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# Expressions with rational numbers

Learn to compare expressions with positive and negative fractions. Created by Sal Khan.

Video transcript

We have four different
expressions here, and what I want
you to do is think about which of these expressions
are equal to negative 2/3. And I encourage you now
to pause this video, and try this on your own. So let's go to this first
expression right over here. I have 1/9, and I'm
going to add to that 5/9. So how many ninths
am I going to have? Well, I had 1/9, now I'm adding
5/9, so I'm going to have 6/9. If I have one of something
and I have five more of that same something--
so in this case, that something is a ninth--
1/9 plus 5/9 is 6/9. Now, can we simplify
this in any way? Well, both six and nine are
divisible by 3, so let's divide them both by 3 to try to
get this fraction in a simpler form. 6 divided by 3 is 2. 9 divided by 3 is 3. So this is 2/3, while what
we're trying to get to is negative 2/3. So these are not equal. This expression does
not equal negative 2/3, so I'll write "no" for that one. Now let's go to this green
expression right over here. Give myself a little bit
more real estate to work in. Now, we have negative
1/6 plus negative 1/2. Now, we can view this
as being the same thing as-- just to clarify, right
now the negative is in front of the entire 1/6,
the negative's in front of the entire 1/2. But this is the same
thing as negative 1/6, plus negative 1/2. Negative 1/2 is the same thing
as negative 1 divided by 2 is one way to think about it. And the whole reason
why I did this is so we can simplify what the
negatives are right now only in our numerator. So whenever we
add two fractions, we want to have the
same denominator. And we see that 6 is
already a multiple of 2, so we could leave this first
fraction the way it is. We can rewrite it
as negative 1/6. And then the second
fraction, we can write it as something over 6. Well, to go from 2 to 6,
we have to multiply by 3. So let's also multiply
the numerator by 3, negative 1 times
3 is negative 3. So if I have negative 1/6 sixth
and I add to that negative 3/6, this is going to be negative 1
plus negative 3 sixths, which is equal to negative 4/6. Now, let's see if
we can simplify it. Both negative 4-- I guess
we can say both four and six are divisible by 2, so
let's divide them both by 2. And in the numerator, we're
left with negative 4 divided by 2 is negative 2. 6 divided by 2 is 3. Negative 2 divided by 3. Well, that's the same thing
as negative 2/3, which is exactly what our goal
value we're trying to get to. So, yes, this thing in green
is equal to negative 2/3. Now let's go over here. So we have negative
1.3 times negative 2. Well, if you multiply a
negative times a negative, we're going to get a
positive, and we're going to get a
positive 1/3 times 2. So one way to
think about this is going to be the same thing
as one third times 2, which is the same thing. And there's a couple of
ways to think about it. If you have 1/3 and now you're
going to multiply it by 2, we now have 2/3. You now have 2/3. Another way to
think about this is that this is the same thing
as 2/3 times 2 over 1. And you know that when we
multiply two fractions, so this time we've expressed
the 2 as a fraction, we can multiply the numerators. So it's 1 times 2 over the
product of their denominators, 3 times 1, which is 2 over 3. So either way you look at it,
this goes to positive 2/3. A negative times a
negative is a positive. So it gets us to positive
2/3 not negative 2/3, so like this first one, no, it
does not equal negative 2/3. Now let's look at this one. Negative 1/3
divided by one half. So when you divide
by a fraction, so when you take
negative 1/3, dividing-- let me write it this way. So negative 1/3
divided by 1/2, this is the same thing
as negative 1/3-- and let me color code it just
so you see what I'm doing. So let me make that green color. Let me make this a blue color. So negative 1/3
divided by 1/2 half is the same thing
as negative 1/3 times the reciprocal of
1/2, so times 2 over 1. And what is this
going to be equal to? Well, we could assume
instead of just doing this as negative 1/3, we could do
this as negative 1 divided by 3, that might help us keep
track of the signs a little bit more. And let me actually
write it that way, just to make it a little bit clearer. Let me write this as
negative 1 divided by 3. So our numerator is now going
to be negative 1 times 2. When you multiply two
fractions, you just multiply the two numerators
to get the new numerator, and it's over 3 times 1. And you normally wouldn't
have to do all these steps, but I'm just doing
them to make sure you understand what's going on. And so this is going to be
equal to negative 1 times 2 is negative 2. And 3 times 1 is positive 3. Negative 2 over 3? Well, that's the same
thing as negative 2/3. So this one works out. It is equal to negative 2/3.