In this video we are going to talk about adding and subtracting fractions.
Now the main thing you need to remember about adding and subtracting fractions is that in
order to add two fractions, or subtract two fractions, the denominators must match each
other.
And if they do, then the process is very straight forward.
I'll give you an example.
If you had two-fifths plus one-fifth when the denominators match together, then the
arithmetic happens in the numerator, and you would just take two plus one, and that would
be your new numerator which is three, and you would keep your common denominator of
five.
Two-fifths plus one fifth would equal three fifths.
Well it gets a little trickier when the denominators don't match each other.
Here's what we have to do.
If the denominators do not match, we need to force those denominators to match.
We're going to use something called the "least common multiple".
Now there's a couple of different ways that you can find the least common multiple.
One's a little informal, and one's a little bit more formal.
In the vast majority of cases that you'll run into, the informal way of finding the
LCL will work just fine.
Now we'll actually have a video coming up after this one, where we talk about how to
find the least common multiple in the more formal way as well.
But ah, in this video we're going do the slightly more informal way.
Let me show you how it works with an example.
Let's say we had three-fourths plus one-sixth.
Well right now we can't add these because the denominators don't agree with one another.
So what we need is the least common multiple of these denominators, four and six.
So first of all, what is a least common multiple?
It's the smallest number that you can have that's both a multiple of four, and a multiple
of six.
So for example, eight.
Eight a multiple of four, but eight is not a multiple of six.
Uhm, you could take something like twenty-four.
Twenty-four is a multiple of four, and it's a multiple of six.
But it's not the lowest common multiple that we can have.
Uh, for this one I think it's actually twelve, and here's how I know that.
The informal way of finding a least common multiple is to start with the larger of the
two numbers, and then start uhm, adding multiples of that on top of itself until you find one
that's also a multiple of the other number.
So all of these, if I take six, and then twelve, and then twelve, and six more make eighteen.
And six more make twenty-four, and six more make thirty.
All of these are multiples of six because I just keep taking larger and larger multiples
of six.
But I'm going to start looking in this list and I am going to stop when I find somebody
who is also a multiple of four.
Six is not a multiple of four, but twelve is, done, we found it.
And so that's all you do, you start with the larger of the two numbers and you keep taking
larger and larger multiples of that, with an eye on the smaller number, and you keep
asking yourself is this a multiple of the other number, is this one, is this one, and
you should run across it pretty quickly.
So the least common multiple of four and six will be twelve.
Now when does this not work so well?
Well if you have two really big numbers, then it can get really cumbersome with this technique.
But you know, if you look in your, your algebra textbook and you look through all the exercises,
the vast majority will be fours, and sixes, and fifteens, and twelves.
And there not going to be very big numbers in the hundreds and what not.
So for the vast majority cases this is just fine.
So now, once we find the LCM, what are we supposed to do with it?
Well we're going to actually force both denominators to look like twelve.
And I know this is possible because twelve is a multiple of four, and twelve is a multiple
if of six.
That's why we chose it.
So let's ask ourselves, four times what, would give us twelve?
Well four times three would give us twelve.
So I am going to multiply the denominator by three.
And remember I also need to multiply the numerator by three.
Now why can I do this, and why does this not change things?
Well, I'll ask you this, what's three over three?
Well three over three, is one.
And when you multiply anything times one, it won't change the quantity.
It might change the way that it looks, but it won't change the meaning of it.
Now what do I have two multiply six by, to get twelve?
Well I think two.
And so I will multiply numerator and denominator by two.
And this is going to give us two new fractions.
The first one you'll get nine, over twelve.
That's three times three, over three times four.
And let me stop for a minute.
Ah, let's be very clear, nine over twelve is the same as three over four.
It's the same ratio it just looks a little bit different it has a different denominator.
And the second fraction will be two over twelve.
Two times one, over two times six.
And so two over twelve, is the same as one over six.
But this has an added bonus.
What's the bonus of writing it this way?
Well we have the same denominator now.
So this is what we do.
If the denominators do not match each other, we force them to match, using the least common
multiple.
Once that's done, everything is down-hill from there.
Uh, nine over twelve, plus two over twelve.
The denominators match, the arithmetic happens in the numerator.
And so we would get eleven over twelve for our final answer.
Alright, let's wrap-up with one more example.
Let's say we have five over fourteen, minus eleven over seven.
The denominators do not match, so I will take fourteen, and I'll take seven, and I'll start
with the larger one and I'll keep making a list of multiples of fourteen.
Fourteen, twenty-eight, so on, and so forth, and I'm going to stop when I get somebody
who's a multiple of seven as well.
Well were done, the very first term, fourteen, fourteen's a multiple of seven.
So fourteen is going to be my least common multiple of these two denominators.
So I'll take seven times two, to get fourteen.
And I'll also need to take eleven times two.
I don't have to do anything with the first fraction because it already has a denominator
of fourteen.
So this would be the same as five over fourteen, minus twenty-two over fourteen.
I have a common denominator now, so I can proceed with the algebra.
See the arithmetic happens in the numerator.
Five minus twenty-two is negative seventeen, divided by the common denominator of fourteen,
and we're finished.
So hopefully that helps you better understand how to add and subtract fractions that have
different denominators.