1
00:00:00,000 --> 00:00:00,330


2
00:00:00,330 --> 00:00:02,900
Let's add some rational
numbers.

3
00:00:02,900 --> 00:00:05,350
And I'm using that word because
that's the word that

4
00:00:05,350 --> 00:00:08,640
this book uses, but in more
popular terminology we'll be

5
00:00:08,640 --> 00:00:10,480
adding fractions.

6
00:00:10,480 --> 00:00:14,100
So let's just go through all
of these, actually, just to

7
00:00:14,100 --> 00:00:15,080
see all of the examples.

8
00:00:15,080 --> 00:00:19,660
So first we're going to
have 3/7 plus 2/7.

9
00:00:19,660 --> 00:00:22,840
Our denominators are the same,
so we can just add the

10
00:00:22,840 --> 00:00:24,070
numerators.

11
00:00:24,070 --> 00:00:28,480
So our denominator is
7, 3 plus 2 is 5.

12
00:00:28,480 --> 00:00:31,060
That is a.

13
00:00:31,060 --> 00:00:31,960
Let me do every other.

14
00:00:31,960 --> 00:00:33,290
It would take forever
to do all of them.

15
00:00:33,290 --> 00:00:36,550
Not forever, but just more time
than I want to spend.

16
00:00:36,550 --> 00:00:42,860
So c is 5/16 plus 5/12.

17
00:00:42,860 --> 00:00:44,900
Our denominators are
not the same.

18
00:00:44,900 --> 00:00:47,700
We have to find a common
denominator, which has to be

19
00:00:47,700 --> 00:00:50,450
the least common-- it actually
could be any common multiple

20
00:00:50,450 --> 00:00:52,050
of these, but for simplicity
let's do the

21
00:00:52,050 --> 00:00:53,770
least common multiple.

22
00:00:53,770 --> 00:00:56,150
So what's the smallest number
that's a multiple

23
00:00:56,150 --> 00:00:58,215
of both 16 and 12?

24
00:00:58,215 --> 00:01:01,700
So let's see, 16 times 2
is 32, not there yet.

25
00:01:01,700 --> 00:01:03,660
Times 3, 48.

26
00:01:03,660 --> 00:01:04,599
That seems to work.

27
00:01:04,599 --> 00:01:06,990
12 goes into 48 four times.

28
00:01:06,990 --> 00:01:09,733
So let's use 48 as our
common denominator.

29
00:01:09,733 --> 00:01:13,960


30
00:01:13,960 --> 00:01:19,415
So we had to multiply 16 times 3
to get to 48, so we're going

31
00:01:19,415 --> 00:01:23,890
to have to multiply
this 5 times 3.

32
00:01:23,890 --> 00:01:25,670
We're just multiplying the
numerator and the denominator

33
00:01:25,670 --> 00:01:28,090
by the same number, so we're
not really changing it.

34
00:01:28,090 --> 00:01:31,370
So 5 times 3 is 15.

35
00:01:31,370 --> 00:01:36,850
And then to get from this 12 to
this 48 right there, we had

36
00:01:36,850 --> 00:01:38,890
to multiply times 4.

37
00:01:38,890 --> 00:01:42,170
So then to get to 5 to this
numerator over here, we have

38
00:01:42,170 --> 00:01:44,120
to multiply times 4.

39
00:01:44,120 --> 00:01:46,690
5 times 4 is 20.

40
00:01:46,690 --> 00:01:49,980
Now we have the same
denominator.

41
00:01:49,980 --> 00:01:54,180
So this is going to be equal
to, our denominator is 48.

42
00:01:54,180 --> 00:02:01,150
And so we can add 15 plus
20, which is 35.

43
00:02:01,150 --> 00:02:02,670
And can we reduce this?

44
00:02:02,670 --> 00:02:04,950
Let's see, 5 does
not go into 48.

45
00:02:04,950 --> 00:02:06,620
7 does not go into 48.

46
00:02:06,620 --> 00:02:08,330
It looks like we're all done.

47
00:02:08,330 --> 00:02:13,940
Let's do e right there.

48
00:02:13,940 --> 00:02:19,790
8/25 plus 7 over 10.

49
00:02:19,790 --> 00:02:23,570
Once again, we don't have
a common denominator.

50
00:02:23,570 --> 00:02:25,850
But we can solve that.

51
00:02:25,850 --> 00:02:28,890
Let's make, let's see, 50 is the
smallest number that both

52
00:02:28,890 --> 00:02:29,800
of these go into.

