College Algebra: Working with Fractions

This lesson has not been reviewed.

Please purchase the lesson to review.

Okay, so fractions. This always gives me problems and probably gives other people problems as well, and so let me just quickly review this with you really fast and make sure we're all on the same page.
So suppose you want to take a look at something like--I'll start really simple. You have and you want to add . So what do you do? Well, if you add the tops you get 2, you add the bottoms you get 6, and so this looks like 1/3. Great, except this is wrong. So this is actually not...
Okay, what you've got to do is you've got to get a common bottom? Well, why? Well, if you have like half, half of something, and then you take a quarter of something, if you put them together, that doesn't look like a third, does it? So what you have to do is you have to sort of get everything in the same language, in the same terms so we can combine them. So in this case what I would do is I would take a half and say, "Don't think of it as a half. Think of it as two-fourths." Then you add it with and you say, "Ah, the answer is 3/4." Okay, not a big deal. But what's the point? The point is we have to always get a common denominator whenever you're adding or subtracting fractions.
So let me do an example for you right now live. Let's suppose you have 5/6 and you want to subtract 1/10. So how would you do that? Well, what I would do, I would actually factor the bottom so I can see all its factors. That would be just 2 times 3, and then I'd factor the bottom here, which is 2 times 5. And now, to get a common factor, I want the same bottom. So what do I do? Well, what I would do is I would say, "Okay, how do I make all these things the same? So here I don't have a 5, but I have a 5 here. So let me multiply top and bottom by 5. See, multiplying the top and the bottom doesn't change the fraction, because they would cancel out and I would just have 1. So if I multiply the top here by 5 and the bottom here by 5, then I've got something that has 2 times 3 times 6 all on the bottom. Now, what do I have on the top? Well, I'm sorry. On the top I just have 5 times 5. Who cares? Well, what do I have on the second fraction? Well, I have 2 times 5, but I need a 3. So I multiply top and bottom by 3. Now, what do I see? What I see is 5 times 5 is 25, divided by... It's going to be 15 times 2 is 30, and then I subtract off 3 divided by 30. There's my common bottom. And now I can just subtract very nicely. If I have 25 of something, I take away 3 of those somethings, then of course, I'm left with 22 of that something. So 22/30, which if you simplify, is just 11/15. Terrific. So adding and subtracting fractions, not a problem. Always remember, though, common bottoms.
Now, multiplication is a piece of cake. I mean, if only addition and subtraction were as easy as multiplication, we'd be home free. So let me just remind you how multiplication works with a really simple example. Suppose I asked you just to take half of 6. Now, remember "of" is the same thing as multiplying. So taking half of 6 is taking multiplied by 6, but we know the answer is 3. And the answer is 3 because it's 6 divided by 2, which equals 3. And so this really illustrates the whole point. If you're multiplying fractions, just multiply the tops, and multiply the bottoms, just like you would think. So, for example, if you had something like 5/6 and you want to multiply it by 2/15, not a problem. Just multiply the tops, 5 times 2, and then multiply the bottoms. So you've got to be saying, "Gee Ed, why are you just wasting our time? Why don't you actually say that's 10? Well, I know it's 10. But the thing is, it's easier to cancel sometimes. Because I don't know, 10 is a big number and I can't subtract it. So what I do - what I notice is, that I can cancel this 5 away with this 15 and that leaves me with a 3 downstairs, and this 2 and the invisible 2 here can cancel, so if I cancel this I get a 3. Now, if I cancel this, some people say, "Hey, it's zero on top. There's nothing left." That's not quite right, because remember, there's always, always in life, an invisible one factor. Whenever you think you're alone and you're depressed, remember, you always have a factor of one right by your side. Really, there's a 1 right there, and so you really have which is 9. So multiplication, not a problem.
Okay, what about division? Well, division is not a problem as long as you look at it the right way. So let me remind you how division works. Now, division comes in a couple of flavors and we'll see all sorts of flavors in this course. So, for example, we could see a problem like this: We could see 2/3 and then it's all divided by 5/7. Some people call this a complex fraction. This has no identity at all, so it's complex. Well, how do you deal with it? Well, it's not a problem. There's a couple of ways of thinking about it. One way is just to write this down using the divided sign - you know, this little divided sign that we learned about. If we write it down that way then it's real easy to see what's going on, because I could write it this way. I could say 2/3 divided by, and then 5/7. That's the same thing as this. And now just to divide them, here's the trick. You take the second guy, you invert it, and multiply. So what that would look like is the following. That would just look like 2/3, but now times, and then the flip of this would be 7/5, and then we could just do that now in the old way. Multiply tops and get 14; multiply the bottoms and get 15. And so there's the answer.
So let me show you that again now in general. So suppose that you want to take and you want to divide it by . What do you do? Well, the trick to this, remember, is to take the bottom and invert it and multiply. So what you want to do is you want to take the bottom and invert it... Now, watch this. This is high tech, folks. This can only happen on the web. Don't try this at home. There, you got it. And you take this away, throw it away, and what you do... What used to be a divided is now a multiplication. And now you multiply the top and multiply the bottom. So dividing, not a problem, as long as you remember that the denominator has to be inverted and brought to the top and multiplied, and multiplication, not a problem, tops and bottoms, just put them together. Addition and subtraction - always get a common bottom. Please, please, please.
Okay, that's all I have to say about fractions. See you soon.
Prerequisites
Rational Expressions
Working with Fractions Page [2 of 2]