1
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--
2
00:00:00,740 --> 00:00:03,420
Use less than, greater than, or equal
3
00:00:03,420 --> 00:00:09,170
to compare the two fractions 21/28, or 21 over 28, and 6/9,
4
00:00:09,170 --> 00:00:11,060
or 6 over 9.
5
00:00:11,060 --> 00:00:13,110
So there's a bunch of ways to do this.
6
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The easiest way is if they had the same denominator,
7
00:00:16,690 --> 00:00:18,450
you could just compare the numerators.
8
00:00:18,450 --> 00:00:22,040
Unlucky for us, we do not have the same denominator.
9
00:00:22,040 --> 00:00:24,600
So what we could do is we can find a common denominator
10
00:00:24,600 --> 00:00:26,840
for both of them and convert both of these fractions
11
00:00:26,840 --> 00:00:30,250
to have the same denominator and then compare the numerators.
12
00:00:30,250 --> 00:00:34,340
Or even more simply, we could simplify them first and then
13
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try to do it.
14
00:00:34,970 --> 00:00:36,511
So let me do that last one, because I
15
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have a feeling that'll be the fastest way to do it.
16
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So 21/28-- you can see that they are both divisible by 7.
17
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So let's divide both the numerator
18
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and the denominator by 7.
19
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So we could divide 21 by 7.
20
00:00:51,190 --> 00:00:53,600
And we can divide-- so let me make the numerator--
21
00:00:53,600 --> 00:00:55,489
and we can divide the denominator by 7.
22
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We're doing the same thing to the numerator
23
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and the denominator, so we're not
24
00:00:58,655 --> 00:01:00,950
going to change the value of the fraction.
25
00:01:00,950 --> 00:01:05,650
So 21 divided by 7 is 3, and 28 divided by 7 is 4.
26
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So 21/28 is the exact same fraction as 3/4.
27
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3/4 is the simplified version of it.
28
00:01:11,770 --> 00:01:14,680
Let's do the same thing for 6/9.
29
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6 and 9 are both divisible by 3.
30
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So let's divide them both by 3 so we
31
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can simplify this fraction.
32
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So let's divide both of them by 3.
33
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6 divided by 3 is 2, and 9 divided by 3 is 3.
34
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So 21/28 is 3/4.
35
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They're the exact same fraction, just written a different way.
36
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This is the more simplified version.
37
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And 6/9 is the exact same fraction as 2/3.
38
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So we really can compare 3/4 and 2/3.
39
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So this is really comparing 3/4 and 2/3.
40
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And the real benefit of doing this is now this
41
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is much easier to find a common denominator for than 28 and 9.
42
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Then we would have to multiply big numbers.
43
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Here we could do fairly small numbers.
44
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The common denominator of 3/4 and 2/3
45
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is going to be the least common multiple of 4 and 3.
46
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And 4 and 3 don't share any prime factors with each other.
47
00:02:13,570 --> 00:02:15,520
So their least common multiple is really
48
00:02:15,520 --> 00:02:17,430
just going to be the product of the two.
49
00:02:17,430 --> 00:02:20,890
So we can write 3/4 as something over 12.
50
00:02:20,890 --> 00:02:24,800
And we can write 2/3 as something over 12.
51
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And I got the 12 by multiplying 3 times 4.
52
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They have no common factors.
53
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Another way you could think about it is 4,
54
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if you do a prime factorization, is 2 times 2.
55
00:02:34,470 --> 00:02:36,870
And 3-- it's already a prime number,
56
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so you can't prime factorize it any more.
57
00:02:39,239 --> 00:02:41,030
So what you want to do is think of a number
58
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that has all of the prime factors of 4 and 3.
59
00:02:44,120 --> 00:02:46,970
So it needs one 2, another 2, and a 3.
60
00:02:46,970 --> 00:02:49,744
Well, 2 times 2 times 3 is 12.
61
00:02:49,744 --> 00:02:51,160
And either way you think about it,
62
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that's how you would get the least common multiple
63
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or the common denominator for 4 and 3.
64
00:02:57,450 --> 00:03:04,360
Well, to get from 4 to 12, you've got to multiply by 3.
65
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So we're multiplying the denominator by 3 to get to 12.
66
00:03:07,460 --> 00:03:10,710
So we also have to multiply the numerator by 3.
67
00:03:10,710 --> 00:03:13,740
So 3 times 3 is 9.
68
00:03:13,740 --> 00:03:15,640
Over here, to get from 3 to 12, we
69
00:03:15,640 --> 00:03:18,320
have to multiply the denominator by 4.
70
00:03:18,320 --> 00:03:21,080
So we also have to multiply the numerator by 4.
71
00:03:21,080 --> 00:03:24,540
So we get 8.
72
00:03:24,540 --> 00:03:26,250
And so now when we compare the fractions,
73
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it's pretty straightforward.
74
00:03:27,770 --> 00:03:33,270
21/28 is the exact same thing as 9/12,
75
00:03:33,270 --> 00:03:40,140
and 6/9 is the exact same thing as 8/12.
76
00:03:40,140 --> 00:03:43,960
So which of these is a greater quantity?
77
00:03:43,960 --> 00:03:46,210
Well, clearly, we have the same denominator right now.
78
00:03:46,210 --> 00:03:50,120
We have 9/12 is clearly greater than 8/12.
79
00:03:50,120 --> 00:03:54,180
So 9/12 is clearly greater than 8/12.
