0:00:00.000,0:00:00.330


0:00:00.330,0:00:02.900
Let's add some rational[br]numbers.

0:00:02.900,0:00:05.350
And I'm using that word because[br]that's the word that

0:00:05.350,0:00:08.640
this book uses, but in more[br]popular terminology we'll be

0:00:08.640,0:00:10.480
adding fractions.

0:00:10.480,0:00:14.100
So let's just go through all[br]of these, actually, just to

0:00:14.100,0:00:15.080
see all of the examples.

0:00:15.080,0:00:19.660
So first we're going to[br]have 3/7 plus 2/7.

0:00:19.660,0:00:22.840
Our denominators are the same,[br]so we can just add the

0:00:22.840,0:00:24.070
numerators.

0:00:24.070,0:00:28.480
So our denominator is[br]7, 3 plus 2 is 5.

0:00:28.480,0:00:31.060
That is a.

0:00:31.060,0:00:31.960
Let me do every other.

0:00:31.960,0:00:33.290
It would take forever[br]to do all of them.

0:00:33.290,0:00:36.550
Not forever, but just more time[br]than I want to spend.

0:00:36.550,0:00:42.860
So c is 5/16 plus 5/12.

0:00:42.860,0:00:44.900
Our denominators are[br]not the same.

0:00:44.900,0:00:47.700
We have to find a common[br]denominator, which has to be

0:00:47.700,0:00:50.450
the least common-- it actually[br]could be any common multiple

0:00:50.450,0:00:52.050
of these, but for simplicity[br]let's do the

0:00:52.050,0:00:53.770
least common multiple.

0:00:53.770,0:00:56.150
So what's the smallest number[br]that's a multiple

0:00:56.150,0:00:58.215
of both 16 and 12?

0:00:58.215,0:01:01.700
So let's see, 16 times 2[br]is 32, not there yet.

0:01:01.700,0:01:03.660
Times 3, 48.

0:01:03.660,0:01:04.599
That seems to work.

0:01:04.599,0:01:06.990
12 goes into 48 four times.

0:01:06.990,0:01:09.733
So let's use 48 as our[br]common denominator.

0:01:09.733,0:01:13.960


0:01:13.960,0:01:19.415
So we had to multiply 16 times 3[br]to get to 48, so we're going

0:01:19.415,0:01:23.890
to have to multiply[br]this 5 times 3.

0:01:23.890,0:01:25.670
We're just multiplying the[br]numerator and the denominator

0:01:25.670,0:01:28.090
by the same number, so we're[br]not really changing it.

0:01:28.090,0:01:31.370
So 5 times 3 is 15.

0:01:31.370,0:01:36.850
And then to get from this 12 to[br]this 48 right there, we had

0:01:36.850,0:01:38.890
to multiply times 4.

0:01:38.890,0:01:42.170
So then to get to 5 to this[br]numerator over here, we have

0:01:42.170,0:01:44.120
to multiply times 4.

0:01:44.120,0:01:46.690
5 times 4 is 20.

0:01:46.690,0:01:49.980
Now we have the same[br]denominator.

0:01:49.980,0:01:54.180
So this is going to be equal[br]to, our denominator is 48.

0:01:54.180,0:02:01.150
And so we can add 15 plus[br]20, which is 35.

0:02:01.150,0:02:02.670
And can we reduce this?

0:02:02.670,0:02:04.950
Let's see, 5 does[br]not go into 48.

0:02:04.950,0:02:06.620
7 does not go into 48.

0:02:06.620,0:02:08.330
It looks like we're all done.

0:02:08.330,0:02:13.940
Let's do e right there.

0:02:13.940,0:02:19.790
8/25 plus 7 over 10.

0:02:19.790,0:02:23.570
Once again, we don't have[br]a common denominator.

0:02:23.570,0:02:25.850
But we can solve that.

0:02:25.850,0:02:28.890
Let's make, let's see, 50 is the[br]smallest number that both

0:02:28.890,0:02:29.800
of these go into.

