Expressions with rational numbers

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 2/3 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 1. 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.