WEBVTT 00:00:00.000 --> 00:00:00.740 -- 00:00:00.740 --> 00:00:03.420 Use less than, greater than, or equal 00:00:03.420 --> 00:00:09.170 to compare the two fractions 21/28, or 21 over 28, and 6/9, 00:00:09.170 --> 00:00:11.060 or 6 over 9. 00:00:11.060 --> 00:00:13.110 So there's a bunch of ways to do this. 00:00:13.110 --> 00:00:16.690 The easiest way is if they had the same denominator, 00:00:16.690 --> 00:00:18.450 you could just compare the numerators. 00:00:18.450 --> 00:00:22.040 Unlucky for us, we do not have the same denominator. 00:00:22.040 --> 00:00:24.600 So what we could do is we can find a common denominator 00:00:24.600 --> 00:00:26.840 for both of them and convert both of these fractions 00:00:26.840 --> 00:00:30.250 to have the same denominator and then compare the numerators. 00:00:30.250 --> 00:00:34.340 Or even more simply, we could simplify them first and then 00:00:34.340 --> 00:00:34.970 try to do it. 00:00:34.970 --> 00:00:36.511 So let me do that last one, because I 00:00:36.511 --> 00:00:39.190 have a feeling that'll be the fastest way to do it. 00:00:39.190 --> 00:00:45.050 So 21/28-- you can see that they are both divisible by 7. 00:00:45.050 --> 00:00:46.650 So let's divide both the numerator 00:00:46.650 --> 00:00:48.570 and the denominator by 7. 00:00:48.570 --> 00:00:51.190 So we could divide 21 by 7. 00:00:51.190 --> 00:00:53.600 And we can divide-- so let me make the numerator-- 00:00:53.600 --> 00:00:55.489 and we can divide the denominator by 7. 00:00:55.489 --> 00:00:57.280 We're doing the same thing to the numerator 00:00:57.280 --> 00:00:58.655 and the denominator, so we're not 00:00:58.655 --> 00:01:00.950 going to change the value of the fraction. 00:01:00.950 --> 00:01:05.650 So 21 divided by 7 is 3, and 28 divided by 7 is 4. 00:01:05.650 --> 00:01:09.510 So 21/28 is the exact same fraction as 3/4. 00:01:09.510 --> 00:01:11.770 3/4 is the simplified version of it. 00:01:11.770 --> 00:01:14.680 Let's do the same thing for 6/9. 00:01:14.680 --> 00:01:17.520 6 and 9 are both divisible by 3. 00:01:17.520 --> 00:01:19.410 So let's divide them both by 3 so we 00:01:19.410 --> 00:01:21.430 can simplify this fraction. 00:01:21.430 --> 00:01:24.080 So let's divide both of them by 3. 00:01:24.080 --> 00:01:30.380 6 divided by 3 is 2, and 9 divided by 3 is 3. 00:01:30.380 --> 00:01:33.060 So 21/28 is 3/4. 00:01:33.060 --> 00:01:35.760 They're the exact same fraction, just written a different way. 00:01:35.760 --> 00:01:37.700 This is the more simplified version. 00:01:37.700 --> 00:01:41.580 And 6/9 is the exact same fraction as 2/3. 00:01:41.580 --> 00:01:44.590 So we really can compare 3/4 and 2/3. 00:01:44.590 --> 00:01:49.230 So this is really comparing 3/4 and 2/3. 00:01:49.230 --> 00:01:51.990 And the real benefit of doing this is now this 00:01:51.990 --> 00:01:55.870 is much easier to find a common denominator for than 28 and 9. 00:01:55.870 --> 00:01:58.680 Then we would have to multiply big numbers. 00:01:58.680 --> 00:02:00.850 Here we could do fairly small numbers. 00:02:00.850 --> 00:02:03.780 The common denominator of 3/4 and 2/3 00:02:03.780 --> 00:02:08.810 is going to be the least common multiple of 4 and 3. 00:02:08.810 --> 00:02:13.570 And 4 and 3 don't share any prime factors with each other. 