0:02:29.800,0:02:32.340
25 times 2, so that's 50.

0:02:32.340,0:02:37.050
8 over 25, to go to 50[br]we multiply by 2.

0:02:37.050,0:02:39.990
So the 8, we're going to[br]have to multiply by 2.

0:02:39.990,0:02:42.640
So it's going to[br]be 16 over 50.

0:02:42.640,0:02:45.945
And then the 7 over 10,[br]we're going to want

0:02:45.945,0:02:47.930
to put it over 50.

0:02:47.930,0:02:51.750
We multiply the 10 times[br]5, so we have to

0:02:51.750,0:02:54.605
multiply the 7 times 5.

0:02:54.605,0:02:57.720
So it's going to[br]be 35 over 50.

0:02:57.720,0:03:01.560
Now that our denominators are[br]the same, we have it over 50.

0:03:01.560,0:03:05.550
16 plus 35, what is that?

0:03:05.550,0:03:10.690
10 plus 35 is 45,[br]plus 6 is 51.

0:03:10.690,0:03:14.770
So it is 51 over 50.

0:03:14.770,0:03:16.992
Problem g.

0:03:16.992,0:03:19.700
Let me do it in a new color.

0:03:19.700,0:03:22.410
Problem g.

0:03:22.410,0:03:28.470
So here we have 7 over 15-- I'll[br]write the second one in a

0:03:28.470,0:03:33.530
different color--[br]plus 2 over 9.

0:03:33.530,0:03:35.620
Once again, the denominators[br]are different.

0:03:35.620,0:03:37.490
Find a common denominator.

0:03:37.490,0:03:41.540
What is the smallest number that[br]both 15 and 9 go into?

0:03:41.540,0:03:43.260
Let's see, 15 times 2 is 30.

0:03:43.260,0:03:44.940
Nope, not divisible by 9.

0:03:44.940,0:03:47.670
15 times 3 is 45, that works.

0:03:47.670,0:03:50.220
45 is divisible by 9.

0:03:50.220,0:03:52.590
So we use 45.

0:03:52.590,0:03:59.810
15 times 3 is 45, so[br]7 times 3 is 21.

0:03:59.810,0:04:02.850
These two fractions[br]are equivalent.

0:04:02.850,0:04:06.680
Plus we're going over 45.

0:04:06.680,0:04:11.520
To get from 9 to 45, we have[br]to multiply times 5.

0:04:11.520,0:04:14.420
So to get our numerator[br]over here, we have to

0:04:14.420,0:04:15.980
multiply it times 5.

0:04:15.980,0:04:18.420
So 2 times 5 is 10.

0:04:18.420,0:04:22.422
2/9 is the same thing[br]as 10/45.

0:04:22.422,0:04:24.710
So now we can add.

0:04:24.710,0:04:27.130
We're adding fractions of 45.

0:04:27.130,0:04:33.130
21 plus 10 is 31,[br]and we are done.

0:04:33.130,0:04:36.900
Let's do one more problem down[br]here, a word problem.

0:04:36.900,0:04:40.070
Nadia, Peter and Ian are pooling[br]their money to buy a

0:04:40.070,0:04:41.640
gallon of ice cream.

0:04:41.640,0:04:44.630
Nadia's the oldest and gets[br]the greatest allowance.

0:04:44.630,0:04:49.740
She contributes 1/2 the cost.[br]So Nadia is contributing 1/2

0:04:49.740,0:04:53.750
the cost. So that is[br]Nadia right there.

0:04:53.750,0:04:58.850
Ian is next oldest and[br]contributes 1/3 of the cost.

0:04:58.850,0:05:02.280
So Ian contributes 1/3.

0:05:02.280,0:05:03.820
That is Ian.

0:05:03.820,0:05:06.360
Peter, the youngest, gets the[br]smallest allowance and

0:05:06.360,0:05:13.730
contributes 1/4 of the cost.[br]So Peter gives 1/4 of the

0:05:13.730,0:05:17.560
cost. Peter contributes[br]1/4 of cost.

