Level 4 Subtraction

WEBVTT
Kind: captions
Language: en

00:00:01.090 --> 00:00:04.020
Welcome to the presentation
on level 4 subtraction.

00:00:04.020 --> 00:00:06.640
Let's get started
with some problems.

00:00:06.640 --> 00:00:22.970
First problem I have here
is 33,220 minus 399.

00:00:22.970 --> 00:00:26.250
So this like we did with, I
believe, that we also did

00:00:26.250 --> 00:00:29.140
borrowing in the level
subtraction, we have to go

00:00:29.140 --> 00:00:31.920
through all of the digits in
the top number starting with

00:00:31.920 --> 00:00:36.220
the top right digit and make
sure that they are larger

00:00:36.220 --> 00:00:37.320
than the digit below it.

00:00:37.320 --> 00:00:42.080
Because you can only attract
a smaller number from

00:00:42.080 --> 00:00:42.710
a larger number.

00:00:42.710 --> 00:00:44.630
You can't do it the other
way, at least, until we

00:00:44.630 --> 00:00:45.850
learn negative numbers.

00:00:45.850 --> 00:00:47.710
So let's go through this and
check to make sure all the

00:00:47.710 --> 00:00:48.660
top numbers are larger.

00:00:48.660 --> 00:00:49.640
Well immediately, we see no.

00:00:49.640 --> 00:00:52.650
Well this 0 is not
larger than 9.

00:00:52.650 --> 00:00:56.300
So we have to borrow
to make the 0 bigger.

00:00:56.300 --> 00:01:00.170
So what we do is we borrow
1-- well some people say

00:01:00.170 --> 00:01:01.010
we're borrowing a 1.

00:01:01.010 --> 00:01:03.450
Some people say, I mean,
borrowing a 1 from the

00:01:03.450 --> 00:01:06.535
10's place is really
like borrowing a 10.

00:01:06.535 --> 00:01:09.840
So for simplicity let's just
say we're borrowing a 1.

00:01:09.840 --> 00:01:15.710
So you borrow a 1 from this 2
and this 10 will become a-- the

00:01:15.710 --> 00:01:18.580
0 sorry, will become a 10.

00:01:18.580 --> 00:01:23.040
And since we borrowed that
1, this 2 will become a 1.

00:01:23.040 --> 00:01:26.940
We took 1 away from this 2 and
we gave it to the 0 to make 10.

00:01:26.940 --> 00:01:29.560
We actually took 10 away
from this 2 because this

00:01:29.560 --> 00:01:30.230
2's in the 10's place.

00:01:30.230 --> 00:01:32.230
I don't want to
confuse you too much.

00:01:32.230 --> 00:01:35.600
If just the mechanics of it are
we took 1 away from the 2 and

00:01:35.600 --> 00:01:39.410
we put it in front of
the 0 to make 10.

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Now let's keep checking.

00:01:40.320 --> 00:01:42.400
So now we have a
1 in this place.

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1 is smaller than 9, so
we have to borrow again.

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So we borrow 1 from this 2 now.

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So this 2 now becomes 1.

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And this 1 will
now become an 11.

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So now we have a 10
is larger than a 9.

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11 is larger than a 9.

00:02:02.250 --> 00:02:06.810
1 is not larger than 3, so
we have to borrow again.

00:02:06.810 --> 00:02:08.920
This is a good problem.

00:02:08.920 --> 00:02:10.360
Maybe I should've warmed you
all up a little bit more.

00:02:10.360 --> 00:02:11.540
It involves a lot of borrowing.

00:02:11.540 --> 00:02:13.790
So in order to borrow we do
the same thing over again.

00:02:13.790 --> 00:02:17.630
This 1 will become 11.

00:02:17.630 --> 00:02:21.082
And it's going to borrow from
this 3, which will become a 2.

00:02:21.082 --> 00:02:22.270
I think we're done now.

00:02:22.270 --> 00:02:25.900
10 is larger than 9, 11 is
larger than 9, 11 is larger

00:02:25.900 --> 00:02:28.620
than 3, 2 is larger than
nothing, 3 is larger

00:02:28.620 --> 00:02:29.800
than nothing.