00:02:13.570 --> 00:02:15.520 So their least common multiple is really 00:02:15.520 --> 00:02:17.430 just going to be the product of the two. 00:02:17.430 --> 00:02:20.890 So we can write 3/4 as something over 12. 00:02:20.890 --> 00:02:24.800 And we can write 2/3 as something over 12. 00:02:24.800 --> 00:02:27.570 And I got the 12 by multiplying 3 times 4. 00:02:27.570 --> 00:02:29.150 They have no common factors. 00:02:29.150 --> 00:02:31.500 Another way you could think about it is 4, 00:02:31.500 --> 00:02:34.470 if you do a prime factorization, is 2 times 2. 00:02:34.470 --> 00:02:36.870 And 3-- it's already a prime number, 00:02:36.870 --> 00:02:39.239 so you can't prime factorize it any more. 00:02:39.239 --> 00:02:41.030 So what you want to do is think of a number 00:02:41.030 --> 00:02:44.120 that has all of the prime factors of 4 and 3. 00:02:44.120 --> 00:02:46.970 So it needs one 2, another 2, and a 3. 00:02:46.970 --> 00:02:49.744 Well, 2 times 2 times 3 is 12. 00:02:49.744 --> 00:02:51.160 And either way you think about it, 00:02:51.160 --> 00:02:53.243 that's how you would get the least common multiple 00:02:53.243 --> 00:02:57.450 or the common denominator for 4 and 3. 00:02:57.450 --> 00:03:04.360 Well, to get from 4 to 12, you've got to multiply by 3. 00:03:04.360 --> 00:03:07.460 So we're multiplying the denominator by 3 to get to 12. 00:03:07.460 --> 00:03:10.710 So we also have to multiply the numerator by 3. 00:03:10.710 --> 00:03:13.740 So 3 times 3 is 9. 00:03:13.740 --> 00:03:15.640 Over here, to get from 3 to 12, we 00:03:15.640 --> 00:03:18.320 have to multiply the denominator by 4. 00:03:18.320 --> 00:03:21.080 So we also have to multiply the numerator by 4. 00:03:21.080 --> 00:03:24.540 So we get 8. 00:03:24.540 --> 00:03:26.250 And so now when we compare the fractions, 00:03:26.250 --> 00:03:27.770 it's pretty straightforward. 00:03:27.770 --> 00:03:33.270 21/28 is the exact same thing as 9/12, 00:03:33.270 --> 00:03:40.140 and 6/9 is the exact same thing as 8/12. 00:03:40.140 --> 00:03:43.960 So which of these is a greater quantity? 00:03:43.960 --> 00:03:46.210 Well, clearly, we have the same denominator right now. 00:03:46.210 --> 00:03:50.120 We have 9/12 is clearly greater than 8/12. 00:03:50.120 --> 00:03:54.180 So 9/12 is clearly greater than 8/12. 00:03:54.180 --> 00:03:56.450 Or if you go back and you realize 00:03:56.450 --> 00:03:59.650 that 9/12 is the exact same thing as 21/28, 00:03:59.650 --> 00:04:04.750 we could say 21/28 is definitely greater than-- and 8/12 00:04:04.750 --> 00:04:10.130 is the same thing as 6/9-- is definitely greater than 6/9. 00:04:10.130 --> 00:04:11.867 And we are done. 00:04:11.867 --> 00:04:13.450 Another way we could have done it-- we 00:04:13.450 --> 00:04:15.370 didn't necessarily have to simplify that. 00:04:15.370 --> 00:04:17.310 And let me show you that just for fun. 00:04:17.310 --> 00:04:21.450 So if we were doing it with-- if we didn't think 00:04:21.450 --> 00:04:24.920 to simplify our two numbers first. 00:04:24.920 --> 00:04:27.270 I'm trying to find a color I haven't used yet. 00:04:27.270 --> 00:04:32.270 So 21/28 and 6/9. 00:04:32.270 --> 00:04:35.460 So we could just find a least common multiple 00:04:35.460 --> 00:04:37.980 in the traditional way without simplifying first. 