Fractions
8 Adding and Subtracting Fractions
Chad Flinn and Mark Overgaard
Adding Fractions with Common Denominators
Abigail, Hanna, and Naomi are studying for their midterm exam. The material they are required to study consists of 16 chapters of reading. The three of them realize that 16 chapters is a lot of reading for each of them to do, so they decide to study in a more efficient manner. They come up with a plan in which each of them reads a certain number of chapters and then summarizes it for the other two. They will share notes, and each will find online videos corresponding to their particular set of chapters.
Now, the chapters are not created equally. Some are quite easy, while others are much tougher. Their goal is to spread the workload evenly between the three of them. Remember that there are 16 chapters.
Abigail has the highest number of chapters to go through with 6. Hanna has 5, while Naomi has only 4. If you were to add those up, you would notice that that only comes to 15 chapters. The last chapter in the book is about troubleshooting electrical systems, and the apprentices decide that they will go through that one together.
We can represent each of their workloads as a fraction of a whole:
[latex]\LARGE\text{Abigail has }\dfrac{6}{16}[/latex]
[latex]\LARGE\text{Hanna has }\dfrac{5}{16}[/latex]
[latex]\LARGE\text{Naomi has }\dfrac{4}{16}[/latex]
What if were to add those fractions? It would look something like this:
[latex]\LARGE\dfrac{6}{16}+\dfrac{5}{16}+\dfrac{4}{16}=?[/latex]
What you’ll note is that the numerators are all different, while the denominators are all the same (16). When adding or subtracting fractions, the denominators must be the same. We refer to this as having a common denominator.
So, in order to get the answer to the above question, you just add all the numerators. Adding fractions is very simple in this respect.
Notice that the denominator in the final answer is the same as that in the fractions being added. By the end, the apprentices will have gone through 15 of the 16 chapters separately, and then they will go through the last chapter together.
The concept of adding fractions with common denominators is easy enough, and we did enough adding whole numbers that going through examples at this point might not be worth it (but if you need a review, see Adding Whole Numbers). What we will do instead is write down some examples of adding fractions so you can see the idea.
[latex]\LARGE\dfrac{1}{8}+\dfrac{2}{8}=\dfrac{3}{8}[/latex]
[latex]\LARGE\dfrac{5}{16}+\dfrac{6}{16}=\dfrac{11}{16}[/latex]
[latex]\LARGE\dfrac{13}{32}+\dfrac{11}{32}=\dfrac{24}{32}[/latex]
Do you notice anything about the answer to the last one? It can be reduced.
[latex]\LARGE\dfrac{24}{32}\longrightarrow\dfrac{3}{4}[/latex]
Before we get going any further with work on fractions, this might be a good time to state that, when working with fractions, we generally want to put the answer in lowest terms.
Subtracting Fractions with Common Denomi