Distributing First vs. Adding First: Differences & Examples

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  • 0:02 What Comes First?
  • 1:00 Questions to Ask
  • 1:55 Decimals
  • 2:33 Fractions
  • 4:19 Practice
  • 6:01 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

When we need to simplify an expression, how do we know if we should distribute first or add first? In this lesson, we'll look at when this matters and how to make the best decision.

What Comes First?

When you get ready to go out in the morning, do you put your socks on first or your shoes? Socks, right? What about your pants or your shirt? You probably have a habit of doing one or the other first, but it doesn't really matter, does it?

What about simplifying algebraic expressions? We know the distributive property tells us that a(b + c) = ab + ac. But what if we have 3(2 + 4)? Should we do 3 * 2, which is 6, then 3 * 4, which is 12, to get 6 + 12, which is 18? Or should we do 2 + 4, which is 6, then 3 * 6, which is, again, 18?

This is a shirt vs. pants kind of example. It really doesn't matter. We'll get the same answer, 18, just as we'll still end up with our shirt and pants on no matter what we do first.

How do we know if we should add first or distribute first? There are a few factors to consider.

Questions to Ask

Let's start with the most crucial question: can you add first? We're looking to see if we can combine like terms. If the answer is no, well, then it's time to distribute first.

If you have different variables, like 2(3x + 2y), you can't add first. You can't combine 3x and 2y. That's like trying to put your belt on before your pants. Likewise, if you have something like 5(1 + 2x), you have one term with a variable, 2x, and one term that's just a constant with no variable, 1. You can't combine those either. Sorry, belt, you have to wait.

Next question: is adding or distributing first easier? This is subjective. Is it easier to pour the milk after putting the cereal in the bowl? Yes. Then you know how much milk to add. Can you start with the milk? I guess so, though that would be kind of weird.


Here's an expression: 2x(1.7 + 3.3). If we distribute first, we do 2x * 1.7, which is 3.4x, and 2x * 3.3, which is 6.6x. 3.4x + 6.6x = 10x. That's not so bad. But we had to do two steps with decimals. If we added first, we'd do 1.7 + 3.3, which is 5. Then 2x * 5 is 10x. That's easier. Adding first is often easier with decimals.


What about fractions? Do we go milk or cereal first? Look at 3y(1/4 + 1/2). If we distribute first, it's 3y * 1/4, which is 3y/4, then 3y * 1/2, which is 3y/2. Now we have 3y/4 + 3y/2. We need to get common denominators, so we multiply the second fraction by 2/2 to get 6y/4. Now we can add them to get 9y/4. Whew.

What if we added first? 1/4 + 1/2. Ok, that's 1/4 + 2/4, which is 3/4. 3/4 * 3y? 9y/4. Adding first is often easier with fractions, too.

This isn't always the case, though. Look at this one: 9x(2/3 + 1/9). We can make 2/3 into 6/9, so 6/9 + 1/9 is 7/9. Then we can multiply that by 9x to get 63x/9. That simplifies to 7x. But what if we distribute first? We do 9x * 2/3, which is 18x/3, or 6x. Then 9x * 1/9, which is 9x/9, or just x. 6x + x is 7x. Distributing first really wasn't any harder, was it?

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