00:04:37.980 --> 00:04:40.340 So what's the prime factorization of 28? 00:04:40.340 --> 00:04:42.720 It's 2 times 14. 00:04:42.720 --> 00:04:45.710 And 14 is 2 times 7. 00:04:45.710 --> 00:04:47.230 That's its prime factorization. 00:04:47.230 --> 00:04:50.490 Prime factorization of 9 is 3 times 3. 00:04:50.490 --> 00:04:53.070 So the least common multiple of 28 and 9 00:04:53.070 --> 00:04:56.570 have to contain a 2, a 2, a 7, a 3 and a 3. 00:04:56.570 --> 00:04:59.830 Or essentially, it's going to be 28 times 9. 00:04:59.830 --> 00:05:04.232 So let's over here multiply 28 times 9. 00:05:04.232 --> 00:05:05.940 There's a couple of ways you could do it. 00:05:05.940 --> 00:05:07.870 You could multiply in your head 28 times 00:05:07.870 --> 00:05:11.079 10, which would be 280, and then subtract 28 00:05:11.079 --> 00:05:12.370 from that, which would be what? 00:05:12.370 --> 00:05:14.140 252. 00:05:14.140 --> 00:05:16.850 Or we could just multiply it out if that confuses you. 00:05:16.850 --> 00:05:18.580 So let's just do the second way. 00:05:18.580 --> 00:05:22.110 9 times 8 is 72. 00:05:22.110 --> 00:05:24.380 9 times 2 is 18. 00:05:24.380 --> 00:05:26.370 18 plus 7 is 25. 00:05:26.370 --> 00:05:28.880 So we get 252. 00:05:28.880 --> 00:05:31.660 So I'm saying the common denominator here 00:05:31.660 --> 00:05:37.650 is going to be 252. 00:05:37.650 --> 00:05:40.270 Least common multiple of 28 and 9. 00:05:40.270 --> 00:05:46.370 Well, to go from 28 to 252, we had to multiply it by 9. 00:05:46.370 --> 00:05:48.590 We had to multiply 28 times 9. 00:05:48.590 --> 00:05:50.570 So we're multiplying 28 times 9. 00:05:50.570 --> 00:05:53.240 So we also have to multiply the numerator times 9. 00:05:53.240 --> 00:05:55.336 So what is 21 times 9? 00:05:55.336 --> 00:05:56.710 That's easier to do in your head. 00:05:56.710 --> 00:05:59.080 20 times 9 is 180. 00:05:59.080 --> 00:06:00.710 And then 1 times 9 is 9. 00:06:00.710 --> 00:06:04.080 So this is going to be 189. 00:06:04.080 --> 00:06:09.100 To go from 9 to 252, we had to multiply by 28. 00:06:09.100 --> 00:06:11.630 So we also have to multiply the numerator by 28 00:06:11.630 --> 00:06:14.290 if we don't want to change the value of the fraction. 00:06:14.290 --> 00:06:18.512 So 6 times 28-- 6 times 20 is 120. 00:06:18.512 --> 00:06:22.420 6 times 8 is 48. 00:06:22.420 --> 00:06:24.414 So we get 168. 00:06:24.414 --> 00:06:25.830 Let me write that out just to make 00:06:25.830 --> 00:06:27.290 sure I didn't make a mistake. 00:06:27.290 --> 00:06:36.180 So 28 times 6-- 8 times 6 is 48. 00:06:36.180 --> 00:06:40.040 2 times 6 is 12, plus 4 is 16. 00:06:40.040 --> 00:06:41.770 So right, 168. 00:06:41.770 --> 00:06:44.819 So now we have a common denominator here. 00:06:44.819 --> 00:06:46.860 And so we can really just compare the numerators. 00:06:46.860 --> 00:06:50.380 And 189 is clearly greater than 168. 00:06:50.380 --> 00:06:56.110 So 189/252 is clearly greater than 168/252. 00:06:56.110 --> 00:06:58.710 Or that's the same thing as saying 21/28, 00:06:58.710 --> 00:07:01.580 because that's what this is over here. 00:07:01.580 --> 00:07:04.400 The left-hand side is 21/28, is clearly greater 00:07:04.400 --> 00:07:07.710 than the right-hand side, which is really 6/9.