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### Course: Arithmetic > Unit 9

Lesson 2: Common denominators# Finding common denominators

Learn all about finding common denominators in fractions. Watch how to use the least common multiple of the denominators to rewrite fractions, making them easier to compare or add. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- How do I find LCM with 3 fractions(300 votes)
- Its quite simple you find out what the three numbers are, lets say the numbers are: 2,3,4.

then you wright the numbers they all have in common.

2: 2,4,6,8,10,12

3: 3,6,9,12

4: 4,8,12

Then circle or look to find witch they all have In common. If you look hard they all have or are equal to the number 12. I hope my answer helped you answer your question :)(172 votes)

- What if one of the denominators is a 1?(54 votes)
- If the denominator is a 1, then it's still the same formula: you simply divide the denominator by the numerator (ex: 4/1 = 4 divided by 1 = 4). It works no matter what the numerator or denominators are, even if the denominator isn't a 1 (ex: 7/3 = 7 divided by 3 = 2 1/3). With 4/1, it's easy to know that it will be 4, but with bigger numbers (ex: 282/47), it can be a lot harder to mentally figure out what your answer will be, so the dividing trick can help a lot in those scenarios. I know this answer is really late -- four years late; sorry -- but I hope either you see this, Noah, or somebody else who needed help sees this and gets the help that they need. :)(23 votes)

- how is 2/8 the same as 6/24 and 5/6 the same as 20/24(4 votes)
- Well, 2/8 is equivalent to 6/24 because the ratio is the same. The ratio is 1/4, and any same-numerator-denominator pair that's multiplied with it just changes the numerator and denominator. The hard-boiled value of the fraction is the same: 1/4.

It's a little hard to visualize, but here:

1/4 times 2/2 is what? 2/8.

2/8 times 3/3 is what? 6/24.

6/24 times 4/4 is what? 24/96.

24/96 divided by 24/24? 1/4!

So long as you continue to multiply the fraction by ONLY same-numerator-denominator fractions the ratio and value will stay the same, and thus the fractions will equal each other.

Tip: This means that you can simplify larger fractions down using a numerator-denominator to its smallest form. Keep in mind, though, that it helps to find the common factor(preferably greatest) of both the numerator and denominator before creating your same-numerator-denominator pair.

For example: 100/120 + 43 (Don't mind the +43, this is solely to simplify the fraction 100/240)

100 and 120 are both divisible by these numbers: 1, 2, 4, 5, 10, and 20. To simplify the fraction, you just need to divide the fraction by a same-numerator-denominator pair like this:

100/120 / 20/20 = 5/24

Thus, the "true" and simplified value of 100/240 is 5/24.

I know this was really winded, but I hope it helps. Let me know if you need help on anything else!(3 votes)

- How do I find the LCD of two rational equations? If there is a video for this can you point me to it?(3 votes)
- Good question! Let's say that your two equations or expressions were x^2 - 9 and x + 3. You would have to factor the equations. The second expression (x + 3) is already completely factored, so you can leave that. Now for x^2 - 9, you'd need to find what times what will get you that. We have x^2 which means x times x, and we know that 9 is being subtracted. The square root of 9 is 3. Our two factors are (x - 3) and (x + 3). Our LCD is going to be x + 3, because both equations have this in common. I hope this helps. It's a little advanced. Here is an extra example you can view: https://www.brainfuse.com/curriculumupload/1378832857500.pdf

Also, here is a video that uses somewhat the same example as I made: http://virtualnerd.com/algebra-2/rational-functions/add-subtract-expressions/add-subtract-different-denominators/find-least-common-denominator-example

:) Have an amazing day!(3 votes)

- I have a question. How do you find the least common denominator by using the prime factorization method?(1 vote)
- Maybe an example would answer your question. There are three denominators, 6, 8, and 9 and you want the least common denominator. 6=3*2, 8=2*2*2, and 9=3*3. Start with any of the numbers, 3*2. If the next has any in common, eliminate it, so one 2 is eliminated giving 3*2*2*2. The next has one 3 in common, so eliminate one 3, final LCM is 3*2*2*2*3. Try 10, 18, and 24. 10=5*2, 18=3*3*2, 24=3*2*2*2. So you start with 5*2, one two in common with 18, so eliminate it to get 5*2*3*3, last (24) has both a 3 and 2 in common now that you can eliminate, so 5*2*3*3*2*2=360. Hope this makes sense.(2 votes)

- Can someone explane this to me pls?(20 votes)
- So you have to find a common denominator, right? For this one, you have to multiply to get the common denominator. So what times 6 equals 24? Also, what times 8 equals 24? Then that is how you get your common denominator.(7 votes)