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So now we're ready to subtract.

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This is the easy part.

00:02:32.550 --> 00:02:35.220
10 minus 9 is 1.

00:02:35.220 --> 00:02:39.230
11 minus 9 is 2.

00:02:39.230 --> 00:02:45.950
11 minus 3 is 8.

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2 minus nothing is 2.

00:02:48.560 --> 00:02:51.160
3 minus nothing is 3.

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So we get 32,821.

00:02:56.930 --> 00:02:59.480
The only thing that makes this
harder than just normal

00:02:59.480 --> 00:03:01.470
subtraction is that you have to
know how to do the borrowing.

00:03:01.470 --> 00:03:03.600
And the way I do the borrowing
might be different than the way

00:03:03.600 --> 00:03:05.490
you learned in school, but I
think it's easier because you

00:03:05.490 --> 00:03:08.680
do all of the borrowing at once
instead of switching back and

00:03:08.680 --> 00:03:11.250
forth between borrowing
and subtracting.

00:03:11.250 --> 00:03:14.040
So all we did here, we said
that 0 is less than 9.

00:03:14.040 --> 00:03:15.320
Let's borrow 1.

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The 0 becomes a 10 because
we got this 1 right here.

00:03:19.100 --> 00:03:22.630
We got this 1 from this 2
and this 2 became a 1.

00:03:22.630 --> 00:03:25.390
I think you might see the
pattern if we do a couple

00:03:25.390 --> 00:03:26.460
of more problems.

00:03:26.460 --> 00:03:29.370
So let's do a couple of more.

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If I had 25,633 minus 578.

00:03:47.571 --> 00:03:49.860
So name drill.

00:03:49.860 --> 00:03:52.970
Start at the top right and we
make sure that the digits on

00:03:52.970 --> 00:03:55.630
top are larger than
the digit below it.

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Immediately we see 3 is smaller
than 8, so we have to borrow.

00:04:00.530 --> 00:04:02.445
So this 3 will become 13.

00:04:05.820 --> 00:04:11.430
And then we borrow from this
3, which will now become 2.

00:04:11.430 --> 00:04:13.695
We took a 1 away from this
3, it became a 2, and

00:04:13.695 --> 00:04:16.250
this 1 is right here.

00:04:16.250 --> 00:04:19.790
13 is now larger than 8, but
2 is now smaller than 7.

00:04:19.790 --> 00:04:21.120
So we have to borrow again.

00:04:21.120 --> 00:04:24.640
This 2 becomes a 12.

00:04:24.640 --> 00:04:29.290
And this 6 will become a 5.

00:04:29.290 --> 00:04:33.920
13 is larger than 8, 12 is
larger than 7, 5 is the same

00:04:33.920 --> 00:04:37.420
as 5, so you can actually
do the subtraction.

00:04:37.420 --> 00:04:39.560
Because 5 minus 5 is 0.

00:04:39.560 --> 00:04:41.430
As long as the top number's
not smaller than the

00:04:41.430 --> 00:04:42.180
number below it.

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And then obviously this 5
is larger than this 0 and

00:04:44.290 --> 00:04:46.740
this 2 is larger than
this nothing here.

00:04:46.740 --> 00:04:48.450
So now we're ready to subtract.

00:04:48.450 --> 00:04:53.890
13 minus 8 is 5.

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12 minus 7 is 5.

00:04:58.320 --> 00:05:01.530
5 minus 5 is 0.

00:05:01.530 --> 00:05:04.780
5 minus nothing is 5.

00:05:04.780 --> 00:05:05.710
Bring down the 2.

00:05:05.710 --> 00:05:13.090
So the answer is 25,055.

00:05:13.090 --> 00:05:16.470
So let's do a problem now that
I think will confuse you a

00:05:16.470 --> 00:05:19.900
little bit more because the
borrowing isn't as easy.

00:05:19.900 --> 00:05:22.480
You have to actually borrow
from a couple places.

00:05:22.480 --> 00:05:39.800
Let's say I had 37,002
minus-- let's say I had 155.

00:05:39.800 --> 00:05:42.090
So the same drill.

