This tutorial covers adding and subtracting fractions, through the exploration of:
- Review of PEMDAS, the Order of Operations
- Adding and Subtracting Fractions with Common Denominators
- Adding and Subtracting Fractions with Uncommon Denominators
- Finding Least Common Denominators of Fractions
- Using the Order of Operations with Fractions
1. Review of PEMDAS, the Order of Operations
To review, PEMDAS is an acronym used to remember the order of operations, which is the order in which you use operations when simplifying or solving problems, including problems involving fractions. PEMDAS stands for:
It’s important to remember that multiplication and division are done together from left to right, and addition and subtraction are also done together from left to right.
2. Adding and Subtracting Fractions with Common Denominators
When you add or subtract fractions with the same, or common, denominators, you simply add or subtract the numerators.
In the example below, you are adding two fractions with common, or the same, denominators (the 5s). Therefore, you simply add the numerators of each fraction (2 and 1), which equals 3. The denominators stay the same.
To think about this problem in a different way, look at the picture representation below. The bar is split into five pieces, which matches the 5 in the denominator. The pieces are all the same size because the denominators are all the same size. You start with 2/5, or 2 out of 5 pieces, shaded in, and then you add 1/5, or 1 piece, which equals a total of 3/5, or 3 out of 5 pieces, shaded in.
You can subtract the two fractions in the same way. When you subtract fractions, you subtract the numerators.
Referring to our original equation, 2 minus 1 is 1. Again, the denominators stay the same.
Looking at the bar representation, you start with 2 out of 5 pieces, or 2/5, shaded in, then subtract 1 piece (1/5), leaving 1 piece, or 1/5 of the bar, shaded in.
3. Adding and Subtracting Fractions with Uncommon Denominators
Suppose you want to add 1/2 and 3/4. Looking at the picture representation of these two fractions, you can see that there is a problem.
You can’t simply combine the pieces together as you did in the last example, because the pieces are different sizes. However, if the denominators were the same, the pieces would be the same size, and you could add or subtract your numerators.
When adding or subtracting fractions with uncommon denominators, you need to convert them into equivalent fractions with common denominators.
Step 1: To find a common denominator, you start by multiplying your two denominators together. In this case, your common denominator will be 8, because 2 times 4 is 8. To convert to a denominator of 8 in your first fraction, you multiply the 2 in the denominator by 4. Note, however, that if you multiply by 4 in the denominator, you must multiply by 4 in the numerator as well so that you do not change the value of the fraction. Now you have the equivalent fraction 4/8.
- You can multiply by 4 in the denominator and the numerator without changing the value of the fraction because 4/4 is the same as 1, and multiplying by 1 does not change the value of your fraction.
Step 2: To convert to a denominator of 8 in your second fraction, you multiply the 4 in the denominator by 2. Remember, if you multiply by 2 in the denominator, you have to multiply by 2 in the numerator as well. Now you have another equivalent fraction, 6/8.
- Again, you can multiply by 2 in the denominator and numerator without changing the value of the fraction because 2/2 is the same as 1, and multiplying by 1 does not change the value of your fraction.
Step 3: Now that your denominators are both 8, you can add your numerators together and leave your denominator the same, providing an answer of 10/8. 10/8 can be simplified, or reduced, by dividing the numerator and the denominator by 2, which equals a final answer of 5/4.
This method of finding a common denominator will always work, but it may not give you the least or smallest common denominator, and some simplification might be necessary at the end.
4. Finding Least Common Denominators of Fractions
Finding the least common denominator between fractions can eliminate some common factors, making simplification easier. To find the least common denominator, you find the smallest number that is a multiple of the denominators of each fraction.
Consider the last example again to see how you can find the least common denominator.
First, look at the multiples of your first denominator, 2. The multiples of 2 are 2, 4, 6, 8, 10, 12, and so on. The multiples of your second denominator, 4, are 4, 8, 12, 16, 20, 24, and so on. Looking at both lists, the smallest common multiple is 4. Therefore, 4 is your least common denominator. Now you can rewrite your fractions using 4 as your common denominator.
- To convert the denominator of the fraction 1/2 to 4, you multiply by 2 in the denominator and numerator. This gives you an equivalent fraction of 2/4. Notice that the second fraction, 3/4, already has a denominator of 4, so you can leave it unchanged. Now that your denominators are the same, you can add the numerators together and leave the denominators as 4, which equals a final answer of 5/4. Note that you did not need to simplify your answer because you used the least common denominator.
5. Using the Order of Operations with Fractions
You can also use order of operations, or PEMDAS, with fractions. Suppose you are solving the following problem:
You start with multiplication and multiply 2 times 1/3. The whole number 2 can be written as a fraction with a denominator of 1, or 2/1. Proceeding with the order of operations, you can add and subtract your fractions from left to right.
You know that you need a common denominator, and looking at your denominators 4, 3, and 2, the least common denominator will be 12, because 12 is the least common multiple of 4, 3, and 2. For the first fraction, you need to multiply by 3 in the denominator and the numerator. In the second fraction, you need to multiply by 4 in the denominator and numerator. For the last fraction, you need to multiply by 6 in the denominator and numerator. Now your expression is:
Since the denominators are all the same, you can simply add and subtract your numerators (9 + 8 – 6 = 11), and leave your denominator as 12. This equals a final answer of 11/12, which is in its simplest form, so you do not have to reduce it.
Today you learned about adding and subtracting fractions, and that to do so, fractions must have the same or common denominator. You also learned that to find a common denominator, you can always multiply the denominators together, but you may need to simplify your answer. Lastly, you learned that the order of operations (PEMDAS) must be used when simplifying all expressions, including expressions with fractions.