Understanding How to Add and Subtract Fractions
Adding and subtracting fractions can be confusing for a lot of people. However, finding the sum and difference of two fractions is not difficult once you understand the process. In this tutorial, we’ll provide some essential tips that will help you learn how to add and subtract fractions effectively.
The main idea when you add or subtract fractions is to have the same denominators.
For example, you can add:
\large \frac{1}{5}+\frac{2}{5}
because the denominators (the bottom numbers in the fractions) are “common” or the same. All we need to do is add the respective numerators to get the answer, in this case:
\large \frac{3}{5}
To add or subtract fractions with different denominators, such as:
\large \frac{2}{9}+\frac{5}{12}
we need to find the LCD (Lowest Common Denominator) so we can rewrite each fraction with the same denominator.
Most people only learn how to add and subtract fractions by finding the LCD. However, there is also a great shortcut for adding and subtracting fractions without finding the LCD, which we’ll cover below.
First, let’s review the standard method using the LCD.
Steps to Add and Subtract Fractions Using the LCD
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- Look at the denominators.
If the denominators are the same, simply add or subtract the numerators. - If the denominators are different, find the LCD.
The LCD (Lowest Common Denominator) is the smallest number both denominators can divide into. Please see this article on how to find the LCD. - Rewrite each fraction with the LCD.
Once you have the LCD, rewrite each fraction so that both denominators are the same. - Add or subtract the numerators.
Once the fractions have the same denominator, combine the numerators. - Simplify the final answer if possible.
- Look at the denominators.
Example: Adding Fractions with Different Denominators
Let’s solve the following problem:
\large \frac{2}{9}+\frac{5}{12}
Step 1: Look at the denominators.
The denominators are 9 and 12. Since they are not the same, we need to find the LCD.
Step 2: Find the LCD for the denominators 9 and 12.
The LCD is the product of each unique prime factor. If the same prime factor appears more than once, use the highest power.
Prime factors of 9:
\large 9=3^2
Prime factors of 12:
\large 12=2^2\times3
Now use each unique prime factor with the highest power:
\large LCD = 2^2\times3^2 = 4\times9 = 36
Step 3: Rewrite each fraction with the LCD.
Convert each fraction so the denominator becomes 36.
\large \frac{2}{9}=\frac{2\times4}{9\times4}=\frac{8}{36}
\large \frac{5}{12}=\frac{5\times3}{12\times3}=\frac{15}{36}
Step 4: Add the numerators.
Now the fractions have the same denominator, so add the numerators.
\large \frac{8}{36}+\frac{15}{36}=\frac{23}{36}
Step 5: Simplify if possible.
The fraction cannot be reduced, so the final answer is:
\large \frac{23}{36}
Watch Step-by-Step Example
Watch John explain how to subtract fractions using the LCD:
The Bow-Tie Method
There is also a great shortcut for adding and subtracting fractions called the Bow-Tie Method.
This method avoids finding the LCD first, although you may need to simplify your answer at the end.
Let’s solve the same problem:
\large \frac{2}{9}+\frac{5}{12}
The Bow-Tie method follows this pattern:
\large \frac{(12)(2)+(9)(5)}{9\times12}
Step 1: Cross multiply the numerators.
\large (12)(2)=24
\large (9)(5)=45
Step 2: Add the results.
\large 24+45=69
Step 3: Multiply the denominators.
\large 9\times12=108
So the fraction becomes:
\large \frac{69}{108}
Step 4: Simplify the final answer.
Both numbers divide by 3:
\large \frac{69}{108}=\frac{23}{36}
Final Answer:
\large \frac{23}{36}
Watch the Bow-Tie Method in Action
Check out this video to watch the Bow-Tie method step-by-step:
Need More Help with Fractions?
If you want more help with fractions, check out John’s full courses. They include clear instruction, practice problems, and step-by-step solution videos.
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👉 Explore the Foundations Math Course