Adding and subtracting fractions can look intimidating at first glance. Not only are you working with fractions, which are notoriously confusing, but suddenly you have to contend with converting numerators and denominators, too.
But adding and subtracting fractions is a useful skill. Once you know the vocabulary and the basics, you’ll be adding and subtracting fractions with ease. This guide will walk you through everything you need to know for adding and subtracting fractions, including some example problems to test your skills.
Key Vocabulary for Adding and Subtracting Fractions
Before we can get into the math for adding and subtracting fractions, you need to know the terminology. We’ll be using these terms throughout, so brush up on them to be sure you always know what part of the fraction we’re referring to.
Fraction: A number that is not a whole number; a part of a whole. For our purposes, a fraction will refer to a number written with a numerator and a denominator, such as $1/5$ or $147/4$.
Numerator: The top number in a fraction, reflecting the number of parts of a whole, such as the 1 in $1/5$.
Denominator: The bottom number in a fraction, representing the total number of parts, such as the 5 in $1/5$.
Common Denominator: When two fraction share the same denominator, such as $1/3$ and $2/3$.
Least Common Denominator: The smallest denominator two fractions can share. For example, the least common denominator of $1/2$ and $1/5$ is 10, because the smallest number both 2 and 5 go into is 10.
Pies make great fractions.
How Do You Add and Subtract Fractions?
Now that you have the vocabulary, it’s time to put that into action. You can’t simply add or subtract fractions as you would a whole number $1/4 - 1/2$ doesn’t equal $0/2$, for example.
Instead, you’ll need to find a common denominator before you add or subtract. There are many ways to find a common denominator, some of which are easier or more efficient than others.
One of the easiest ways to find a common denominator, though not necessarily the best, is to simply multiply the two denominators together.
For example, a possible least common denominator for $1/2$ and $1/12$ would be 24, which you find by multiplying the 2 denominator by the 12 denominator. You can solve a problem using the common denominator of 24 using the steps below, but if you do, you’ll run into a problem—your fraction will need to be reduced.
To eliminate the need to reduce once you’ve added or subtracted, instead try to find the least common denominator. Sometimes that will be the same as multiplying two denominators together, but it often won’t be.
However, finding the least common denominator isn’t hard—you’ll just need to be familiar with your multiplication tables. For example, let’s try to find the least common denominator, rather than just a common denominator, for the same fractions we used above:
$$1/2\: \and \: 1/12$$.
To do this, list out a few multiples of each denominator
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24
Multiples of 12: 12, 24, 36, 48, 60
Then, look at both lists of multiples and find the lowest number both share. In this case, both 2 and 12 share the multiple 12. If we kept going, we would end up with other multiples they share, such as 24, but 12 is the smallest, meaning it’s the least common multiple.
You can do this with any pair of numbers, though larger numbers may present more of a challenge. For adding or subtracting, you can always return to simply multiplying one denominator by the other if you’re having trouble finding the least common denominator, but do keep in mind that you will likely have to reduce.
Fractions are the tastiest part of math.
How to Add Fractions — Method 1
Now that you know how to find a common denominator, you’re ready to start adding and subtracting.
Let’s return to the example of $1/2$ and $1/12$—in this case, let's look at this problem:
$$1/2 + 1/12$$
Remember, you can’t add straight across; $1/2 + 1/12$ does not equal $2/14$.
#1: Find a Common Denominator
We’ll find the least common denominator first, since that’s generally the best way to go about it.
We already did the work above, but as a reminder, you’ll want to write out a series of multiples of each number until you find a match. In this case, both 2 and 12 have a multiple of 12.
#2: Multiply to Get Each Numerator Over the Same Denominator
Always remember that anything you do to the denominator must also be done to the numerator. So let’s take a look at these two fractions we need to get over the denominator 12.
$1/12$ is easy—it’s already over the denominator of 12, so we don’t have to do anything to it.
$1/2$ will need some work. What number multiplied by 2 will equal 12?
To rephrase that question as a problem we can solve, $2*?=12$. Or, even simpler, we can invert the operation to get $12/2=?$, which we can easily solve.
So now we know that to go from a denominator of 2 to a denominator of 12, we need to multiply by 6. Again, remember that everything you do to the denominator needs to be done to the numerator as well, so multiply the top and bottom by 6 to get $6/12$.
#3: Add the Numerators, but Leave the Denominators Alone
Now that you have the same denominators, you can add the numerators straight across.
In this case, that will mean that $6/12 + 1/12 = 7/12$. Ask yourself if you can reduce the fraction by diving both the numerator and the denominator by the same number. In this case, you can’t, so your answer is a simple $7/12$.
How to Add Fractions — Method 2
Alternatively, we could simply multiply the two denominators together to find a different common denominator. This is a different way to solve the problem, but will end up with the same answer.
#1: Multiply the Denominators Together
No fancy tricks here—simply multiply 2 by 12 to get 24. That will be your common denominator.
#2: Multiply to Get Each Numerator Over the Same Denominator
Just as we did when we found the least common denominator, we’ll need to multiply both the top and bottom number of each fraction. In this case, use inverse operations to find out what number you’ll need to multiply.
If $1/2$ needs to be $?/24$, you can do $24÷2$ to figure out what number you’ll need to multiply by—12. Multiply the top and the bottom by 12 to get $12/24$.
Repeat the process with $1/12$. If $1/12$ needs to be $?/24$, solve $24÷12$ to get 2. Now multiply the numerator and denominator of $1/12$ by 2 to get $2/24$.
#3: Add the Numerators Together
Now you can simply add straight across. $$12/24 + 2/24 = 14/24$$.