53
00:02:29,800 --> 00:02:32,340
25 times 2, so that's 50.

54
00:02:32,340 --> 00:02:37,050
8 over 25, to go to 50
we multiply by 2.

55
00:02:37,050 --> 00:02:39,990
So the 8, we're going to
have to multiply by 2.

56
00:02:39,990 --> 00:02:42,640
So it's going to
be 16 over 50.

57
00:02:42,640 --> 00:02:45,945
And then the 7 over 10,
we're going to want

58
00:02:45,945 --> 00:02:47,930
to put it over 50.

59
00:02:47,930 --> 00:02:51,750
We multiply the 10 times
5, so we have to

60
00:02:51,750 --> 00:02:54,605
multiply the 7 times 5.

61
00:02:54,605 --> 00:02:57,720
So it's going to
be 35 over 50.

62
00:02:57,720 --> 00:03:01,560
Now that our denominators are
the same, we have it over 50.

63
00:03:01,560 --> 00:03:05,550
16 plus 35, what is that?

64
00:03:05,550 --> 00:03:10,690
10 plus 35 is 45,
plus 6 is 51.

65
00:03:10,690 --> 00:03:14,770
So it is 51 over 50.

66
00:03:14,770 --> 00:03:16,992
Problem g.

67
00:03:16,992 --> 00:03:19,700
Let me do it in a new color.

68
00:03:19,700 --> 00:03:22,410
Problem g.

69
00:03:22,410 --> 00:03:28,470
So here we have 7 over 15-- I'll
write the second one in a

70
00:03:28,470 --> 00:03:33,530
different color--
plus 2 over 9.

71
00:03:33,530 --> 00:03:35,620
Once again, the denominators
are different.

72
00:03:35,620 --> 00:03:37,490
Find a common denominator.

73
00:03:37,490 --> 00:03:41,540
What is the smallest number that
both 15 and 9 go into?

74
00:03:41,540 --> 00:03:43,260
Let's see, 15 times 2 is 30.

75
00:03:43,260 --> 00:03:44,940
Nope, not divisible by 9.

76
00:03:44,940 --> 00:03:47,670
15 times 3 is 45, that works.

77
00:03:47,670 --> 00:03:50,220
45 is divisible by 9.

78
00:03:50,220 --> 00:03:52,590
So we use 45.

79
00:03:52,590 --> 00:03:59,810
15 times 3 is 45, so
7 times 3 is 21.

80
00:03:59,810 --> 00:04:02,850
These two fractions
are equivalent.

81
00:04:02,850 --> 00:04:06,680
Plus we're going over 45.

82
00:04:06,680 --> 00:04:11,520
To get from 9 to 45, we have
to multiply times 5.

83
00:04:11,520 --> 00:04:14,420
So to get our numerator
over here, we have to

84
00:04:14,420 --> 00:04:15,980
multiply it times 5.

85
00:04:15,980 --> 00:04:18,420
So 2 times 5 is 10.

86
00:04:18,420 --> 00:04:22,422
2/9 is the same thing
as 10/45.

87
00:04:22,422 --> 00:04:24,710
So now we can add.

88
00:04:24,710 --> 00:04:27,130
We're adding fractions of 45.

89
00:04:27,130 --> 00:04:33,130
21 plus 10 is 31,
and we are done.

90
00:04:33,130 --> 00:04:36,900
Let's do one more problem down
here, a word problem.

91
00:04:36,900 --> 00:04:40,070
Nadia, Peter and Ian are pooling
their money to buy a

92
00:04:40,070 --> 00:04:41,640
gallon of ice cream.

93
00:04:41,640 --> 00:04:44,630
Nadia's the oldest and gets
the greatest allowance.

94
00:04:44,630 --> 00:04:49,740
She contributes 1/2 the cost.
So Nadia is contributing 1/2

95
00:04:49,740 --> 00:04:53,750
the cost. So that is
Nadia right there.

96
00:04:53,750 --> 00:04:58,850
Ian is next oldest and
contributes 1/3 of the cost.

97
00:04:58,850 --> 00:05:02,280
So Ian contributes 1/3.

98
00:05:02,280 --> 00:05:03,820
That is Ian.

99
00:05:03,820 --> 00:05:06,360
Peter, the youngest, gets the
smallest allowance and

100
00:05:06,360 --> 00:05:13,730
contributes 1/4 of the cost.
So Peter gives 1/4 of the

101
00:05:13,730 --> 00:05:17,560
cost. Peter contributes
1/4 of cost.

