[Script Info]
Title:
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.74,Default,,0000,0000,0000,,--
Dialogue: 0,0:00:00.74,0:00:03.42,Default,,0000,0000,0000,,Use less than, greater than, or equal
Dialogue: 0,0:00:03.42,0:00:09.17,Default,,0000,0000,0000,,to compare the two fractions 21/28, or 21 over 28, and 6/9,
Dialogue: 0,0:00:09.17,0:00:11.06,Default,,0000,0000,0000,,or 6 over 9.
Dialogue: 0,0:00:11.06,0:00:13.11,Default,,0000,0000,0000,,So there's a bunch of ways to do this.
Dialogue: 0,0:00:13.11,0:00:16.69,Default,,0000,0000,0000,,The easiest way is if they had the same denominator,
Dialogue: 0,0:00:16.69,0:00:18.45,Default,,0000,0000,0000,,you could just compare the numerators.
Dialogue: 0,0:00:18.45,0:00:22.04,Default,,0000,0000,0000,,Unlucky for us, we do not have the same denominator.
Dialogue: 0,0:00:22.04,0:00:24.60,Default,,0000,0000,0000,,So what we could do is we can find a common denominator
Dialogue: 0,0:00:24.60,0:00:26.84,Default,,0000,0000,0000,,for both of them and convert both of these fractions
Dialogue: 0,0:00:26.84,0:00:30.25,Default,,0000,0000,0000,,to have the same denominator and then compare the numerators.
Dialogue: 0,0:00:30.25,0:00:34.34,Default,,0000,0000,0000,,Or even more simply, we could simplify them first and then
Dialogue: 0,0:00:34.34,0:00:34.97,Default,,0000,0000,0000,,try to do it.
Dialogue: 0,0:00:34.97,0:00:36.51,Default,,0000,0000,0000,,So let me do that last one, because I
Dialogue: 0,0:00:36.51,0:00:39.19,Default,,0000,0000,0000,,have a feeling that'll be the fastest way to do it.
Dialogue: 0,0:00:39.19,0:00:45.05,Default,,0000,0000,0000,,So 21/28-- you can see that they are both divisible by 7.
Dialogue: 0,0:00:45.05,0:00:46.65,Default,,0000,0000,0000,,So let's divide both the numerator
Dialogue: 0,0:00:46.65,0:00:48.57,Default,,0000,0000,0000,,and the denominator by 7.
Dialogue: 0,0:00:48.57,0:00:51.19,Default,,0000,0000,0000,,So we could divide 21 by 7.
Dialogue: 0,0:00:51.19,0:00:53.60,Default,,0000,0000,0000,,And we can divide-- so let me make the numerator--
Dialogue: 0,0:00:53.60,0:00:55.49,Default,,0000,0000,0000,,and we can divide the denominator by 7.
Dialogue: 0,0:00:55.49,0:00:57.28,Default,,0000,0000,0000,,We're doing the same thing to the numerator
Dialogue: 0,0:00:57.28,0:00:58.66,Default,,0000,0000,0000,,and the denominator, so we're not
Dialogue: 0,0:00:58.66,0:01:00.95,Default,,0000,0000,0000,,going to change the value of the fraction.
Dialogue: 0,0:01:00.95,0:01:05.65,Default,,0000,0000,0000,,So 21 divided by 7 is 3, and 28 divided by 7 is 4.
Dialogue: 0,0:01:05.65,0:01:09.51,Default,,0000,0000,0000,,So 21/28 is the exact same fraction as 3/4.
Dialogue: 0,0:01:09.51,0:01:11.77,Default,,0000,0000,0000,,3/4 is the simplified version of it.
Dialogue: 0,0:01:11.77,0:01:14.68,Default,,0000,0000,0000,,Let's do the same thing for 6/9.
Dialogue: 0,0:01:14.68,0:01:17.52,Default,,0000,0000,0000,,6 and 9 are both divisible by 3.
