0:00:00.000,0:00:00.740
--

0:00:00.740,0:00:03.420
Use less than, greater than, or equal

0:00:03.420,0:00:09.170
to compare the two fractions 21/28, or 21 over 28, and 6/9,

0:00:09.170,0:00:11.060
or 6 over 9.

0:00:11.060,0:00:13.110
So there's a bunch of ways to do this.

0:00:13.110,0:00:16.690
The easiest way is if they had the same denominator,

0:00:16.690,0:00:18.450
you could just compare the numerators.

0:00:18.450,0:00:22.040
Unlucky for us, we do not have the same denominator.

0:00:22.040,0:00:24.600
So what we could do is we can find a common denominator

0:00:24.600,0:00:26.840
for both of them and convert both of these fractions

0:00:26.840,0:00:30.250
to have the same denominator and then compare the numerators.

0:00:30.250,0:00:34.340
Or even more simply, we could simplify them first and then

0:00:34.340,0:00:34.970
try to do it.

0:00:34.970,0:00:36.511
So let me do that last one, because I

0:00:36.511,0:00:39.190
have a feeling that'll be the fastest way to do it.

0:00:39.190,0:00:45.050
So 21/28-- you can see that they are both divisible by 7.

0:00:45.050,0:00:46.650
So let's divide both the numerator

0:00:46.650,0:00:48.570
and the denominator by 7.

0:00:48.570,0:00:51.190
So we could divide 21 by 7.

0:00:51.190,0:00:53.600
And we can divide-- so let me make the numerator--

0:00:53.600,0:00:55.489
and we can divide the denominator by 7.

0:00:55.489,0:00:57.280
We're doing the same thing to the numerator

0:00:57.280,0:00:58.655
and the denominator, so we're not

0:00:58.655,0:01:00.950
going to change the value of the fraction.

0:01:00.950,0:01:05.650
So 21 divided by 7 is 3, and 28 divided by 7 is 4.

0:01:05.650,0:01:09.510
So 21/28 is the exact same fraction as 3/4.

0:01:09.510,0:01:11.770
3/4 is the simplified version of it.

0:01:11.770,0:01:14.680
Let's do the same thing for 6/9.

0:01:14.680,0:01:17.520
6 and 9 are both divisible by 3.

0:01:17.520,0:01:19.410
So let's divide them both by 3 so we

0:01:19.410,0:01:21.430
can simplify this fraction.

0:01:21.430,0:01:24.080
So let's divide both of them by 3.

0:01:24.080,0:01:30.380
6 divided by 3 is 2, and 9 divided by 3 is 3.

0:01:30.380,0:01:33.060
So 21/28 is 3/4.

0:01:33.060,0:01:35.760
They're the exact same fraction, just written a different way.

0:01:35.760,0:01:37.700
This is the more simplified version.

0:01:37.700,0:01:41.580
And 6/9 is the exact same fraction as 2/3.

0:01:41.580,0:01:44.590
So we really can compare 3/4 and 2/3.

0:01:44.590,0:01:49.230
So this is really comparing 3/4 and 2/3.

0:01:49.230,0:01:51.990
And the real benefit of doing this is now this

0:01:51.990,0:01:55.870
is much easier to find a common denominator for than 28 and 9.

0:01:55.870,0:01:58.680
Then we would have to multiply big numbers.

0:01:58.680,0:02:00.850
Here we could do fairly small numbers.

0:02:00.850,0:02:03.780
The common denominator of 3/4 and 2/3

0:02:03.780,0:02:08.810
is going to be the least common multiple of 4 and 3.

0:02:08.810,0:02:13.570
And 4 and 3 don't share any prime factors with each other.

0:02:13.570,0:02:15.520
So their least common multiple is really

0:02:15.520,0:02:17.430
just going to be the product of the two.

0:02:17.430,0:02:20.890
So we can write 3/4 as something over 12.

0:02:20.890,0:02:24.800
And we can write 2/3 as something over 12.

0:02:24.800,0:02:27.570
And I got the 12 by multiplying 3 times 4.

0:02:27.570,0:02:29.150
They have no common factors.

0:02:29.150,0:02:31.500
Another way you could think about it is 4,

0:02:31.500,0:02:34.470
if you do a prime factorization, is 2 times 2.

0:02:34.470,0:02:36.870
And 3-- it's already a prime number,

0:02:36.870,0:02:39.239
so you can't prime factorize it any more.

0:02:39.239,0:02:41.030
So what you want to do is think of a number

0:02:41.030,0:02:44.120
that has all of the prime factors of 4 and 3.

0:02:44.120,0:02:46.970
So it needs one 2, another 2, and a 3.

0:02:46.970,0:02:49.744
Well, 2 times 2 times 3 is 12.

0:02:49.744,0:02:51.160
And either way you think about it,

0:02:51.160,0:02:53.243
that's how you would get the least common multiple

0:02:53.243,0:02:57.450
or the common denominator for 4 and 3.

0:02:57.450,0:03:04.360
Well, to get from 4 to 12, you've got to multiply by 3.

0:03:04.360,0:03:07.460
So we're multiplying the denominator by 3 to get to 12.

0:03:07.460,0:03:10.710
So we also have to multiply the numerator by 3.

0:03:10.710,0:03:13.740
So 3 times 3 is 9.

0:03:13.740,0:03:15.640
Over here, to get from 3 to 12, we

0:03:15.640,0:03:18.320
have to multiply the denominator by 4.

0:03:18.320,0:03:21.080
So we also have to multiply the numerator by 4.

0:03:21.080,0:03:24.540
So we get 8.