- is LCM n LCD are same?(17 votes)
- The LCM is used to find the LCD but yes they are the same(17 votes)

- What's a common denominator?(14 votes)
- A common denominator is a denominator that you can reach by both denominators. For example in the problem 3/4+ 5/6 a common denominator is 12 because it is the lowest number that both 4 and 6 can reach by multiplying with whole numbers. Thanks for reading hope it helps!(14 votes)

- How to know if you are finding the common denominators with some choices like; 3/4+1/3

A.3/4+1/3 = 9/12+4/12

B.3/4+1/3=9/12+3/12

C.3/4+1/3=12/12+4/12

D.3/4+1/3=3/12+1/12(9 votes)- The key is to remember that if you multiply the denominator by a value, you also have to multiply the numerator by that same value. In this example, it's clear the common denominator is 12. For the first fraction, to get 12 in the denominator, you have to multiply top and bottom by 3, which gives you 9/12, so this limits your answer choices to A and B. For the second fraction, you have to multiply by 4 on the top and bottom to get 12 in the denominator, so we end up with 4/12. This means our answer is A!(13 votes)

- find the the lcd of 1/5, 1/4 and 1/9?(17 votes)
- Then you can convert it to a mix fraction.(0 votes)

## Video transcript

We're asked to rewrite
the following two fractions as fractions with
a least common denominator. So a least common
denominator for two fractions is really just going to be the
least common multiple of both of these denominators over here. And the value of
doing that is then if you can make these
a common denominator, then you can add
the two fractions. And we'll see that
in other videos. But first of all, let's just
find the least common multiple. Let me write it out
because sometimes LCD could meet other things. So least common denominator
of these two things is going to be the same thing
as the least common multiple of the two
denominators over here. The least common
multiple of 8 and 6. And a couple of ways to think
about least common multiple-- you literally could just
take the multiples of 8 and 6 and see what they're
smallest common multiple is. So let's do it that way first. So multiples of six
are 6, 12, 18, 24 30. And I could keep going if we
don't find any common multiples out of this group here with
any of the multiples in eight. And the multiples of
eight are 8, 16, 24, and it looks like we're done. And we could keep
going obviously-- 32, so on and so forth. But I found a common
multiple and this is their smallest
common multiple. They have other common
multiples-- 48 and 72, and we could keep adding
more and more multiple. But this is their
smallest common multiple, their least common multiple. So it is 24. Another way that you could have
found at least common multiple is you could have taken the
prime factorization of six and you say, hey,
that's 2, and 3. So the least common multiple has
to have at least 1, 2, and 1, 3 in its prime factorization
in order for it to be divisible by 6. And you could have said, what's
the prime factorization of 8? It is 2 times 4
and 4 is 2 times 2. So in order to be
divisible by 8, you have to have at least three
2's in the prime factorization. So to be divisible by 6, you
have to have a 2 times a 3. And then to be divisible by 8,
you have to have at least three 2's. You have to have two
times itself three times I should say. Well, we have one 2 and
let's throw in a couple more. So then you have another
2 and then another 2. So this part right over here
makes it divisible by 8. And this part right over
here makes it divisible by 6. If I take 2 times 2 times 2
times 3, that does give me 24. So our least common
multiple of 8 and 6, which is also the least common
denominator of these two fractions is going to be 24. So what we want to do is
rewrite each of these fractions with 24 as the denominator. So I'll start with 2 over 8. And I want to write that
as something over 24. Well, to get the
denominator be 24, we have to multiply it by 3. 8 times 3 is 24. And so if we don't
want to change the value of the
fraction, we have to multiply the numerator and
denominator by the same thing. So let's multiply the
numerator by 3 as well. 2 times 3 is 6. So 2/8 is the exact
same thing as 6/24. To see that a
little bit clearer, you say, look, if I have 2/8,
and if I multiply this times 3 over 3, that gives me 6/24. And this are the same
fraction because 3 over 3 is really just 1. It's one whole. So 2/8 is 6/24 let's do
the same thing with 5/6. So 5 over 6 is equal
to something over 24. Let me do that in
a different color. I'll do it in blue. Something over 24. To get the denominator
from 6 to 24, we have to multiply it by 4. So if we don't want to
change the value of 5/6, we have to multiply the
numerator and denominator by the same thing. So let's multiply the
numerator times 4. 5 times 4 is 20. 5/6 is the same thing as 20/24. So we're done. We've written 2/8 as 6/24 and
we've written 5/6 as 20/24. If we wanted to add them
now, we could literally just add 6/24 to 20/24. And I'll leave you
there because they didn't ask us to
actually do that.