Dialogue: 0,0:01:17.52,0:01:19.41,Default,,0000,0000,0000,,So let's divide them both by 3 so we
Dialogue: 0,0:01:19.41,0:01:21.43,Default,,0000,0000,0000,,can simplify this fraction.
Dialogue: 0,0:01:21.43,0:01:24.08,Default,,0000,0000,0000,,So let's divide both of them by 3.
Dialogue: 0,0:01:24.08,0:01:30.38,Default,,0000,0000,0000,,6 divided by 3 is 2, and 9 divided by 3 is 3.
Dialogue: 0,0:01:30.38,0:01:33.06,Default,,0000,0000,0000,,So 21/28 is 3/4.
Dialogue: 0,0:01:33.06,0:01:35.76,Default,,0000,0000,0000,,They're the exact same fraction, just written a different way.
Dialogue: 0,0:01:35.76,0:01:37.70,Default,,0000,0000,0000,,This is the more simplified version.
Dialogue: 0,0:01:37.70,0:01:41.58,Default,,0000,0000,0000,,And 6/9 is the exact same fraction as 2/3.
Dialogue: 0,0:01:41.58,0:01:44.59,Default,,0000,0000,0000,,So we really can compare 3/4 and 2/3.
Dialogue: 0,0:01:44.59,0:01:49.23,Default,,0000,0000,0000,,So this is really comparing 3/4 and 2/3.
Dialogue: 0,0:01:49.23,0:01:51.99,Default,,0000,0000,0000,,And the real benefit of doing this is now this
Dialogue: 0,0:01:51.99,0:01:55.87,Default,,0000,0000,0000,,is much easier to find a common denominator for than 28 and 9.
Dialogue: 0,0:01:55.87,0:01:58.68,Default,,0000,0000,0000,,Then we would have to multiply big numbers.
Dialogue: 0,0:01:58.68,0:02:00.85,Default,,0000,0000,0000,,Here we could do fairly small numbers.
Dialogue: 0,0:02:00.85,0:02:03.78,Default,,0000,0000,0000,,The common denominator of 3/4 and 2/3
Dialogue: 0,0:02:03.78,0:02:08.81,Default,,0000,0000,0000,,is going to be the least common multiple of 4 and 3.
Dialogue: 0,0:02:08.81,0:02:13.57,Default,,0000,0000,0000,,And 4 and 3 don't share any prime factors with each other.
Dialogue: 0,0:02:13.57,0:02:15.52,Default,,0000,0000,0000,,So their least common multiple is really
Dialogue: 0,0:02:15.52,0:02:17.43,Default,,0000,0000,0000,,just going to be the product of the two.
Dialogue: 0,0:02:17.43,0:02:20.89,Default,,0000,0000,0000,,So we can write 3/4 as something over 12.
Dialogue: 0,0:02:20.89,0:02:24.80,Default,,0000,0000,0000,,And we can write 2/3 as something over 12.
Dialogue: 0,0:02:24.80,0:02:27.57,Default,,0000,0000,0000,,And I got the 12 by multiplying 3 times 4.
Dialogue: 0,0:02:27.57,0:02:29.15,Default,,0000,0000,0000,,They have no common factors.
Dialogue: 0,0:02:29.15,0:02:31.50,Default,,0000,0000,0000,,Another way you could think about it is 4,
Dialogue: 0,0:02:31.50,0:02:34.47,Default,,0000,0000,0000,,if you do a prime factorization, is 2 times 2.
Dialogue: 0,0:02:34.47,0:02:36.87,Default,,0000,0000,0000,,And 3-- it's already a prime number,
Dialogue: 0,0:02:36.87,0:02:39.24,Default,,0000,0000,0000,,so you can't prime factorize it any more.
Dialogue: 0,0:02:39.24,0:02:41.03,Default,,0000,0000,0000,,So what you want to do is think of a number
Dialogue: 0,0:02:41.03,0:02:44.12,Default,,0000,0000,0000,,that has all of the prime factors of 4 and 3.
