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# How to add fractions that have different denominators

To solve this problem, we need to find the least common multiple to get at the common denominator. Can you help? We bet you can! Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

We're asked to add 4/9 and 11/12
and to write our answer as a mixed number, and then
simplify and write our answer as a mixed number. So here we have two fractions
we're adding together, but we have different denominators. So whenever you add fractions,
the first thing you have to do is check the denominators. If they're the same, you can
add, but if they're different like this, you have to make
them have the same denominator. So what we have to do is find
a number that both 9 and 12 will divide into, and that
will be our common denominator, and you'll see
why both 9 and 12 have to divide into it. So let's think about what that
number is, and there's two ways of coming up with that
what we could call a least common multiple, the smallest
multiple of both 9 and 12 that is common. One way is just to kind of look
at the multiples of 9 and see if any of them are
divisible by 12. So if you start with 9--
we can do it over here. So you have 9, that's
not divisible by 12. 18 isn't divisible by 12. 27 isn't divisible by 12. 36, well, that is
divisible by 12. That is 12 times 3. So 9 goes into 36 and
12 goes into 36. So what we want to do is write
a common denominator. So we're going to write 4/9 as
something over 36, and we're going to write 11/12 as
something over 36. Now, to turn your 9 into
a 36, you have to multiply it by 4, right? 9 times 4 is equal to 36. Now, you can't just multiply
the denominator by 4. You also have to multiply the
numerator by the same thing. So if you multiply the numerator
by 4, you get 4 times 4 is 16. So 4/9 is the exact same
thing as 16/36. If you wanted to simplify this
one to 4/9, you divide the numerator and the denominator
by 4. Now, we do the same
thing over here. 36, 12 times 3, so we're
multiplying 12 by 3 to get 36. Well, if we did that to the
denominator, we also have to do that to the numerator,
so 11 times 3 is 33. And just like that, we've now
rewritten each of the fractions so that they have
the same denominator. Both of their denominators
is 36. So now we're ready to add. If you add these two things,
we'll have 36, because we're considering kind of parts of
36 or fractions of 36, and then we have 16 plus 33
in the numerator. Let me write that down. 16 plus 33 in the numerator. And 16 plus 33 is what? 6 plus 33 would be 39
and then you have another 10, so it's 49. So it's equal to 49/36. Now, can we simplify this? 49, it's 7 squared, so it has
1, 7 and 49 as factors. This has 1-- it has a bunch
of numbers, but it's not divisible by 7, so this is
actually in simplest form, but this is an improper fraction. The numerator is larger
than the denominator. So let's write it as
a proper fraction. To do that, we divide
36 into 49. 36 goes into 49 how
many times? Well, it only goes one
time, so it equals 1. And how much will
be left over? If I divide 36 into 49 one time,
or 1 times 36 is 36, then I have 13 left
over to get to 49. So it's 1 and 13/36. And you can do that manually,
if you like you. You'd say 36 into 49. 36 goes into 49 one time. 1 times 36 is 36, and
then you subtract. 9 minus 6 is 3. 4 minus 3 is 1. You have a remainder of 13. So that's our answer:
1 and 13/36.