00:05:42.090 --> 00:05:45.130
So this 2 is less than 5,
so we have to borrow.

00:05:45.130 --> 00:05:48.820
So this 2 will become a 12.

00:05:48.820 --> 00:05:51.930
But huh, there's a 0 here, so
you can't borrow from the 0.

00:05:51.930 --> 00:05:53.940
Some people will let you borrow
from the 0, but I think that

00:05:53.940 --> 00:05:56.103
just confuses things because
you can't borrow from the

00:05:56.103 --> 00:05:57.450
0, there's nothing there.

00:05:57.450 --> 00:05:59.710
So instead of borrowing from
the 0 we look to this 0.

00:05:59.710 --> 00:06:01.470
Well, there's still
nothing there.

00:06:01.470 --> 00:06:04.250
So now we look, oh,
there's a 7 here.

00:06:04.250 --> 00:06:07.620
So what we do is instead of
borrowing a 1 from the 0, which

00:06:07.620 --> 00:06:15.490
is hard to do, we borrow 1 from
this 700, from this whole 700.

00:06:15.490 --> 00:06:18.330
And what is 700 minus 1?

00:06:18.330 --> 00:06:19.480
Right, it's 699.

00:06:19.480 --> 00:06:26.710
So that 700 becomes 699.

00:06:26.710 --> 00:06:28.770
Cross all of this out.

00:06:28.770 --> 00:06:31.320
And now let's check
our numbers again.

00:06:31.320 --> 00:06:36.870
12 is larger than 5, nine is
larger than 5, 9 is larger than

00:06:36.870 --> 00:06:40.500
1, 6 is larger than nothing,
and 3 is larger than nothing,

00:06:40.500 --> 00:06:42.410
so we're ready to subtract.

00:06:42.410 --> 00:06:45.670
12 minus 5 is 7.

00:06:45.670 --> 00:06:48.730
9 minus 5 is 4.

00:06:48.730 --> 00:06:51.606
9 minus 1 is 8.

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6 minus nothing is 6.

00:06:54.180 --> 00:06:55.795
3 minus nothing is 3.

00:06:55.795 --> 00:06:57.630
So there, we're done.

00:06:57.630 --> 00:07:02.100
The answer is 36,847.

00:07:02.100 --> 00:07:05.851
I think we could have time
for one more problem.

00:07:05.851 --> 00:07:21.870
Let's say I had 3,201
minus-- let's say it's 502.

00:07:21.870 --> 00:07:22.950
Same drill.

00:07:22.950 --> 00:07:27.040
1 is less than 2, so
we have to borrow.

00:07:27.040 --> 00:07:28.370
Turn that into an 11.

00:07:28.370 --> 00:07:30.380
But you can't borrow from the
0, so you're going to have to

00:07:30.380 --> 00:07:33.000
borrow from this entire 20.

00:07:33.000 --> 00:07:34.800
Well, what's 20 minus 1?

00:07:34.800 --> 00:07:37.060
Right, it's 19.

00:07:37.060 --> 00:07:38.080
This becomes a 19.

00:07:40.660 --> 00:07:41.660
So let's check again.

00:07:41.660 --> 00:07:43.480
11 is greater than 2.

00:07:43.480 --> 00:07:44.320
Check.

00:07:44.320 --> 00:07:45.570
9 is greater than 0.

00:07:45.570 --> 00:07:46.620
Check.

00:07:46.620 --> 00:07:46.970
Uh-oh.

00:07:46.970 --> 00:07:49.310
1 is not greater than 5.

00:07:49.310 --> 00:07:50.840
So we have to borrow again.

00:07:50.840 --> 00:07:53.000
This 1 becomes an 11.

00:07:53.000 --> 00:07:56.450
We borrowed from this
3, which becomes a 2.

00:07:56.450 --> 00:08:00.630
11 is greater than 2, 9 is
greater than 0, 11 is greater

00:08:00.630 --> 00:08:04.200
than 5, 2 is obviously greater
than nothing below it.

00:08:04.200 --> 00:08:06.100
So we're ready to subtract.