Dialogue: 0,0:02:44.12,0:02:46.97,Default,,0000,0000,0000,,So it needs one 2, another 2, and a 3.
Dialogue: 0,0:02:46.97,0:02:49.74,Default,,0000,0000,0000,,Well, 2 times 2 times 3 is 12.
Dialogue: 0,0:02:49.74,0:02:51.16,Default,,0000,0000,0000,,And either way you think about it,
Dialogue: 0,0:02:51.16,0:02:53.24,Default,,0000,0000,0000,,that's how you would get the least common multiple
Dialogue: 0,0:02:53.24,0:02:57.45,Default,,0000,0000,0000,,or the common denominator for 4 and 3.
Dialogue: 0,0:02:57.45,0:03:04.36,Default,,0000,0000,0000,,Well, to get from 4 to 12, you've got to multiply by 3.
Dialogue: 0,0:03:04.36,0:03:07.46,Default,,0000,0000,0000,,So we're multiplying the denominator by 3 to get to 12.
Dialogue: 0,0:03:07.46,0:03:10.71,Default,,0000,0000,0000,,So we also have to multiply the numerator by 3.
Dialogue: 0,0:03:10.71,0:03:13.74,Default,,0000,0000,0000,,So 3 times 3 is 9.
Dialogue: 0,0:03:13.74,0:03:15.64,Default,,0000,0000,0000,,Over here, to get from 3 to 12, we
Dialogue: 0,0:03:15.64,0:03:18.32,Default,,0000,0000,0000,,have to multiply the denominator by 4.
Dialogue: 0,0:03:18.32,0:03:21.08,Default,,0000,0000,0000,,So we also have to multiply the numerator by 4.
Dialogue: 0,0:03:21.08,0:03:24.54,Default,,0000,0000,0000,,So we get 8.
Dialogue: 0,0:03:24.54,0:03:26.25,Default,,0000,0000,0000,,And so now when we compare the fractions,
Dialogue: 0,0:03:26.25,0:03:27.77,Default,,0000,0000,0000,,it's pretty straightforward.
Dialogue: 0,0:03:27.77,0:03:33.27,Default,,0000,0000,0000,,21/28 is the exact same thing as 9/12,
Dialogue: 0,0:03:33.27,0:03:40.14,Default,,0000,0000,0000,,and 6/9 is the exact same thing as 8/12.
Dialogue: 0,0:03:40.14,0:03:43.96,Default,,0000,0000,0000,,So which of these is a greater quantity?
Dialogue: 0,0:03:43.96,0:03:46.21,Default,,0000,0000,0000,,Well, clearly, we have the same denominator right now.
Dialogue: 0,0:03:46.21,0:03:50.12,Default,,0000,0000,0000,,We have 9/12 is clearly greater than 8/12.
Dialogue: 0,0:03:50.12,0:03:54.18,Default,,0000,0000,0000,,So 9/12 is clearly greater than 8/12.
Dialogue: 0,0:03:54.18,0:03:56.45,Default,,0000,0000,0000,,Or if you go back and you realize
Dialogue: 0,0:03:56.45,0:03:59.65,Default,,0000,0000,0000,,that 9/12 is the exact same thing as 21/28,
Dialogue: 0,0:03:59.65,0:04:04.75,Default,,0000,0000,0000,,we could say 21/28 is definitely greater than-- and 8/12
Dialogue: 0,0:04:04.75,0:04:10.13,Default,,0000,0000,0000,,is the same thing as 6/9-- is definitely greater than 6/9.
Dialogue: 0,0:04:10.13,0:04:11.87,Default,,0000,0000,0000,,And we are done.
Dialogue: 0,0:04:11.87,0:04:13.45,Default,,0000,0000,0000,,Another way we could have done it-- we
Dialogue: 0,0:04:13.45,0:04:15.37,Default,,0000,0000,0000,,didn't necessarily have to simplify that.
