Simplifying Fractions
The work that we did understanding factors (see Factors and Multiples and Greatest Common Factor) will really help us with simplifying fractions. When you think about it, a number is just simply made up of its factors (that’s one way to look at a number). If you find a number that is large or complicated, you can usually break it down into its smaller factors to make sense of it—you can simplify it.
Think about the fraction above, 12/15 (twelve fifteenths), it can really be simplified into a more commonly recognized fraction that tells us what it represents more easily. Now, if I wanted to I could just divide 12 by 15 and come up with an answer that’s in decimal form, but that isn’t always what I might want to do. So, we can simplify it another way.
When I look at the number 12 and the number 15, I immediately recognize that they have factors in common and since a number is made up of it’s factors, I can simplify this fraction by using the greatest factor they have in common.
The highest factor they have in common is 3. If I take 3 and divide the numerator by 3 and the denominator by 3, I will get 4 for the new numerator and 5 for the new denominator.
12/15 is really 4/5 —If you took a calculator and put in 12 ÷ 15 and then 4 ÷ 5, you would see that you got the same answer = 0.80.
All you have to do is identify the greatest common factor between the numerator and denominator of a fraction problem, then divide both the numerator and denominator by that and you will simply the fraction.
(See Greatest Common Factor)
You will be asked to do this a lot in high school and in college so understanding this topic is important. We will take about other strategies to do this, but for right now let’s look at more examples.
In the above fraction, 15/30 can be simplified to 1/2. It’s really useful to know that we are dealing with half of something, sometimes when we are dealing with numbers.
Let’s see if you can figure out the answers to these problems below. What is the simplest fraction that can be written with each of these fractions?
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Answer: 2/3
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Answer: 1/2
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Answer: 5/7
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Answer: 7/10
Summary
To simplify a fraction find the greatest (or largest) factor the numerator and denominator have in common and divide them both by it.