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We cover the what, why, and how of simplifying fractions, then run through some examples, looking for common factors of the numerator and denominator so that we can divide them both by the same common factor and create a simpler fraction.
Let’s talk about simplifying fractions. To do that, we’re gonna talk about three parts. One, what does it mean to simplify a fraction? Two, why we would do it? And three, how would we simplify a fraction?
Number one, what does simplifying fractions mean? To simplify a fraction is to make it as simple as possible. You might also hear people use the word reduce. Reducing fractions and simplifying fractions are the same thing.
Onto question two, why would we simplify a fraction? We simplify a fraction for two main reasons; it clarifies communication and it helps us avoid misunderstandings. You might be wondering, what in the world does she mean clarify communication; isn’t this math? And yes, I am talking about communication in math because good mathematicians communicate clearly. Good mathematicians explain and present their data in simplified terms.
Here’s an example in clarity. Sara and Sam both took a math test and their teacher gave them the grades you see on the screen. She gave Sara a one hundred and thirty out of one hundred and eighty-two and Sam got ninety out of one hundred and twenty-six. Whose score is higher? Did they do okay? Is this a good score for a math test? It’s not really clear. The simplified form of both of these fractions is five-sevenths. Sam and Sara actually got the same score. But it wasn’t clear because they were using very complex fractions, instead of simplifying them. Taking the time to simplify fractions makes your communication more clear and avoids misunderstandings between you and the people you’re communicating your math to.
There is a third and more practical reason, and that would be easier computations. For example, one hundred and thirty over one eighty-two 𝑥 equals five doesn’t seem like that simple of a problem to solve. But five-sevenths 𝑥 equals five is a much easier problem to solve.
Before we get to our third, how to simplify fractions portion, there’s one more thing we need to note. Simplifying does not change the value of a fraction; it does not change how much we’re talking about. Here is an illustration of the fact that the value doesn’t change. The first pizza is four-eighths, the second pizza is two-forths of a pizza, the third pizza is one-half. Here the only thing changing is the number of slices but not the amount of pizza that you ate. One-half is four-eighths in a simplified form. And one-half is also the simplified form of two-fourths.
Now onto how we simplify fractions. Let’s take the example four-eighths from the pizza problem I’ve just showed you and try and simplify it. To do that, I’m going to consider a number that’s a factor of both, the numerator and the denominator. In other words, I’m looking for a number that the top and the bottom are both divisible by. So in this problem, I recognize that both the numerator and the denominator are divisible by four. Now I just divide the numerator by four and the denominator by four, one-half. This was a very simple example and probably one you can recognize immediately. You might even memorize that four-eighths equals one-half. Not all fractions are that easy to simplify.
Here’s a slightly harder one but simplify twelve over twenty-eight. Again, we need to consider a number that both twelve and twenty-eight are divisible by. I’m going to start with four. The simplified form of twelve over twenty-eight is three-sevenths.
Here’s another example to simplify, thirty over seventy-two. Our process is still the same. We need to come up with a number that thirty and seventy-two are both divisible by. I recognize that thirty and seventy-two are both even numbers, so I’m going to start there and divide them both by two. This gives me fifteen over thirty-six. Fifteen over thirty-six, however, is not the simplest form of this fraction. If you look closely, there’s a number that fifteen and thirty-six are both divisible by, and that number is three. When I divide the top and bottom by three, I get five twelfths. And I know that five twelfths is the simplest form of this fraction because five and twelve do not share any common factors.
Here’s our final example. Adults need eight hours of sleep per day. What fraction of the day is this? Give your answer in simplest form.
We can identify that in our problem. We have these two pieces of information, eight hours of sleep is what they need per day. We also know that that means twenty-four hours. So we can start with our fraction of eight twenty-fourths, with a need to reduce the fraction just like all the others we’ve practiced. Eight and twenty-four share a common factor of eight, so we’re gonna divide the numerator and the denominator by eight here. When you divide the numerator and the denominator by eight, you get the simplified fraction one-third. Based on the information given, that would mean that adults need to sleep one-third of the day.
Simplifying fractions is a simple process that helps us communicate effectively. And now you have the tools to simplify fractions and communicate more clearly to the people around you.
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