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Addition and subtraction of fractions

Video transcript

We're asked to subtract, simplify the answer and write as a mixed number. So we have two mixed numbers here, and like when you're adding mixed numbers, you can do it two ways. You can either turn both of these into improper fractions and then subtract, or you can subtract the whole number part and then the fraction part, so let's do it that way. So just to remind ourselves, 5 and 5/8 is the exact same thing-- let me write it here in yellow-- as 5 plus 5/8. These are the exact same thing. And subtracting-- let me write this subtraction sign in blue-- and then we're subtracting 2 and 1/5, but 2 and 1/5 is the exact same thing as 2 plus 1/5. That's what 2 and 1/5 really is. Now if we're subtracting, that means we have to subtract 2 and we're subtracting the 1/5. Distributive property. We're subtracting both of them. So we get 5 plus 5/8, and then we distribute the negative sign, minus 2 minus 1/5. And now we can rearrange it. We can say this is the same thing as-- let me do it in orange-- as 5 minus 2, that's the whole number parts, and then you have plus 5/8 minus 1/5. Now, 5 minus 2, that's pretty straightforward. That's just 3. But what's 5/8 minus 1/5? Let's do that over here. So we have 5/8 minus 1/5. Now, just like when you're adding fractions, when you're subtracting fractions, you have to have a common denominator. 8 and 5 are not the same denominator so we have to find the least common multiple of these two numbers. If we do a prime factorization of 8, you get 2 times 4. 4 is 2 times 2. So 8 is 2 times 2 times 2. That's if you break it down into its constituent primes. 5 is just 5, because it is a prime number. So the least common multiple of 8 and 5 has to have a 5 and three 2's in it. Or another way of saying it, it has to be 5 times 2 times 2 times 2, which is the same thing as 5 times 8. So the least common multiple is going to be-- because these guys share no common factors other than 1 really. The least common multiple is just going to be 8 times 5. So the least common multiple is going to be 40. It's going to be 40. Now, to go from 8 to 40, you have to multiply by 5. Then we have to do the same thing to the numerator. If we multiplied 8 times 5, we have to multiply 5 times 5 to get 25. So in either case, we're multiplying times 5, the numerator and the denominator. Now, to go from 5 to 40, we have to multiply by 8, so we also have to do that for the numerator, so 1 times 8 is 8. So now we have 25/40 minus 40/40, which would be equal to-- we're going to have 40 as the denominator. 25 minus 8 is 17. So 5/8 minus 1/5, which is the same thing as 25/40 minus 8/40, is equal to 17/40. And 17 is a prime number so it's not going to share any common factors with 40, or at least 40 isn't divisible by 17 either, so this is as simplified as we can get. So this part, the fraction part, is 3 and the fraction part is 17/40. And we're done. And this worked. We were able to separate this out and do it pretty straightforward, because in this situation, the fraction part that we're subtracting from the other fraction part was smaller. If it was larger, it would've gotten a little bit messy. Maybe we would have ended up with a negative number and we would've done all this other stuff, so that's why sometimes it's easier to go straight to an improper fraction for both of them. But in this case, it worked out doing it this way.