102
00:05:17,560 --> 00:05:19,920
They figure that this will
be enough money.

103
00:05:19,920 --> 00:05:22,480
When they get to the checkout,
they realize that they forgot

104
00:05:22,480 --> 00:05:24,000
about sales tax and
worry there will

105
00:05:24,000 --> 00:05:25,340
not be enough money.

106
00:05:25,340 --> 00:05:28,370
Amazingly, they have exactly
the right amount of money.

107
00:05:28,370 --> 00:05:32,460
What fraction of the cost of
ice cream was added as tax?

108
00:05:32,460 --> 00:05:35,640
Well, let's see, if we add 1/2
plus 1/3, plus 1/4 of the

109
00:05:35,640 --> 00:05:37,640
cost, let's see what we get.

110
00:05:37,640 --> 00:05:41,100
So we have to find a common
denominator, some number that

111
00:05:41,100 --> 00:05:44,250
is the least common multiple
of 2, 3, and 4.

112
00:05:44,250 --> 00:05:46,970
And let's see, 4, it would
have to be 12, right?

113
00:05:46,970 --> 00:05:49,150
12 is divisible by 2, it's
divisible by 3, and it's

114
00:05:49,150 --> 00:05:50,400
divisible by 4.

115
00:05:50,400 --> 00:05:56,480
So 1/2 is the same
thing as 6/12.

116
00:05:56,480 --> 00:05:58,750
2 times 6 is 12.

117
00:05:58,750 --> 00:06:00,420
1 times 6 is 6.

118
00:06:00,420 --> 00:06:01,240
These are equivalent.

119
00:06:01,240 --> 00:06:03,720
6 is 1/2 of 12.

120
00:06:03,720 --> 00:06:09,440
1/3, if we use 12 as a common
denominator, to go from 3 to

121
00:06:09,440 --> 00:06:11,570
12 you have to multiply by 4.

122
00:06:11,570 --> 00:06:14,190
So you take that 4 and
you multiply it by 1.

123
00:06:14,190 --> 00:06:17,620
4/12 is the same thing as 1/3.

124
00:06:17,620 --> 00:06:24,280
And then 1/4, if you use your
denominator 12, to go from 4

125
00:06:24,280 --> 00:06:27,410
to 12 you have to multiply by
3, so multiply the numerator

126
00:06:27,410 --> 00:06:30,080
by 3 as well, you get 3.

127
00:06:30,080 --> 00:06:31,360
So let's add these.

128
00:06:31,360 --> 00:06:36,660
So 6/12 plus 4/12, plus 3/12 is
going to be equal to-- our

129
00:06:36,660 --> 00:06:40,670
denominator's going to be 12--
it's going to be 6 plus 4,

130
00:06:40,670 --> 00:06:47,560
plus 3, which is equal to 6 plus
4 is 10, plus 3 is 13.

131
00:06:47,560 --> 00:06:50,980
So it's going to be
equal to 13/12.

132
00:06:50,980 --> 00:06:53,000
And this is as an improper
fraction.

133
00:06:53,000 --> 00:06:55,950
Or we could say that this is the
same thing, this is equal

134
00:06:55,950 --> 00:07:02,880
to 12/12 plus 1/12, or we could
say the same thing as

135
00:07:02,880 --> 00:07:04,420
12/12 is just 1, right?

136
00:07:04,420 --> 00:07:05,770
12 divided by 12 is 1.

137
00:07:05,770 --> 00:07:10,050
So this is 1 and 1/12.

138
00:07:10,050 --> 00:07:13,950
So when they pool their money,
they get 1 and 1/12 of the

139
00:07:13,950 --> 00:07:19,180
price of the ice cream that
they want to buy.

140
00:07:19,180 --> 00:07:21,480
So they say what fraction of
the cost of ice cream was

141
00:07:21,480 --> 00:07:22,310
added as tax?

142
00:07:22,310 --> 00:07:24,620
This is the exact amount that
they needed to pay.

143
00:07:24,620 --> 00:07:29,740
So clearly, 1 is the non-taxed
price of the ice cream, so

144
00:07:29,740 --> 00:07:32,760
this 1/12 was the amount
added as tax.

145
00:07:32,760 --> 00:07:35,740
So the answer to the question
is 1/12 of the price

146
00:07:35,740 --> 00:07:39,290
was added as tax.

147
00:07:39,290 --> 00:07:39,466