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How to Add Fractions with Common and Unlike Denominators
Do you struggle with fractions?
If you have a hard time with fractions, you're not the only one. They even make shirts that say things like "4 out of 3 people are confused by fractions." My hope is that this lesson will help you better understand what a fraction is and how to add fractions together.
What is a Fraction?
Before we look at how to add fractions, let's make sure you understand what a fraction is. We use fractions to represent parts of a whole. For example, the chocolate bar below is divided into 6 equal pieces. The fraction 1/6 means you have 1 piece out of the 6 total.
What are numerators and denominators?
The bottom number of a fraction is called the denominator. The denominator tells you the total number of equal parts that make up the whole. In the chocolate bar example, the denominator is 6 because there are 6 pieces total that make up the whole bar.
The top number of a fraction is called the numerator. The numerator tells you how many of the smaller equal parts you have. If you ate 1/6 of the candy bar, that means you ate 1 of the 6 pieces. If you ate 5/6 of the candy bar, that would mean you ate 5 of the 6 pieces of chocolate.
Practice Quiz: Identifying Parts of a Fraction
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Using Manipulatives to Work With Fractions
When I taught middle school, I had a set of magnetic fraction tiles very similar to the set shown below. This set of tiles is an excellent way to help you visualize fractions with different denominators.
Fraction tiles are a great way to help you see the "size" of the fraction and to help you see how many pieces it takes to make a whole. For example, the fraction 1/3 means you have 1 out of 3 pieces. In the fraction tile set above, the 1/3 pieces are the orange pieces. Do you see that three of them together are the same length as the red 1 piece at the top? Three of the 1/3 pieces together form a whole.
When the denominator gets larger, that means that the whole has been split into more pieces. Can you see that the black 1/12 pieces are a lot smaller than the two purple 1/2 pieces? The larger the denominator, the smaller the relative size of each fraction piece will be.
Adding Fractions with a Common Denominator
When fractions have a common denominator, it just means the fractions have the same number in the bottom. For example, 2/7 and 3/7 have a common denominator of 7.
When you're adding fractions with the same denominator, you're basically counting up the pieces. Let's take a look at the 1/6 pieces. The fraction 1/6 means you have 1 out of 6 total pieces. If you put all 6 together, it forms a whole.
So let's say you're adding 2/6 + 3/6. The fraction 2/6 means we have 2 of the 1/6 pieces and the fraction 3/6 means we have 3 of the 1/6 pieces.
What does this equal? When the denominators of two fractions are the same, the fractions are all the same relative size so you can just count them up. You have 2 of the 1/6 pieces plus 3 more 1/6 pieces. How many 1/6 pieces do you have total? 5. This means 2/6 + 3/6 = 5/6.
The denominator does not change when you're adding fractions with a common denominator. You're just counting up the total number of pieces you have of that size.
When you're adding fractions with the same denominator, visualize it as collecting little fraction pieces that are all the same size. You're counting up how many you have total. Your answer will have the same bottom number because the relative size of each piece hasn't changed.
Check out the short video below to see another example of adding fractions with a common denominator.
Practice Quiz: Adding Fractions with a Common Denominator
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Adding Fractions with Unlike Denominators
Can you add fractions together when the bottom numbers are not the same? Yes, but it takes a little bit of extra work. When fractions have different denominators, the pieces do not have the same relative size. 1/4 of a whole chocolate bar is bigger than 1/6 of the bar (If I had to choose, I would rather eat 1/4!).
Let's say we have to add 3/4 + 1/6. The fraction 3/4 means we have three of the orange 1/4 pieces. The fraction 1/6 means we have just one of the blue 1/6 pieces. How do we add these together?
You can't just count them up this time because they're not the same size. In order to add fractions with unlike denominators, you need to first rewrite them with a common denominator.
How to Find a Common Denominator
There are several different ways to find a common denominator. Some teachers may tell you to just multiply the two denominators together to find a common denominator. This will work, but you may end up with a really large denominator and you'll likely have to do some simplifying at the end.
Another way to find a common denominator is to list multiples of each bottom number. Start a list of multiples for each denominator and look for the smallest number that they have in common. This number is called the least common multiple (LCM). It's also called the least common denominator (LCD) because it's the smallest number you can use as the common denominator.
In our example of 3/4 + 1/6, the two denominators are 4 and 6. To find a common denominator, we can list out multiples of 4 and 6 and see if we can find the smallest multiple they both have in common.
The smallest multiple that the two denominators have in common is 12. Did you notice that they also both have 24 in common? You could use 24 as the common denominator, but it's usually easier to use the smallest number possible so you can save some simplifying steps at the end.
Now that we've found a common denominator, we need to rewrite the two fractions with a common denominator of 12. Once the two fractions are rewritten with the same bottom number, we'll be able to add the fractions together.
Practice Quiz: Finding the Least Common Denominator
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How to Rewrite Fractions with a Common Denominator
In order to add 3/4 + 1/6, we need to rewrite both fractions so that they have a common denominator. We just figured out that we can use 12 as the common denominator, so we need to "fix" the fractions so that they both have a 12 in the bottom.
How do you do this? We need to do some multiplying. If you multiply a number by 1, nothing changes. So as long as we multiply the numerator and denominator of the fraction by the same number, we're really just multiplying by a fancy form of 1. Take a look at each fraction and think about what you could multiply the denominator by to get a 12.
For the first fraction 3/4, we need to change the 4 to a 12. To do this, we can multiply it by 3. If we multiply the bottom of the fraction by 3, we also need to multiply the top of the fraction by 3 so we get an equivalent fraction.
For the second fraction 1/6, we need to change the 6 to a 12. We can do this by multiplying it by 2. To keep the fraction the same, we need to also multiply the numerator by 2.
Sometimes it helps to see a picture of the fractions so you can visualize what's going on. When you rewrite the fractions with a common denominator, you haven't changed the relative size of the fraction. You've just split the whole into a different number of parts and recalculated the numerator to match. 3/4 is the same as 9/12. You can see that each orange 1/4 piece has just been split into 3 purple 1/12 pieces. 1/6 is also the same as 2/12. The blue 1/6 piece has just been split into two purple 1/12 pieces.
Practice Quiz: Rewriting fractions with a new denominator.
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Once you have the fractions written with the same denominator, it's pretty straightforward to add them. Remember, you can just count up the pieces when the denominators are the same. All the 1/12 pieces are the same size, so we're counting up how many there are total. We have 9/12 (nine of the 1/12 pieces) plus 2/12 (two more of the 1/12 pieces). This gives us a total of 11 of the 1/12 pieces: 11/12.
Watch the video below to see another example of adding fractions with different denominators.
Practice Quiz: Adding Fractions with Unlike Denominators
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One last thing to note, make sure to check and see if your answer can be simplified. Look at the numerator and denominator and see if you can divide them both by the same number.
There are many resources available to help you practice adding fractions. I highly recommend using the website IXL. You can practice several problems each day for free and the program provides a detailed explanation if you get a question wrong. There's also an app that you can download if you want to practice on your phone.