Dialogue: 0,0:04:15.37,0:04:17.31,Default,,0000,0000,0000,,And let me show you that just for fun.
Dialogue: 0,0:04:17.31,0:04:21.45,Default,,0000,0000,0000,,So if we were doing it with-- if we didn't think
Dialogue: 0,0:04:21.45,0:04:24.92,Default,,0000,0000,0000,,to simplify our two numbers first.
Dialogue: 0,0:04:24.92,0:04:27.27,Default,,0000,0000,0000,,I'm trying to find a color I haven't used yet.
Dialogue: 0,0:04:27.27,0:04:32.27,Default,,0000,0000,0000,,So 21/28 and 6/9.
Dialogue: 0,0:04:32.27,0:04:35.46,Default,,0000,0000,0000,,So we could just find a least common multiple
Dialogue: 0,0:04:35.46,0:04:37.98,Default,,0000,0000,0000,,in the traditional way without simplifying first.
Dialogue: 0,0:04:37.98,0:04:40.34,Default,,0000,0000,0000,,So what's the prime factorization of 28?
Dialogue: 0,0:04:40.34,0:04:42.72,Default,,0000,0000,0000,,It's 2 times 14.
Dialogue: 0,0:04:42.72,0:04:45.71,Default,,0000,0000,0000,,And 14 is 2 times 7.
Dialogue: 0,0:04:45.71,0:04:47.23,Default,,0000,0000,0000,,That's its prime factorization.
Dialogue: 0,0:04:47.23,0:04:50.49,Default,,0000,0000,0000,,Prime factorization of 9 is 3 times 3.
Dialogue: 0,0:04:50.49,0:04:53.07,Default,,0000,0000,0000,,So the least common multiple of 28 and 9
Dialogue: 0,0:04:53.07,0:04:56.57,Default,,0000,0000,0000,,have to contain a 2, a 2, a 7, a 3 and a 3.
Dialogue: 0,0:04:56.57,0:04:59.83,Default,,0000,0000,0000,,Or essentially, it's going to be 28 times 9.
Dialogue: 0,0:04:59.83,0:05:04.23,Default,,0000,0000,0000,,So let's over here multiply 28 times 9.
Dialogue: 0,0:05:04.23,0:05:05.94,Default,,0000,0000,0000,,There's a couple of ways you could do it.
Dialogue: 0,0:05:05.94,0:05:07.87,Default,,0000,0000,0000,,You could multiply in your head 28 times
Dialogue: 0,0:05:07.87,0:05:11.08,Default,,0000,0000,0000,,10, which would be 280, and then subtract 28
Dialogue: 0,0:05:11.08,0:05:12.37,Default,,0000,0000,0000,,from that, which would be what?
Dialogue: 0,0:05:12.37,0:05:14.14,Default,,0000,0000,0000,,252.
Dialogue: 0,0:05:14.14,0:05:16.85,Default,,0000,0000,0000,,Or we could just multiply it out if that confuses you.
Dialogue: 0,0:05:16.85,0:05:18.58,Default,,0000,0000,0000,,So let's just do the second way.
Dialogue: 0,0:05:18.58,0:05:22.11,Default,,0000,0000,0000,,9 times 8 is 72.
Dialogue: 0,0:05:22.11,0:05:24.38,Default,,0000,0000,0000,,9 times 2 is 18.
Dialogue: 0,0:05:24.38,0:05:26.37,Default,,0000,0000,0000,,18 plus 7 is 25.
Dialogue: 0,0:05:26.37,0:05:28.88,Default,,0000,0000,0000,,So we get 252.
Dialogue: 0,0:05:28.88,0:05:31.66,Default,,0000,0000,0000,,So I'm saying the common denominator here
Dialogue: 0,0:05:31.66,0:05:37.65,Default,,0000,0000,0000,,is going to be 252.
