Welcome to the presentation on adding and
subtracting fractions.
Let's get started.
Let's start with what I hope shouldn't confuse you too much.
This should hopefully be a relatively easy question.
If I were to ask you what 1/4 plus 1/4 is.
Let's think about what that means.
Let's say we had a pie and it was divided into four pieces.
So this is like saying this first 1/4 right here -- let me
do it in a different color.
This 1/4 right here, let's say it's this
1/4 of the pie, right?
And we're going to add it to another 1/4 of the pie.
Let's make it this one -- let me change the color -- pink.
This 1/4, this pink 1/4 is this 1/4 of the pie.
So if I were to eat both 1/4s or 1/4 and then I eat another
1/4, how much have I eaten?
Well, you could look from just the picture, I have now eaten 2
out of the 4 pieces of the pie.
So if I eat 1/4 of a piece of pie or 1/4 of a pie, and then I
eat another 1/4 of a pie, I will have eaten 2/4 of the pie.
And we know from the equivalent fractions module that this is
the same thing as I've eaten 1/2 of the pie,
which makes sense.
If I eat 2 out of 4 pieces of a pie, then I've eaten 1/2 of it.
And if we look at it mathematically, what
happened here?
Well the denominators or the bottom numbers, the bottom
numbers in the fraction stayed the same.
Because that's just the total number of pieces
I have in this example.
Well, I added the numerators, which makes sense.
I had 1 out of the 4 pieces of pie, then I ate another 1 out
of the 4 pieces of pie, so I ate 2 out of the 4 pieces
of pie, which is 1/2.
Let me do a couple more examples.
What is 2/5 plus 1/5?
Well we do the same thing here.
We first check to make sure the denominators are the same --
we'll learn in a second what we do when the denominators
are different.
If the denominators are the same, the denominator of the
answer will be the same.
And we just add the numerators.
2/5 plus 1/5 is just 2 plus 1 over 5, which
is equal to 3 over 5.
And it works the same way with subtraction.
If I had 3 over 7 minus 2 over 7, that just equals 1 over 7.
I just subtracted the 3, I subtracted the 2 from the 3
to get 1 and I kept the denominator the same.
Which makes sense.
If I have 3 out of the 7 pieces of a pie and I were to give
away 2 out of the 7 pieces of a pie, I'd be left with 1 of
the 7 pieces of a pie.
So now let's tackle -- I think it should be pretty
straightforward when we have the same denominator.
Remember, the denominator is just the bottom
number in a fraction.
Numerator is the top number.
What happens when we have different denominators?
Well, hopefully it won't be too difficult.
Let's say I have 1/4 plus 1/2.
Let's go back to that original pie example.
Let me draw that pie.
So this first 1/4 right here, let's just color it in,
that's this 1/4 of the pie.
And now I'm going to eat another 1/2 of the pie.
So I'm going to eat 1/2 of the pie.
So this 1/2.
I'll eat this whole 1/2 of the pie.
So what does that equal?
Well, there's a couple of ways we could think about it.
First we could just re-write 1/2.
1/2 of the pie, that's actually the same thing as 2/4, right?
There's 1/4 here and then another 1/4 here.
So 1/2 is the same thing as 2/4, and we know that from the
equivalent fractions module.
So we know that 1/4 plus 1/2, this is the same thing as
saying 1/4 plus 2/4, right?
And all I did here is I changed the 1/2 to a 2/4 by essentially
multiplying the numerator and the denominator of
this fraction by 2.
And you can do that to any fraction.
As long as you multiply the numerator and the denominator
by the same number, you can multiply by anything.
That makes sense because 1/2 times 1 is equal
to 1/2, you know that.
Well another way of writing 1 is 1/2 times 2/2.
2 over 2 is the same thing as 1, and that equals 2 over 4.
The reason why I picked 2 is because I wanted to get
the same denominator here.
I hope I'm not completely confusing you.
Well, let's just finish up this problem.
So we have 1/4 plus 2/4, so we know that we just add the
numerators, 3, and the denominators are the same, 3/4.
And if we look at the picture, true enough, we have
eaten 3/4 of this pie.
Let's do another one.
Let's do 1/2 plus 1/3.
Well once again, we want to get both denominators to be the
same, but you can't just multiply one of them to get --
there's nothing I can multiply 3 by to get 2, or there's no,
at least, integer I can multiply 3 by to get 2.
And there's nothing I can multiply 2 by to get 3.
So I have to multiply both of them so they equal each other.
It turns out that what we want for, what we'll call the common
denominator, it turns out to be the least common
multiple of 2 and 3.
Well what's the least common multiple of 2 and 3?
Well that's the smallest number that's a multiple
of both 2 and 3.
Well the smallest number that's a multiple of
both 2 and 3 is 6.
So let's convert both of these fractions to something over 6.
So 1/2 is equal to what over 6.
You should know this from the equivalent fractions module.
Well if I eat 1/2 of a pizza with 6 pieces, I would have
eaten 3 pieces, right?
That make sense.
1 is 1/2 of 2, 3 is 1/2 of 6.
Similarly, if I eat 1/3 of a pizza with 6 pieces, it's
the same thing as 2 over 6.
So 1/2 plus 1/3 is the same thing as 3/6 plus 2/6.
Notice I didn't do anything crazy.
All I did is I re-wrote both of these fractions with
different denominators.
I essentially changed the number of pieces in the
pie, if that helps at all.
Now that we're at this point then the problem
becomes very easy.
We just add the numerators, 3 plus 2 is 5, and we keep
the denominators the same.
3 over 6 plus 2 over 6 equals 5/6.
And subtraction is the same thing.
1/2 minus 1/3, well that's the same thing as 3
over 6 minus 2 over 6.
Well that equals 1 over 6.
Let's do a bunch more problems and hopefully you'll
start to get it.
And always remember you can re-watch the presentation, or
you can pause it and try to do the problems yourself, because
I think sometimes I talk fast.
Let me throw you a curve ball.
What's 1/10 minus 1?
Well, one doesn't even look like a fraction.
But you can write it as a fraction.
Well that's the same thing as 1/10 minus -- how could we
write 1 so it has the denominator of 10?
Right.
It's the same thing as 10 over 10, right?
10 over 10 is 1.
So 1/10 minus 10 over 10 is the same thing as 1 minus 10 --
remember, we only subtract the numerators and we keep the
denominator 10, and that equals negative 9 over 10.
1/10 minus 1 is equal to negative 9 over 10.
Let's do another one.
Let's do one more.
I think that's all I have time for.
Let's do minus 1/9 minus 1 over 4.
Well the least common multiple of 0 and 4 is 36.
So that's equal to 36.
So what's negative 1/9 where we change the denominator
from 9 to 36?
Well, we multiply 9 times 4 to get 36.
We have to multiply the numerator times 4 as well.
So we have negative 1, so it becomes negative 4.
Then minus over 36.
Well to go from 4 to 36, we have to multiply this fraction
by 9, or we have to multiply the denominator by 9, so you
also have to multiply the numerator by 9.
1 times 9 is 9.
So this equals minus 4 minus 9 over 36, which equals
minus 13 over 36.
I think that's all I have time for right now, and I'll
probably add a couple more modules, but I think you might
be ready now to do the adding and subtracting module.
Have fun.