Ordering Fractions

Finding common denominators of multiple fraction to order them

What I want to do in this video is order these fractions from least to greatest

And, the easiest way--and the way that people are sure to get the right answer--

is to find a common denominator, because if we can't find a common denominator,

these fractions are difficult to compare: 4/9 v. 3/4 v. 4/5, 11/12, 13/15.

You can try to estimate them, but you'll be able to directly compare them if

they all have the same denominator. So, the trick here is to first find the common denominator.

And there is many ways to do it, you could just pick one of these numbers,

and take all of its multiples until you find a multiple that is divisible by all of the rest.

Another way to do it is to look at the prime factorization of each of these numbers.

and then the 'least common multiple' of them would have each of those prime numbers in it.

Let's do it that second way, and then verify it.

So, 9 is 33, so our LCM is going to have at least one 33 in it.

And then 4 is the same thing as 2*2.

So, we will also have 2*2 in our prime factorization (LCM).

5 is a prime number, so we'll put 5 right there.

And then, 12 is the same thing as 26, and 6 = 23.

So, in our LCM, we have to have two 2's, but we already have two 2's, and we already have one 3.

Another way to think about it, is that something that is divisible by both 9 and 4

is going to be divisible by 12.

And then finally, we need it to be divisible by 15's prime factors.

15 is the same thing as 3*5.

So once again, we already have 3 and 5.

So, this is our least common multiple (LCM).

So, LCM is going to be equal to 33225 =180

So, our LCM is 180. So, we want to rewrite all of these fractions with 180 in the denominator.

So, our first fraction, 4/9, is what over 180?

To go from 9 to 180, we have to multiple 9 by 20.

So, to get the denominator to equal 180, we multiple by 20.

Since we don't want to change the value of the fraction, we should also multiple by the 4 by 20.

4*20 = 80. So, 4/9 is the same thing as 80/180.

Now, let's do 3/4. What do we have to multiple the denominator by to equal 180?

You can divide 4 into 180 (180/4 = x) to figure that out.

4*45 = 180. Now, you also have to multiple the numerator by 45.

3*45 = 135. So, 3/4 equals 135/180.

Now let's do 4/5. To get 180 from 5, multiple 5 by 36.

Have to multiple numerator by same number, 36.

So, 144/180.

And then we have only two more to do.

180/12 = 15. Same for numerator, 15. So, 11/12 = 165/180.

And then finally, we have 13/15.

To get 180 from 15, multiply 15 by 12--1510 = 150, 30 remaining for 180. 152 = 30. So, 15*12 = 180.

Multiple numerator by same number, 13.

We know 12*12 = 144, so just add one more 12 = 156.

So, we've rewritten each of these fractions with the new common denominator.

Now, it is very easy to compare them. We only have to look at their numerators.

Foe example, the smallest numerator is 80, so 4/9 is the least of these numbers.

The next smallest number looks like 135, which was 3/4.

And then the next one is going to be the 144/180, which was 4/5.

Next is 156/180, which was 13/15.

Finally, we have 165/180, which was 11/12.

And, we're done! We have finished our ordering.