Let's start by dividing the denominator of the first fraction and the numerator of the second fraction by 5, because 5 is the greatest common factor of 5 and 15. Because = 3 and = 1, the problem becomes . Next, let's divide the numerator and the denominator of the first fraction by 3, because 3 is the greatest common factor of 3 and 6. Because = 2 and = 1, the problem becomes: . Finally, let's divide the numerator of the first fraction and the denominator of the second fraction by 2, because 2 is the greatest common factor of 2 and 14. = 1 and = 7. We have reduced the problem to , or 1 × , which is . After all that simplifying, multiplication was a snap!
Caution!
You can divide the numerator and denominator of the same fraction by the same number, and you can divide the numerator of one fraction and the denominator of another fraction by the same number before multiplying. However, you CANNOT divide the numerator of one fraction and the numerator of another fraction by the same number before multiplying. You also cannot divide the denominator of one fraction and the denominator of another fraction by the same number before multiplying. The problem cannot be simplified before multiplying.

Fractions as Division
It's easy to forget that the fraction bar in a fraction actually represents division. The fraction means "3 divided by 4." That is why we can simplify fractions such as into 4: because means "8 divided by 2."
Dividing Fractions
As with multiplying fractions, we do not need common denominators to divide one fraction by another fraction. Here's what's really strange: When we divide fractions, we don't even use division—we use multiplication! What's the catch? Before multiplying, we find the reciprocal of the divisor. The reciprocal of a fraction is easy to find—just flip it over.
Before we divide fractions, let's look at division of whole numbers.
Example
8 ÷ 2
In this example, 8 is the dividend and 2 is the divisor. The dividend always comes before the division symbol, and the divisor always comes after the division symbol: 8 ÷ 2 = 4.
Now that we've reviewed the names of the parts of a division problem, let's divide some fractions!
Example
In this example, is the dividend and is the divisor. The first step in solving a fraction division problem is to find the reciprocal of the divisor. Switch the numerator and denominator of the fraction ; its reciprocal is . Next, switch the division symbol to a multiplication symbol: ÷ becomes × —these two math problems have exactly the same answer! You know what to do next: Multiply the numerators and multiply the denominators: 1 × 3 = 3 and 4 × 5 = 20, which means that × , and ÷ , both equal .
Example
This problem can be solved two ways—in fact, all fraction division problems can be solved two ways. The easiest method is the one we just saw—take the reciprocal of the divisor, and multiply. However, if you already have common denominators, you can simply divide the numerator of the dividend by the numerator of the divisor. Because 8 ÷ 4 = 2, = 2. Now let's use the first method to check our answer. Because is the divisor, find its reciprocal. Switching the numerator and the denominator, we find that its reciprocal is . Next, switch the division symbol to a multiplication symbol. becomes × . We can cancel the 9 in the denominator of with the 9 in the numerator of . Now the problem becomes . Multiply the numerators and multiply the denominators. 8 × 1 = 8 and 1 × 4 = 4, which means that , and , equal , or 2. Both methods give us the same answer.
Example
Because these are not like fractions, we'll solve this problem using the first method. The reciprocal of is , so becomes × . Now that it is a multiplication problem, we can simplify these fractions. By dividing the numerator of and the denominator of by 5, and by dividing the denominator of and the numerator of by 3, × becomes , so now the problem becomes × 3, which is equal to .
Now we've mastered the four major operations (addition, subtraction, multiplication, and division) with like and unlike fractions.
Find practice problems and solutions for these concepts at Operations with Like and Unlike Fractions Practice Problems.