0:05:17.560,0:05:19.920
They figure that this will[br]be enough money.

0:05:19.920,0:05:22.480
When they get to the checkout,[br]they realize that they forgot

0:05:22.480,0:05:24.000
about sales tax and[br]worry there will

0:05:24.000,0:05:25.340
not be enough money.

0:05:25.340,0:05:28.370
Amazingly, they have exactly[br]the right amount of money.

0:05:28.370,0:05:32.460
What fraction of the cost of[br]ice cream was added as tax?

0:05:32.460,0:05:35.640
Well, let's see, if we add 1/2[br]plus 1/3, plus 1/4 of the

0:05:35.640,0:05:37.640
cost, let's see what we get.

0:05:37.640,0:05:41.100
So we have to find a common[br]denominator, some number that

0:05:41.100,0:05:44.250
is the least common multiple[br]of 2, 3, and 4.

0:05:44.250,0:05:46.970
And let's see, 4, it would[br]have to be 12, right?

0:05:46.970,0:05:49.150
12 is divisible by 2, it's[br]divisible by 3, and it's

0:05:49.150,0:05:50.400
divisible by 4.

0:05:50.400,0:05:56.480
So 1/2 is the same[br]thing as 6/12.

0:05:56.480,0:05:58.750
2 times 6 is 12.

0:05:58.750,0:06:00.420
1 times 6 is 6.

0:06:00.420,0:06:01.240
These are equivalent.

0:06:01.240,0:06:03.720
6 is 1/2 of 12.

0:06:03.720,0:06:09.440
1/3, if we use 12 as a common[br]denominator, to go from 3 to

0:06:09.440,0:06:11.570
12 you have to multiply by 4.

0:06:11.570,0:06:14.190
So you take that 4 and[br]you multiply it by 1.

0:06:14.190,0:06:17.620
4/12 is the same thing as 1/3.

0:06:17.620,0:06:24.280
And then 1/4, if you use your[br]denominator 12, to go from 4

0:06:24.280,0:06:27.410
to 12 you have to multiply by[br]3, so multiply the numerator

0:06:27.410,0:06:30.080
by 3 as well, you get 3.

0:06:30.080,0:06:31.360
So let's add these.

0:06:31.360,0:06:36.660
So 6/12 plus 4/12, plus 3/12 is[br]going to be equal to-- our

0:06:36.660,0:06:40.670
denominator's going to be 12--[br]it's going to be 6 plus 4,

0:06:40.670,0:06:47.560
plus 3, which is equal to 6 plus[br]4 is 10, plus 3 is 13.

0:06:47.560,0:06:50.980
So it's going to be[br]equal to 13/12.

0:06:50.980,0:06:53.000
And this is as an improper[br]fraction.

0:06:53.000,0:06:55.950
Or we could say that this is the[br]same thing, this is equal

0:06:55.950,0:07:02.880
to 12/12 plus 1/12, or we could[br]say the same thing as

0:07:02.880,0:07:04.420
12/12 is just 1, right?

0:07:04.420,0:07:05.770
12 divided by 12 is 1.

0:07:05.770,0:07:10.050
So this is 1 and 1/12.

0:07:10.050,0:07:13.950
So when they pool their money,[br]they get 1 and 1/12 of the

0:07:13.950,0:07:19.180
price of the ice cream that[br]they want to buy.

0:07:19.180,0:07:21.480
So they say what fraction of[br]the cost of ice cream was

0:07:21.480,0:07:22.310
added as tax?

0:07:22.310,0:07:24.620
This is the exact amount that[br]they needed to pay.

0:07:24.620,0:07:29.740
So clearly, 1 is the non-taxed[br]price of the ice cream, so

0:07:29.740,0:07:32.760
this 1/12 was the amount[br]added as tax.

0:07:32.760,0:07:35.740
So the answer to the question[br]is 1/12 of the price

0:07:35.740,0:07:39.290
was added as tax.

0:07:39.290,0:07:39.466