Dialogue: 0,0:05:37.65,0:05:40.27,Default,,0000,0000,0000,,Least common multiple of 28 and 9.
Dialogue: 0,0:05:40.27,0:05:46.37,Default,,0000,0000,0000,,Well, to go from 28 to 252, we had to multiply it by 9.
Dialogue: 0,0:05:46.37,0:05:48.59,Default,,0000,0000,0000,,We had to multiply 28 times 9.
Dialogue: 0,0:05:48.59,0:05:50.57,Default,,0000,0000,0000,,So we're multiplying 28 times 9.
Dialogue: 0,0:05:50.57,0:05:53.24,Default,,0000,0000,0000,,So we also have to multiply the numerator times 9.
Dialogue: 0,0:05:53.24,0:05:55.34,Default,,0000,0000,0000,,So what is 21 times 9?
Dialogue: 0,0:05:55.34,0:05:56.71,Default,,0000,0000,0000,,That's easier to do in your head.
Dialogue: 0,0:05:56.71,0:05:59.08,Default,,0000,0000,0000,,20 times 9 is 180.
Dialogue: 0,0:05:59.08,0:06:00.71,Default,,0000,0000,0000,,And then 1 times 9 is 9.
Dialogue: 0,0:06:00.71,0:06:04.08,Default,,0000,0000,0000,,So this is going to be 189.
Dialogue: 0,0:06:04.08,0:06:09.10,Default,,0000,0000,0000,,To go from 9 to 252, we had to multiply by 28.
Dialogue: 0,0:06:09.10,0:06:11.63,Default,,0000,0000,0000,,So we also have to multiply the numerator by 28
Dialogue: 0,0:06:11.63,0:06:14.29,Default,,0000,0000,0000,,if we don't want to change the value of the fraction.
Dialogue: 0,0:06:14.29,0:06:18.51,Default,,0000,0000,0000,,So 6 times 28-- 6 times 20 is 120.
Dialogue: 0,0:06:18.51,0:06:22.42,Default,,0000,0000,0000,,6 times 8 is 48.
Dialogue: 0,0:06:22.42,0:06:24.41,Default,,0000,0000,0000,,So we get 168.
Dialogue: 0,0:06:24.41,0:06:25.83,Default,,0000,0000,0000,,Let me write that out just to make
Dialogue: 0,0:06:25.83,0:06:27.29,Default,,0000,0000,0000,,sure I didn't make a mistake.
Dialogue: 0,0:06:27.29,0:06:36.18,Default,,0000,0000,0000,,So 28 times 6-- 8 times 6 is 48.
Dialogue: 0,0:06:36.18,0:06:40.04,Default,,0000,0000,0000,,2 times 6 is 12, plus 4 is 16.
Dialogue: 0,0:06:40.04,0:06:41.77,Default,,0000,0000,0000,,So right, 168.
Dialogue: 0,0:06:41.77,0:06:44.82,Default,,0000,0000,0000,,So now we have a common denominator here.
Dialogue: 0,0:06:44.82,0:06:46.86,Default,,0000,0000,0000,,And so we can really just compare the numerators.
Dialogue: 0,0:06:46.86,0:06:50.38,Default,,0000,0000,0000,,And 189 is clearly greater than 168.
Dialogue: 0,0:06:50.38,0:06:56.11,Default,,0000,0000,0000,,So 189/252 is clearly greater than 168/252.
Dialogue: 0,0:06:56.11,0:06:58.71,Default,,0000,0000,0000,,Or that's the same thing as saying 21/28,
Dialogue: 0,0:06:58.71,0:07:01.58,Default,,0000,0000,0000,,because that's what this is over here.
Dialogue: 0,0:07:01.58,0:07:04.40,Default,,0000,0000,0000,,The left-hand side is 21/28, is clearly greater
Dialogue: 0,0:07:04.40,0:07:07.71,Default,,0000,0000,0000,,than the right-hand side, which is really 6/9.