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# Subtracting mixed numbers

Subtracting Mixed Numbers. Created by Sal Khan and Monterey Institute for Technology and Education.

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.