#4: Reduce
Here’s where the extra step comes in. $14/24$ is not a fraction in its lowest form, so we’ll need to reduce it. To reduce, we need to divide both the numerator and the denominator by the same number.
To do so, we’ll need to find the greatest common factor. Much like finding the least common multiple, this means listing out numbers until we find two factors that both the numerator and the denominator have in common, excluding 1, like so:
14: 2, 7
24: 2, 3, 4, 6, 8, 12
What number do they have in common? 2. That means that 2 is our greatest common factor, and therefore the number we’ll be dividing the numerator and denominator by.
$14÷2=7$ and $24÷2=12$ giving us the answer of $7/12$.
The answer is the same as when we solved using the least common multiple, and can’t be reduced any further, so that’s our final answer!
If you ever find yourself writing out lots of factors without much luck, there are some quick ways to figure out potential factors.
- If a number is even, it can be divided by 2.
- If you can add a number's digits a number that is divisible by 3, the number is divisible by 3—such as 96 ($9+6=15$ and $1+5=6$, which is divisible by 3).
- If the number ends in a 5 or a 0, it is divisible by 5.
- If you’re not sure when to stop looking for factors, subtract the smaller number from the larger one. That number will be the largest possible common factor, but not the greatest common factor itself.
For example, let’s take 50 and 32. Sure, we could just divide both by 2 and keep reducing from there, but if you do $50-32$ you get 18, telling us to stop looking for the greatest common factor once we hit 18.
In practice, that looks like this:
50: 2, 5, 10
32: 2, 4, 8, 16
Instead of continuing on, we know to stop when the next factor would be 18 or above, stopping us from spending more time figuring out factors we don’t need. We can see a lot quicker that the greatest common factor is 2 and move on with the problem!
$1/1 - 1/? = yum$
How to Subtract Fractions
Once you’ve mastered adding fractions, subtracting fractions will be a breeze! The process is exactly the same, though you’ll naturally be subtracting instead of adding.
#1: Find a Common Denominator
Let’s look at the following example:
$$2/3-3/10$$
We need to find the least common multiple for the denominators, which will look like this:
3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
10: 10, 20, 30
The first number they have in common is 30, so we’ll be putting both numerators over a denominator of 30.
#2: Multiply to Get Both Numerators Over the Same Denominator
First, we need to figure out how much we’ll need to multiply both the numerator and denominator of each fraction by to get a denominator of 30. For $2/3$, what number times 3 equals 30? In equation form:
$$30÷3=?$$
Our answer is 10, so we’ll multiply both the numerator and denominator by 10 to get $20/30$.
Next, we’ll repeat the process for the second fraction. What number do we need to multiply by 10 to get 30? Well, $30÷10=3$, so we’ll multiply the top and bottom by 3 to get $9/30$.
This makes our problem $20/30-9/30$, which means we’re ready to continue!
#3: Subtract the Numerators
Just as we did with addition, we’ll subtract one numerator from the other but leave the denominators alone.
$$20/30-9/30=11/30$$.
Since we found the least common multiple, we already know that the problem can’t be reduced any further.
However, let’s say that we just multiplied 3 by 10 to get the denominator of 30, so we need to check if we can reduce. Let’s use that little trick we learned to find the greatest possible common factor. Whatever factors 11 and 30 share, they can’t be greater than $30-11$, or 19.
11: 11
30: 2, 3, 5, 6, 10, 15
Since they don’t share any common factors, the answer cannot be reduced any further.
$1/10$ pizza is still $10/10$ tasty.
Adding and Subtracting Fractions Examples
Let’s go over a few more sample problems!
$$8/15-4/9$$
#1: Find a common denominator
15: 15, 30, 45, 60
9: 9, 18, 27, 26, 45
#2: Multiply to get both numerators over the same denominator
$$45/15=\bo3$$
$$8÷3=24$$
$$15*3=45$$
$$24/45$$
$$45÷9=\bo5$$
$$4*5=20$$
$$9*5=45$$
$$20/45$$
#3: Subtract the numerators
$$24/45-20/45=\bo4/\bo45$$
$$6/11+3/4$$
#1: Find a common denominator
11: 11, 22, 33, 44
4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44
#2: Multiply to get both numerators over the same denominator
$$44÷11=\bo4$$
$$6*4=24$$
$$11*4=44$$
$$24/44$$
$$44÷4=\bo11$$
$$3*11=33$$
$$4*11=44$$
$$33/44$$
#3: Add the numerators
$$24/44+33/44=\bo57/\bo44$$ or $$\bo1 \bo13/\bo44$$
$$4/7-11/21$$
#1: Find a common denominator
7: 7, 14, 21
21: 21, 42, 63
#2: Multiply to get both numerators over the same denominator
$$21÷7=\bo3$$
$$3*4=12$$
$$3*7=21$$
$$12/21$$
$11/2$ is already over 21, so we don’t have to do anything.
#3: Subtract the numerators
$$12/21-11/21=\bo1/21$$
$$8/9+7/13$$
#1: Find a common denominator
9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117
13: 13, 26, 39, 52, 65, 78, 91, 104, 117
#2: Multiply to get both numerators over the same denominator
$$117÷9=\bo13$$
$$8*13=104$$
$$9*13=117$$
$$104/117$$
$$117÷13=\bo9$$
$$7*9=63$$
$$13*9=117$$
$$63/117$$
#3: Add the numerators
$$104/117+63/117=\bo167/\bo117$$
What’s Next?
Adding and subtracting fractions can get even more simple if you start converting decimals to fractions!
If you're unsure what high school math classes you should be taking, this guide will help you figure out your schedule to be sure you're ready for college!
Now that you're an expert in adding and subtracting fractions, challenge yourself by learning how to convert Celsius to Fahrenheit!
