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

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

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

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these fractions are difficult to compare: 4/9 v. 3/4 v. 4/5, etc.

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

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

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

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

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

10
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and then the 'least common denominator' would have each of those prime numbers in it.

11
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Let's do it that second way, and then verify it.

12
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So, 9 is 3<i>3, so our LCM is going to have at least 3</i>3 in it.

13
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And then 4 is the same thing as 2*2.

14
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So, we will also have 2*2 in our prime factorization (LCM).

15
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5 is a prime number, so we'll put 5 right there.

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And then, 12 is the same thing as 2<i>6, and 6 = 2</i>3.

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

18
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Another way to think about it, is that something that is divisible by both 9 and 4

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is going to be divisible by 12.

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

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15 is the same thing as 3*5.

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So once again, we already have 3 and 5.

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

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So, LCM is going to be equal to 3<i>3</i>2<i>2</i>5 =180

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

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

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

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

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

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

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

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

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

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

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

36
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Have to multiple numerator by same number, 36.

37
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So, 144/180.

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And then we have only two more to do.

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

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And then finally, we have 13/15.

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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.

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Multiple numerator by same number, 13.

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

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

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

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

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

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

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

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

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