Different denominators
When the denominators are unequal, we have to find a common denominator before we can add or subtract the fractions.
To find a common denominator, we need to find the least common multiple (LCM) of the denominators. Then we can use the LCM as our common denominator. In fact, we call that the “least common denominator,” because it’s the smallest common denominator we can use.
Consider the fractions ???3/5??? and ???1/3???. The LCM of ???5??? and ???3??? is ???15???, so the least common denominator will be ???15???.
Since ???15=3\cdot5???, we have to multiply ???5??? (the denominator of ???3/5???) by ???3??? to get the common denominator of ???15???. And we have to multiply ???3??? (the denominator of ???1/3???) by ???5??? to get the common denominator ???15???.
We aren't allowed to change the value of either of the fractions, so we have to multiply the numerator of each fraction by the same number that we multiplied by its denominator. So we have to multiply the numerator and denominator of ???3/5??? by ???3???, and we have to multiply the numerator and denominator of ???1/3??? by ???5???.
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This is the easiest method we know to add or subtract fractions!
What is a Denominator?
The denominator is the bottom number in a fraction.
It shows how many equal parts the item is divided into.
What is a Common Denominator?
When the denominators of two or more fractions are the same, they are Common Denominators.
Why is it Important?
Before we can add or subtract fractions, the fractions need to have a common denominator
In other words the denominators must be the same.
Making The Denominators the Same
To make the denominators the same we can:
Multiply top and bottom of each fraction by the denominator of the other.
Like in this example (press play button):
This always works, but we often need to simplify the fraction afterwards, as in this example (press play button):
We simplified the fraction 2032 to 1016 , then to 58 by dividing the top and bottom by 2 each time, and that is as simple as it can get!
What Did We Do?
1. We multiplied each fraction by the denominator of the other. Let's use letters instead of numbers:
2. And because they now have the same denominator, we can add them:
In One Step!
We can do those two things in one step like this:
Which we use like this:
Example: What is 23 + 45 ?
23 + 45 = 2×5 + 3×43×5 = 10 + 1215 = 2215
(Note: a was 2, b was 3, c was 4 and d was 5.)
That is how the experts do it!
1698, 1699, 1700, 1701
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.