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## Addition and subtraction of rational numbers

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# Adding and subtracting fractions

## Video transcript

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.