80
00:03:54,180 --> 00:03:56,450
Or if you go back and you realize
81
00:03:56,450 --> 00:03:59,650
that 9/12 is the exact same thing as 21/28,
82
00:03:59,650 --> 00:04:04,750
we could say 21/28 is definitely greater than-- and 8/12
83
00:04:04,750 --> 00:04:10,130
is the same thing as 6/9-- is definitely greater than 6/9.
84
00:04:10,130 --> 00:04:11,867
And we are done.
85
00:04:11,867 --> 00:04:13,450
Another way we could have done it-- we
86
00:04:13,450 --> 00:04:15,370
didn't necessarily have to simplify that.
87
00:04:15,370 --> 00:04:17,310
And let me show you that just for fun.
88
00:04:17,310 --> 00:04:21,450
So if we were doing it with-- if we didn't think
89
00:04:21,450 --> 00:04:24,920
to simplify our two numbers first.
90
00:04:24,920 --> 00:04:27,270
I'm trying to find a color I haven't used yet.
91
00:04:27,270 --> 00:04:32,270
So 21/28 and 6/9.
92
00:04:32,270 --> 00:04:35,460
So we could just find a least common multiple
93
00:04:35,460 --> 00:04:37,980
in the traditional way without simplifying first.
94
00:04:37,980 --> 00:04:40,340
So what's the prime factorization of 28?
95
00:04:40,340 --> 00:04:42,720
It's 2 times 14.
96
00:04:42,720 --> 00:04:45,710
And 14 is 2 times 7.
97
00:04:45,710 --> 00:04:47,230
That's its prime factorization.
98
00:04:47,230 --> 00:04:50,490
Prime factorization of 9 is 3 times 3.
99
00:04:50,490 --> 00:04:53,070
So the least common multiple of 28 and 9
100
00:04:53,070 --> 00:04:56,570
have to contain a 2, a 2, a 7, a 3 and a 3.
101
00:04:56,570 --> 00:04:59,830
Or essentially, it's going to be 28 times 9.
102
00:04:59,830 --> 00:05:04,232
So let's over here multiply 28 times 9.
103
00:05:04,232 --> 00:05:05,940
There's a couple of ways you could do it.
104
00:05:05,940 --> 00:05:07,870
You could multiply in your head 28 times
105
00:05:07,870 --> 00:05:11,079
10, which would be 280, and then subtract 28
106
00:05:11,079 --> 00:05:12,370
from that, which would be what?
107
00:05:12,370 --> 00:05:14,140
252.
108
00:05:14,140 --> 00:05:16,850
Or we could just multiply it out if that confuses you.
109
00:05:16,850 --> 00:05:18,580
So let's just do the second way.
110
00:05:18,580 --> 00:05:22,110
9 times 8 is 72.
111
00:05:22,110 --> 00:05:24,380
9 times 2 is 18.
112
00:05:24,380 --> 00:05:26,370
18 plus 7 is 25.
113
00:05:26,370 --> 00:05:28,880
So we get 252.
114
00:05:28,880 --> 00:05:31,660
So I'm saying the common denominator here
115
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is going to be 252.
116
00:05:37,650 --> 00:05:40,270
Least common multiple of 28 and 9.
117
00:05:40,270 --> 00:05:46,370
Well, to go from 28 to 252, we had to multiply it by 9.
118
00:05:46,370 --> 00:05:48,590
We had to multiply 28 times 9.
119
00:05:48,590 --> 00:05:50,570
So we're multiplying 28 times 9.
120
00:05:50,570 --> 00:05:53,240
So we also have to multiply the numerator times 9.
121
00:05:53,240 --> 00:05:55,336
So what is 21 times 9?
122
00:05:55,336 --> 00:05:56,710
That's easier to do in your head.
123
00:05:56,710 --> 00:05:59,080
20 times 9 is 180.
124
00:05:59,080 --> 00:06:00,710
And then 1 times 9 is 9.
125
00:06:00,710 --> 00:06:04,080
So this is going to be 189.
126
00:06:04,080 --> 00:06:09,100
To go from 9 to 252, we had to multiply by 28.
127
00:06:09,100 --> 00:06:11,630
So we also have to multiply the numerator by 28
128
00:06:11,630 --> 00:06:14,290
if we don't want to change the value of the fraction.
129
00:06:14,290 --> 00:06:18,512
So 6 times 28-- 6 times 20 is 120.
130
00:06:18,512 --> 00:06:22,420
6 times 8 is 48.
131
00:06:22,420 --> 00:06:24,414
So we get 168.
132
00:06:24,414 --> 00:06:25,830
Let me write that out just to make
133
00:06:25,830 --> 00:06:27,290
sure I didn't make a mistake.
134
00:06:27,290 --> 00:06:36,180
So 28 times 6-- 8 times 6 is 48.
135
00:06:36,180 --> 00:06:40,040
2 times 6 is 12, plus 4 is 16.
136
00:06:40,040 --> 00:06:41,770
So right, 168.
137
00:06:41,770 --> 00:06:44,819
So now we have a common denominator here.
138
00:06:44,819 --> 00:06:46,860
And so we can really just compare the numerators.
139
00:06:46,860 --> 00:06:50,380
And 189 is clearly greater than 168.
140
00:06:50,380 --> 00:06:56,110
So 189/252 is clearly greater than 168/252.
141
00:06:56,110 --> 00:06:58,710
Or that's the same thing as saying 21/28,
142
00:06:58,710 --> 00:07:01,580
because that's what this is over here.
143
00:07:01,580 --> 00:07:04,400
The left-hand side is 21/28, is clearly greater
144
00:07:04,400 --> 00:07:07,710
than the right-hand side, which is really 6/9.