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Adding and subtracting fractions with negatives

Use the properties of addition to strategically add and subtract fractions with negatives and unlike denominators. We can rewrite the subtraction as adding the opposite so that the addition properties will apply. Created by Sal Khan.

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Video transcript

- [Instructor] Let's say we wanted to figure out what 3/7 minus negative 7/3 minus 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. And you might be tempted to deal with the negative 7/3 and the 11/3 first because they already have a common denominator. But you have to realize that subtraction, you can't use the associative property. It's not the this, for example which is what you would typically do first 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, minus something, minus something else we could rewrite it in terms of addition. What do I mean by that? Well, if I have 3/7, I'll start with that. Subtracting something is the same thing as adding that something's opposite. So subtracting negative 7/3 is the same thing as adding the opposite of negative 7/3 which is just positive 7/3. And subtracting 11/3 is the same thing as adding the opposite of 11/3 which is negative 11/3. Now addition, you can use the associative property, you could add these two first or you could add these two first. And I like adding these two first because they have the same denominator. So if I have 7/3 plus negative 11/3, what is that going to get me? Well, we have a common denominator. We could rewrite it like this. 3/7 plus common denominator of 3. We could write 7 plus negative 11 in the numerator. And so 7 plus negative 11 is the same thing as 7 minus 11. Because subtracting something's the same thing as adding its opposite. So if we're adding negative 11, same thing as subtracting 11. So 7 plus negative 11, you might, we could get a number line out, but hopefully you've gotten some practice now. That is going to be negative 4. That is negative 4. And so now we have 3/7 plus negative 4/3. And so now we definitely need to find a common denominator. So let me rewrite this. This is equal to 3/7 plus, lemme write this, plus negative 4. Actually yeah, this is fine plus negative 4/3. Or I could write this as even negative 4/3, either way. But if we wanna 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, 3/7. To go from 7 to 21, you multiply by 3. So 3 times 3 is 9. And then to go from 3 to 21, we multiply by 7. So if we have negative 4 times 7, that is negative 28. And so this is going to be equal to 9 plus negative 28/21, which is, this is the same thing as 9 minus 28/21, 'cause subtracting a number is the same thing as adding its opposite. And so this gets us, let's see if 9 minus 9 is 0 and then we're gonna have 19 more to go below 0. So this is negative 19/21 or we could write that as negative 19/21. And we are done.