*Yuanxin (Amy) Yang Alcocer*Show bio

Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.

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6th-8th Grade Math: Practice & Review
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6th-8th Grade Math: Simplifying Whole Number Expressions
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Instructor:
*Yuanxin (Amy) Yang Alcocer*
Show bio

Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.

Mathematical problems involving division can be written as division expressions to be solved easier. Learn how to simplify a division expression by finding factors through examples.
Updated: 11/03/2021

Once you get into the algebra part of math, you will begin to see a lot of problems that look like this:

*Simplify 8x / 4x*

This is essentially a division problem. In math, we call these **division expressions**, mathematical expressions using division. When the problem is asking you to simplify, it is asking you to simply reduce the numbers and letters as much as you can.

You can think of it as a matching game. You know the game where you turn over different cards and if you find a matching pair of cards, you stack them together? In our case, that's what we are looking to do. We are looking for matching pairs that we can stack together. The only difference is that once we've found a matching pair, we can completely take it out of our problem, thus making our problem simpler.

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This whole process is called simplifying. Let's look at how we do this. Right now, we have an 8*x* on the top and a 4*x* on the bottom. To find matching pairs, we will break up our numbers into their factors. For example, our 8 can be broken down into 4 * 2. We see a 4 here, which matches with the 4 on the bottom. That tells us that we don't have to break up our 4 because we have already found a matching pair. When I say 'matching pair', I mean that you have the same thing on the top and the bottom of your problem.

I have rewritten our problem to show the breakdown of the 8 into 4 * 2. We have a matching pair of 4s since we have a 4 in the numerator and a 4 in the denominator. We also have a matching pair of *x*'s since there is an *x* in the numerator and an *x* in the denominator. Because they are matching pairs, we can go ahead and take them out of our problem.

Why can we do this? Well, think about it. What is 4 divided by 4? It's 1. What is anything multiplied by 1? Itself. So, by taking out matching pairs, we are essentially just taking out a multiplication by 1, which doesn't change our problem at all. It is the same with our *x* divided by *x*. Because the numerator and denominator are the same, it will equal 1 and we are actually just taking out a 1, which doesn't change our problem at all.

What do we get after taking out our matching pairs? We get 2 for our answer. And we are done!

Let's look at a couple more examples to get a better grasp of this process.

*Simplify 5x / 10x*

Here we have a 5*x* on the top and a 10*x* on the bottom. We already have a matching pair of *x*'s so we can go ahead and take those out. We are left with 5 / 10. We can't break up the 5 anymore. But we can break apart the 10 into 5 * 2. Now we have a matching pair of 5s. We can take those out. What are we left with? 1/2. That is our answer.

In this problem, we managed to take everything out of the numerator. When this happens, the only number that is left is a 1.

*Simplify 10xy / 2x*

This problem has two different letters now. But we are still just looking for matching pairs. We can break up our 10 into 5 * 2. After doing this, we see that we have a matching pair of 2s and a matching pair of *x*'s that we can take out. What are we left with? We are left with a 5 on the top and a *y* on the top. There is nothing in the bottom. So our answer is 5*y*.

Let's review what we've learned.

**Division expressions** are mathematical expressions using division. To simplify these types of expressions, we look for matching pairs. A matching pair is found when we have the same number or letter in both the numerator and denominator.

When we have a matching pair, we can take it out of the problem or otherwise cancel it out, thus simplifying our expression. When we have two different numbers, we can break up our numbers into their factors to see if there are any matching factors. After all matching pairs have been taken out, then we are done simplifying our expression and we have our answer.

After you have finished this lesson, you should be able to simplify a division expression by finding factors.

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