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Addition and subtraction of fractions

Video transcript

They want us to subtract these two numbers and simplify the answer and write as a mixed number. We're going to do it two ways. The first way, we're going to turn both of these into improper fractions and see what we get. And then the other way, we're going to subtract the whole number parts and then subtract the fraction parts and see what we get. So if we turn them both into improper fractions, let's first turn 8 and 2/3 into an improper fraction. So it has 3 in the denominator, and 3 times 8 is 24. 24 plus 2 is 26. So 8 and 2/3 is the same thing as 26/3. And 5 and 5/6, so we're going to subtract. We're going to have 6 in the denominator. 6 times 5 is 30. 30 plus 5 is 35. So this simplifies as an improper fraction to 35/6. Now to subtract, we have to have a common denominator. 3 and 6 are not the same denominator, but they are both divisible into 6. 6 is the least common multiple of 3 and 6. So let's turn them both into something over 6. So the 26/3, if we put it over 6, to go from 3 to 6 in the denominator, we have to multiply by 2, so we have to do the same thing in the numerator. We have to multiply 2 times 26, which is 52. So it's going to be 52/6 minus-- we don't have to change this one here in orange. It's already over 6-- minus 36/6, which is going to be equal to-- what's 52 minus 35? Let's see, if you add 5 to 35, you get to 40. 15 will get you to 50, and so we want 2 more, so it's going to be 17. So it's 17/6. This is as an improper fraction. Now, if we want to write it as a mixed number, we say how many times does 6 go into 17? Well, 6 times 2 is 12, so that works. 6 times 3 is too big. It's 18, so it goes in two times. Now, if you take it into it two times, you get 12, and then you have 5 left over, right? To go from 12 to 17, you need 5 so you have 5 left over. You could say 6 goes into 17 two times, remainder 5, so 17 over 6 is 2 and 5/6. And we're done! Now this is kind of the improper fraction route, which it'll always work. It'll always work. Sometimes it's easier or harder, but it will always work. Actually, either route will always work, but I tend to do it this way. Let's do it separating out the whole number and the fractional parts. So first we could view this as 8 minus 5. So let's do this. 8 minus 5-- that's subtracting the whole number parts-- plus 2/3 minus 5/6. Now, 8 minus 5, that's easy. That's 3. And then over here, though, we have an interesting-- well, let's try to work it out. Let me do this in a different color. We want a common denominator here, and we already saw that 6 is the least common multiple of 3 and 5, so we could write them both with 6 as the denominator. 5/6 already has 6 as the denominator. We don't have to do anything there. To go from 3 to 6 in the denominator here, we have to multiply by 2, so let's do the same thing with this 2. 2 times 2 is 4. So this results in 4/6 minus 5/6, which will give us-- and we haven't covered it yet-- a negative number. If you take 5 from 4, you're going to go below 0. Now, if we don't want to deal with that negative number, what we could do, and this is essentially the same idea as regrouping or borrowing, although they shouldn't call it borrowing, they should call it taking, is we can rewrite this 3. That's why this can get a little bit tricky sometimes. Let me do it over here. Let me draw a line here so we don't get the two ways of doing the problem confused. We can rewrite the 3 as 2 plus 1, but instead of writing it as 1, we could write it as 6/6, right? 6 over 6 is 1 plus 2 is 3. And then, if we add it to these guys, so plus 4/6 minus 5/6. Now, what happened by doing that? Well, if we throw this 6/6 into the mix and then we had the 4/6 and then we subtract the 5 from it, then we're going to have a positive number, so let's to do it. So we get this 2 right here. So it's 2 plus, and then over 6, we have 6 plus 4-- let me write it. 6 plus 4 minus 5. Now, what's this going to equal? This is going to be this 2 and what over 6? 6 plus 4 is 10 minus 5 is 5. So you have 2 and 5/6, which is exactly what we got the other way.