WEBVTT

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--

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Use less than, greater than, or equal

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to compare the two fractions 21/28, or 21 over 28, and 6/9,

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or 6 over 9.

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So there's a bunch of ways to do this.

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The easiest way is if they had the same denominator,

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you could just compare the numerators.

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Unlucky for us, we do not have the same denominator.

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So what we could do is we can find a common denominator

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for both of them and convert both of these fractions

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to have the same denominator and then compare the numerators.

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Or even more simply, we could simplify them first and then

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try to do it.

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So let me do that last one, because I

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have a feeling that'll be the fastest way to do it.

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So 21/28-- you can see that they are both divisible by 7.

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So let's divide both the numerator

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and the denominator by 7.

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So we could divide 21 by 7.

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And we can divide-- so let me make the numerator--

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and we can divide the denominator by 7.

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We're doing the same thing to the numerator

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and the denominator, so we're not

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going to change the value of the fraction.

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So 21 divided by 7 is 3, and 28 divided by 7 is 4.

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So 21/28 is the exact same fraction as 3/4.

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3/4 is the simplified version of it.

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Let's do the same thing for 6/9.

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6 and 9 are both divisible by 3.

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So let's divide them both by 3 so we

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can simplify this fraction.

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So let's divide both of them by 3.

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6 divided by 3 is 2, and 9 divided by 3 is 3.

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So 21/28 is 3/4.

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They're the exact same fraction, just written a different way.

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This is the more simplified version.

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And 6/9 is the exact same fraction as 2/3.

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So we really can compare 3/4 and 2/3.

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So this is really comparing 3/4 and 2/3.

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And the real benefit of doing this is now this

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is much easier to find a common denominator for than 28 and 9.

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Then we would have to multiply big numbers.

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Here we could do fairly small numbers.

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The common denominator of 3/4 and 2/3

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is going to be the least common multiple of 4 and 3.

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And 4 and 3 don't share any prime factors with each other.

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So their least common multiple is really

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just going to be the product of the two.

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So we can write 3/4 as something over 12.

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And we can write 2/3 as something over 12.

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And I got the 12 by multiplying 3 times 4.

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They have no common factors.

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Another way you could think about it is 4,

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if you do a prime factorization, is 2 times 2.

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And 3-- it's already a prime number,

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so you can't prime factorize it any more.

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So what you want to do is think of a number

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that has all of the prime factors of 4 and 3.

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So it needs one 2, another 2, and a 3.

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Well, 2 times 2 times 3 is 12.

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And either way you think about it,

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that's how you would get the least common multiple

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or the common denominator for 4 and 3.

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Well, to get from 4 to 12, you've got to multiply by 3.

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So we're multiplying the denominator by 3 to get to 12.

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So we also have to multiply the numerator by 3.

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So 3 times 3 is 9.

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Over here, to get from 3 to 12, we

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have to multiply the denominator by 4.

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So we also have to multiply the numerator by 4.

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So we get 8.

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And so now when we compare the fractions,

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it's pretty straightforward.

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21/28 is the exact same thing as 9/12,

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and 6/9 is the exact same thing as 8/12.

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So which of these is a greater quantity?

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Well, clearly, we have the same denominator right now.

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We have 9/12 is clearly greater than 8/12.

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So 9/12 is clearly greater than 8/12.

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Or if you go back and you realize

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that 9/12 is the exact same thing as 21/28,

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we could say 21/28 is definitely greater than-- and 8/12

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is the same thing as 6/9-- is definitely greater than 6/9.

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And we are done.

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Another way we could have done it-- we

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didn't necessarily have to simplify that.

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And let me show you that just for fun.

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So if we were doing it with-- if we didn't think

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to simplify our two numbers first.

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I'm trying to find a color I haven't used yet.

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So 21/28 and 6/9.

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So we could just find a least common multiple

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in the traditional way without simplifying first.

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So what's the prime factorization of 28?

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It's 2 times 14.

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And 14 is 2 times 7.

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That's its prime factorization.

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Prime factorization of 9 is 3 times 3.

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So the least common multiple of 28 and 9

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have to contain a 2, a 2, a 7, a 3 and a 3.

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Or essentially, it's going to be 28 times 9.

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So let's over here multiply 28 times 9.

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There's a couple of ways you could do it.

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You could multiply in your head 28 times

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10, which would be 280, and then subtract 28

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from that, which would be what?

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252.

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Or we could just multiply it out if that confuses you.

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So let's just do the second way.

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9 times 8 is 72.

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9 times 2 is 18.

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18 plus 7 is 25.

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So we get 252.

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So I'm saying the common denominator here

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is going to be 252.

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Least common multiple of 28 and 9.

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Well, to go from 28 to 252, we had to multiply it by 9.

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We had to multiply 28 times 9.

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So we're multiplying 28 times 9.

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So we also have to multiply the numerator times 9.

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So what is 21 times 9?

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That's easier to do in your head.

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20 times 9 is 180.

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And then 1 times 9 is 9.

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So this is going to be 189.

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To go from 9 to 252, we had to multiply by 28.

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So we also have to multiply the numerator by 28

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if we don't want to change the value of the fraction.

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So 6 times 28-- 6 times 20 is 120.

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6 times 8 is 48.

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So we get 168.

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Let me write that out just to make

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sure I didn't make a mistake.

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So 28 times 6-- 8 times 6 is 48.

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2 times 6 is 12, plus 4 is 16.

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So right, 168.

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So now we have a common denominator here.

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And so we can really just compare the numerators.

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And 189 is clearly greater than 168.

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So 189/252 is clearly greater than 168/252.

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Or that's the same thing as saying 21/28,

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because that's what this is over here.

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The left-hand side is 21/28, is clearly greater

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than the right-hand side, which is really 6/9.