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
00:00:13,110 --> 00:00:16,690
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
00:00:34,340 --> 00:00:34,970
try to do it.

14
00:00:34,970 --> 00:00:36,511
So let me do that last one, because I

15
00:00:36,511 --> 00:00:39,190
have a feeling that'll be the fastest way to do it.

16
00:00:39,190 --> 00:00:45,050
So 21/28-- you can see that they are both divisible by 7.

17
00:00:45,050 --> 00:00:46,650
So let's divide both the numerator

18
00:00:46,650 --> 00:00:48,570
and the denominator by 7.

19
00:00:48,570 --> 00:00:51,190
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
00:00:55,489 --> 00:00:57,280
We're doing the same thing to the numerator

23
00:00:57,280 --> 00:00:58,655
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
00:01:05,650 --> 00:01:09,510
So 21/28 is the exact same fraction as 3/4.

27
00:01:09,510 --> 00:01:11,770
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
00:01:17,520 --> 00:01:19,410
So let's divide them both by 3 so we

31
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can simplify this fraction.

32
00:01:21,430 --> 00:01:24,080
So let's divide both of them by 3.

33
00:01:24,080 --> 00:01:30,380
6 divided by 3 is 2, and 9 divided by 3 is 3.

34
00:01:30,380 --> 00:01:33,060
So 21/28 is 3/4.

35
00:01:33,060 --> 00:01:35,760
They're the exact same fraction, just written a different way.

36
00:01:35,760 --> 00:01:37,700
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
00:01:44,590 --> 00:01:49,230
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
00:01:55,870 --> 00:01:58,680
Then we would have to multiply big numbers.

43
00:01:58,680 --> 00:02:00,850
Here we could do fairly small numbers.

44
00:02:00,850 --> 00:02:03,780
The common denominator of 3/4 and 2/3

45
00:02:03,780 --> 00:02:08,810
is going to be the least common multiple of 4 and 3.

46
00:02:08,810 --> 00:02:13,570
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
00:02:24,800 --> 00:02:27,570
And I got the 12 by multiplying 3 times 4.

52
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They have no common factors.

53
00:02:29,150 --> 00:02:31,500
Another way you could think about it is 4,

54
00:02:31,500 --> 00:02:34,470
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
00:02:36,870 --> 00:02:39,239
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
00:02:53,243 --> 00:02:57,450
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
00:03:04,360 --> 00:03:07,460
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
00:03:26,250 --> 00:03:27,770
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
00:05:31,660 --> 00:05:37,650
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.