0:03:24.540,0:03:26.250
And so now when we compare the fractions,

0:03:26.250,0:03:27.770
it's pretty straightforward.

0:03:27.770,0:03:33.270
21/28 is the exact same thing as 9/12,

0:03:33.270,0:03:40.140
and 6/9 is the exact same thing as 8/12.

0:03:40.140,0:03:43.960
So which of these is a greater quantity?

0:03:43.960,0:03:46.210
Well, clearly, we have the same denominator right now.

0:03:46.210,0:03:50.120
We have 9/12 is clearly greater than 8/12.

0:03:50.120,0:03:54.180
So 9/12 is clearly greater than 8/12.

0:03:54.180,0:03:56.450
Or if you go back and you realize

0:03:56.450,0:03:59.650
that 9/12 is the exact same thing as 21/28,

0:03:59.650,0:04:04.750
we could say 21/28 is definitely greater than-- and 8/12

0:04:04.750,0:04:10.130
is the same thing as 6/9-- is definitely greater than 6/9.

0:04:10.130,0:04:11.867
And we are done.

0:04:11.867,0:04:13.450
Another way we could have done it-- we

0:04:13.450,0:04:15.370
didn't necessarily have to simplify that.

0:04:15.370,0:04:17.310
And let me show you that just for fun.

0:04:17.310,0:04:21.450
So if we were doing it with-- if we didn't think

0:04:21.450,0:04:24.920
to simplify our two numbers first.

0:04:24.920,0:04:27.270
I'm trying to find a color I haven't used yet.

0:04:27.270,0:04:32.270
So 21/28 and 6/9.

0:04:32.270,0:04:35.460
So we could just find a least common multiple

0:04:35.460,0:04:37.980
in the traditional way without simplifying first.

0:04:37.980,0:04:40.340
So what's the prime factorization of 28?

0:04:40.340,0:04:42.720
It's 2 times 14.

0:04:42.720,0:04:45.710
And 14 is 2 times 7.

0:04:45.710,0:04:47.230
That's its prime factorization.

0:04:47.230,0:04:50.490
Prime factorization of 9 is 3 times 3.

0:04:50.490,0:04:53.070
So the least common multiple of 28 and 9

0:04:53.070,0:04:56.570
have to contain a 2, a 2, a 7, a 3 and a 3.

0:04:56.570,0:04:59.830
Or essentially, it's going to be 28 times 9.

0:04:59.830,0:05:04.232
So let's over here multiply 28 times 9.

0:05:04.232,0:05:05.940
There's a couple of ways you could do it.

0:05:05.940,0:05:07.870
You could multiply in your head 28 times

0:05:07.870,0:05:11.079
10, which would be 280, and then subtract 28

0:05:11.079,0:05:12.370
from that, which would be what?

0:05:12.370,0:05:14.140
252.

0:05:14.140,0:05:16.850
Or we could just multiply it out if that confuses you.

0:05:16.850,0:05:18.580
So let's just do the second way.

0:05:18.580,0:05:22.110
9 times 8 is 72.

0:05:22.110,0:05:24.380
9 times 2 is 18.

0:05:24.380,0:05:26.370
18 plus 7 is 25.

0:05:26.370,0:05:28.880
So we get 252.

0:05:28.880,0:05:31.660
So I'm saying the common denominator here

0:05:31.660,0:05:37.650
is going to be 252.

0:05:37.650,0:05:40.270
Least common multiple of 28 and 9.

0:05:40.270,0:05:46.370
Well, to go from 28 to 252, we had to multiply it by 9.

0:05:46.370,0:05:48.590
We had to multiply 28 times 9.

0:05:48.590,0:05:50.570
So we're multiplying 28 times 9.

0:05:50.570,0:05:53.240
So we also have to multiply the numerator times 9.

0:05:53.240,0:05:55.336
So what is 21 times 9?

0:05:55.336,0:05:56.710
That's easier to do in your head.

0:05:56.710,0:05:59.080
20 times 9 is 180.

0:05:59.080,0:06:00.710
And then 1 times 9 is 9.

0:06:00.710,0:06:04.080
So this is going to be 189.

0:06:04.080,0:06:09.100
To go from 9 to 252, we had to multiply by 28.

0:06:09.100,0:06:11.630
So we also have to multiply the numerator by 28

0:06:11.630,0:06:14.290
if we don't want to change the value of the fraction.

0:06:14.290,0:06:18.512
So 6 times 28-- 6 times 20 is 120.

0:06:18.512,0:06:22.420
6 times 8 is 48.

0:06:22.420,0:06:24.414
So we get 168.

0:06:24.414,0:06:25.830
Let me write that out just to make

0:06:25.830,0:06:27.290
sure I didn't make a mistake.

0:06:27.290,0:06:36.180
So 28 times 6-- 8 times 6 is 48.

0:06:36.180,0:06:40.040
2 times 6 is 12, plus 4 is 16.

0:06:40.040,0:06:41.770
So right, 168.

0:06:41.770,0:06:44.819
So now we have a common denominator here.

0:06:44.819,0:06:46.860
And so we can really just compare the numerators.

0:06:46.860,0:06:50.380
And 189 is clearly greater than 168.

0:06:50.380,0:06:56.110
So 189/252 is clearly greater than 168/252.

0:06:56.110,0:06:58.710
Or that's the same thing as saying 21/28,

0:06:58.710,0:07:01.580
because that's what this is over here.

0:07:01.580,0:07:04.400
The left-hand side is 21/28, is clearly greater

0:07:04.400,0:07:07.710
than the right-hand side, which is really 6/9.