In this video, we're going to talk about adding and subtracting fractions.
So, recall, umm, when you go to add or subtract a fraction, you have to have common denominators.
And again, that means that the stuff on the bottom has to be exactly the same.
So notice here I have 1/2 plus 5/2. I have the exact same denominator.
When you have common denominators, the arithmetic occurs in the numerator.
So in this case I'll get 1 plus 5 which is 6/2, and 6 divided by 2 is 3.
If you don't have common denominators, with, when you have just plain old numbers,
you're gonna have to get common denominators.
Sometimes you'll only have to manipulate one of the fractions.
Suppose I have 1/5 plus 4/15. Well, certainly the numbers on the bottom are not the same.
One thing that you can do is, you basically find the least common multiple of 15 and 5.
So we need the smallest number that both 15 and 5 go into.
Well, in that case, 15 certainly goes into 15, 5 also goes into 15.
So, then I think, okay, what number would I need to multiply the denominator by so that I have a 15.
Well, three times five is going to give me fifteen.
But whatever you do to the bottom of the fraction, if you multiply the bottom by a number,
you have to multiply the top by that same number.
So in this case, I'm gonna get 3/15, for my first, 3 times 1 is three, 3 times 5 is 15.
I'm not doing anything to the other one.
So I'll be left with 7/15 as my solution.
Likewise, sometimes you may have to change both of them.
Suppose I have 3/4 and 5/7. So again, you have to find the smallest number that both 4 and 7 will go into.
One way to do these problems, sort of, without think about it is, if you look at the other denominator,
in this case seven, you can multiply top and bottom of that fraction by that number.
Likewise, if I look at the original, there' a four in the denominator. I'll multiply top and bottom by four.
And again, notice now you have seven times four. Hey you have seven times four,
you're gonna get the same denominator.
In this case, you'll get 28 on the left side, 7 times 3 is 21. On the right side you'll have 5 times 4 which is 20.
7 times 4 is 28, and again, if you add 21 plus 20, you'll get 41/28.
And if we look at this, this example, umm, one more time,
Let's look, go back to our 1/5 plus 4/15 example.
So 1/5 plus 4/15.
Notice, I can basically factor 15 as 5 times 3.
When you want to get common denominators, if you look at all the denominators of all the fractions,
if you see anything that's in the bottom of one fraction, but is missing from the other,
you'll have to multiply top and bottom of that fraction by that number.
So again, I see a five in the denominator here, but I have a five already.
I see a three down here, but I don't see a three down here.
Again, that means I have to multiply top and bottom of the left term by three.
And again, I'll get 3/15 plus 4/15 which is 7/15 as we found before.
The idea is the same when you have more complicated expressions.
Suppose I have x over x squared, plus 3, and I'm gonna add to that 4x over x squared, plus 3.
Well, I have a common denominator. I see the exact same thing, x squared plus 3, x squared plus 3,
so in the denominator you'll get just x squared plus 3. You don't do anything there.
And then I add my terms, I have 1x plus 4x; I will get 5x.
Now let's suppose that things aren't quite the same.
Suppose I have an x squared minus 1, and suppose I just have x plus 1 in the denominator of my other fraction.
The idea in this one, again you're gonna factor just like in the last example with my number example.
I start off by factoring both sides, again there's nothing really to factor from 5; it's just 5 times 1.
And I'm gonna look at what's missing, and I'm gonna multiply the other part by that thing.
So I've got 5x on top, recall x squared minus 1 is the difference of perfect square,
so I can factor that as x plus 1, x minus 1.
X plus 1 won't factor any further, so I'll just leave it alone. And again, now I look at my fractions.
In the bottom of this fraction on the left side I see an x plus 1, x minus 1.
On the right side, I only have an x plus 1.
So everything that's in the bottom of the right one, I already have in the bottom of the left one.
So I'm not gonna have to do anything in the one on the left.
But notice if I look at the right one and compare that to the left one,
there's an x minus 1 in the denominator of the left one that's not in the denominator of the right one.
Well I would like to have that x minus one there,
so that means I also have to multiply the numerator by x minus 1.
So when I go to simply this, again my common denominator is now the x squared minus 1,
or equivalently, x plus 1 times x minus 1. And now we'll do the arithmetic in the numerator.
So notice I'm just kind of combining everything together
and I'm gonna have to distribute the negative three to each term.
So in the bottom, I will have an x squared minus 1 term.
Notice when I distribute negative 3 times x, I'll get minus 3x. When I take negative 3 times negative 1,
I'll get positive 3. My like terms are the x terms.
5x minus 3x is 2x. I can't do anything with the plus 3 term,
and then I'm left with x squared minus 1 in the denominator.
So the idea is the same whether you're dealing with numbers,
or whether you're dealing with more complicated expressions.
You'll want to factor and then basically multiply by what's missing.
So, I know these are only a couple of very basic examples,
but I hope they get you going in the right direction.