0:00:00.255,0:00:04.714
What I want to do in this video is order these fractions from least to greatest

0:00:04.714,0:00:10.379
And, the easiest way--and the way that people are sure to get the right answer--

0:00:10.379,0:00:14.002
is to find a common denominator, because if we can't find a common denominator,

0:00:14.002,0:00:21.432
these fractions are difficult to compare: 4/9 v. 3/4 v. 4/5, 11/12, 13/15.

0:00:21.432,0:00:25.844
You can try to estimate them, but you'll be able to directly compare them if

0:00:25.844,0:00:32.467
they all have the same denominator. So, the trick here is to first find the common denominator.

0:00:32.467,0:00:36.432
And there is many ways to do it, you could just pick one of these numbers,

0:00:36.432,0:00:42.051
and take all of its multiples until you find a multiple that is divisible by all of the rest.

0:00:42.051,0:00:45.667
Another way to do it is to look at the prime factorization of each of these numbers.

0:00:45.667,0:00:52.067
and then the 'least common multiple' of them would have each of those prime numbers in it.

0:00:52.067,0:00:58.630
Let's do it that second way, and then verify it.

0:00:58.630,0:01:08.429
So, 9 is 3<i>3, so our LCM is going to have at least one 3</i>3 in it.

0:01:08.429,0:01:12.191
And then 4 is the same thing as 2*2.

0:01:12.191,0:01:17.810
So, we will also have 2*2 in our prime factorization (LCM).

0:01:17.810,0:01:22.361
5 is a prime number, so we'll put 5 right there.

0:01:22.361,0:01:31.185
And then, 12 is the same thing as 2<i>6, and 6 = 2</i>3.

0:01:31.185,0:01:40.867
So, in our LCM, we have to have two 2's, but we already have two 2's, and we already have one 3.

0:01:40.867,0:01:48.182
Another way to think about it, is that something that is divisible by both 9 and 4

0:01:48.182,0:01:50.200
is going to be divisible by 12.

0:01:50.200,0:01:58.770
And then finally, we need it to be divisible by 15's prime factors.

0:01:58.770,0:02:03.971
15 is the same thing as 3*5.

0:02:03.971,0:02:09.312
So once again, we already have 3 and 5.

0:02:09.312,0:02:15.163
So, this is our least common multiple (LCM).

0:02:15.163,0:02:45.256
So, LCM is going to be equal to 3<i>3</i>2<i>2</i>5 =180

0:02:45.256,0:02:52.873
So, our LCM is 180. So, we want to rewrite all of these fractions with 180 in the denominator.

0:02:52.873,0:02:59.467
So, our first fraction, 4/9, is what over 180?

0:02:59.467,0:03:04.065
To go from 9 to 180, we have to multiple 9 by 20.

0:03:04.065,0:03:16.836
So, to get the denominator to equal 180, we multiple by 20.

0:03:16.836,0:03:21.851
Since we don't want to change the value of the fraction, we should also multiple by the 4 by 20.

0:03:21.851,0:03:28.863
4*20 = 80. So, 4/9 is the same thing as 80/180.

0:03:28.863,0:03:37.200
Now, let's do 3/4. What do we have to multiple the denominator by to equal 180?

0:03:37.200,0:03:42.656
You can divide 4 into 180 (180/4 = x) to figure that out.

0:03:42.656,0:03:54.452
4*45 = 180. Now, you also have to multiple the numerator by 45.

0:03:54.452,0:04:09.200
3*45 = 135. So, 3/4 equals 135/180.

0:04:09.200,0:04:31.929
Now let's do 4/5. To get 180 from 5, multiple 5 by 36.

0:04:31.929,0:04:35.133
Have to multiple numerator by same number, 36.

0:04:35.133,0:04:46.325
So, 144/180.

0:04:46.325,0:04:50.180
And then we have only two more to do.

0:04:50.180,0:05:25.846
180/12 = 15. Same for numerator, 15. So, 11/12 = 165/180.

0:05:25.846,0:05:28.067
And then finally, we have 13/15.

0:05:28.067,0:05:51.434
To get 180 from 15, multiply 15 by 12--15<i>10 = 150, 30 remaining for 180. 15</i>2 = 30. So, 15*12 = 180.

0:05:51.434,0:05:54.128
Multiple numerator by same number, 13.

0:05:54.128,0:06:01.233
We know 12*12 = 144, so just add one more 12 = 156.

0:06:01.233,0:06:08.431
So, we've rewritten each of these fractions with the new common denominator.

0:06:08.431,0:06:13.029
Now, it is very easy to compare them. We only have to look at their numerators.

0:06:13.029,0:06:21.434
Foe example, the smallest numerator is 80, so 4/9 is the least of these numbers.

0:06:21.434,0:07:04.438
The next smallest number looks like 135, which was 3/4.

0:07:04.438,0:07:08.524
And then the next one is going to be the 144/180, which was 4/5.

0:07:08.524,0:07:20.831
Next is 156/180, which was 13/15.

0:07:20.831,0:07:35.970
Finally, we have 165/180, which was 11/12.

0:07:35.970,9:59:59.000
And, we're done! We have finished our ordering.