Adding and subtracting fractions with negatives | 7th grade | Khan Academy

Let's say we wanted to figure out what (3 \frac{7}{3}) minus (-\frac{7}{3}) minus (\frac{11}{3}) is. Pause this video and see if you can have a go at it before we do it together.

All right, now let's work on this together. You might be tempted to deal with the (-\frac{7}{3}) and the (\frac{11}{3}) first because they already have a common denominator. But you have to realize that with subtraction, you can't use the associative property. It's not this: ((a - b) - c) for example, which is what you would typically do first. It is not the same thing as this right over here. So you have to be very, very, very careful.

But what we could do is rewrite this. Instead of saying "subtracting something minus something else," we could rewrite it in terms of addition. What do I mean by that? Well, if I have (3 \frac{7}{3}), I'll start with that.

Subtracting something is the same thing as adding that something's opposite. So subtracting (-\frac{7}{3}) is the same thing as adding the opposite of (-\frac{7}{3}), which is just (\frac{7}{3}). And subtracting (\frac{11}{3}) is the same thing as adding the opposite of (\frac{11}{3}), which is (-\frac{11}{3}).

Now, with addition, you can use the associative property. You could add these two first or you could add these two first. I like adding these two first because they have the same denominator. So if I have (\frac{7}{3}) plus (-\frac{11}{3}), what is that going to get me?

Well, we have a common denominator. We could rewrite it like this: (3 \frac{7}{3}) plus a common denominator of three. We could write (7 + (-11)) in the numerator. So (7 + (-11)) is the same thing as (7 - 11) because subtracting something is the same thing as adding its opposite.

So, for adding (-11), the same thing as subtracting (11). So (7 + (-11))—you might want to get a number line out—but hopefully, you've gotten some practice. Now, that is going to be (-4). That is (-4).

And so now we have (3 \frac{7}{3}) plus (-\frac{4}{3}). Now we definitely need to find a common denominator. So let me rewrite this. This is equal to (3 \frac{7}{3}) plus (-\frac{4}{3}) or I could write this as even (-\frac{4}{3}). Either way.

But if we want to have a common denominator, it looks like (21) is going to be the least common multiple of (7) and (3). So let's rewrite each of these as something over (21).

From (7) to (21), we multiply by (3). So (3 \times 3 = 9). And then from (3) to (21), we multiply by (7). So if we have (-4) times (7), that is (-28).

And so this is going to be equal to (\frac{9 + (-28)}{21}), which is the same thing as (\frac{9 - 28}{21}) because subtracting a number is the same thing as adding its opposite.

And so this gets us—let's see—if (9 - 9 = 0) and then we're going to have (19) more to go below zero. So this is (-\frac{19}{21}) or we could write that as (-\frac{19}{21}) and we are done.