00:08:06.100 --> 00:08:09.130
11 minus 2 is 9.

00:08:09.130 --> 00:08:12.020
9 minus 0 is 9.

00:08:12.020 --> 00:08:14.470
11 minus 5 is 6.

00:08:14.470 --> 00:08:17.590
And 2 minus nothing is 2.

00:08:17.590 --> 00:08:26.480
So 3,201 minus 502
is equal to 2,699.

00:08:26.480 --> 00:08:28.340
I think you're now ready to
try some of the level 4

00:08:28.340 --> 00:08:29.110
subtraction problems.

00:08:29.110 --> 00:08:31.530
You just always have to
remember, do you're

00:08:31.530 --> 00:08:32.950
borrowing first.

00:08:32.950 --> 00:08:36.030
Make sure all the numbers on
top are larger than, or at

00:08:36.030 --> 00:08:38.620
least as large as all the
numbers on the bottom.

00:08:38.620 --> 00:08:40.720
And then you can just do your
subtraction like a normal

00:08:40.720 --> 00:08:42.010
subtraction problem.

00:08:42.010 --> 00:08:44.500
I hope you have some
fun doing this.

00:08:44.500 --> 00:08:46.090
Talk to you later.


Subtracting decimals (old)

WEBVTT
Kind: captions
Language: en

00:00:01.060 --> 00:00:04.490
Welcome to the presentation on
subtracting decimal numbers.

00:00:04.490 --> 00:00:07.390
Let's get started
with some problems.

00:00:07.390 --> 00:00:23.470
The first problem I have
here says 5.73 minus

00:00:23.470 --> 00:00:30.670
0.0821 equals who knows?

00:00:30.670 --> 00:00:33.260
So the first thing you always
want to do with a decimal like

00:00:33.260 --> 00:00:35.680
this, and I actually kind of
inadvertently did this, is that

00:00:35.680 --> 00:00:37.920
you want to line
up the decimals.

00:00:37.920 --> 00:00:40.250
so you actually want this
decimal to be right

00:00:40.250 --> 00:00:41.540
above this decimal.

00:00:41.540 --> 00:00:43.870
I almost did that when I did
it, it must have been my

00:00:43.870 --> 00:00:44.870
subconscious doing it.

00:00:44.870 --> 00:00:46.290
But let me just do it
a little bit neater.

00:00:46.290 --> 00:00:52.630
So it's 5.73, and I'll
put the decimal here.

00:00:55.840 --> 00:01:00.060
0.0821.

00:01:00.060 --> 00:01:02.580
And some people say it's good
to always put a zero in

00:01:02.580 --> 00:01:03.350
front of the decimal.

00:01:03.350 --> 00:01:05.600
My wife's a doctor and she
says it's critical otherwise

00:01:05.600 --> 00:01:08.310
you might give someone the
wrong amount of medicine.

00:01:08.310 --> 00:01:10.510
So, we've lined up the
decimals and now we're

00:01:10.510 --> 00:01:11.175
ready to subtract.

00:01:13.960 --> 00:01:16.150
So one thing that you have
to think about when you do

00:01:16.150 --> 00:01:20.830
decimals is we're going to
have to subtract this 21

00:01:20.830 --> 00:01:23.890
ten-thousandths or this 2
and this 1 from something.

00:01:23.890 --> 00:01:26.470
We can't just subtract it
from this blank space.

00:01:26.470 --> 00:01:30.910
So we have to add two 0s here.

00:01:30.910 --> 00:01:33.270
And as you know, with the
decimal when you add 0s to the

00:01:33.270 --> 00:01:34.870
end of it, it really doesn't
change the value

00:01:34.870 --> 00:01:35.980
of the decimal.

00:01:35.980 --> 00:01:38.650
So at this point, we just
view this like a level

00:01:38.650 --> 00:01:40.660
four subtraction problem.

00:01:40.660 --> 00:01:44.610
So the first thing we do in any
subtraction problem is see if

00:01:44.610 --> 00:01:46.970
any of the numbers on top are
smaller than any of the

00:01:46.970 --> 00:01:47.770
numbers on the bottom.

00:01:47.770 --> 00:01:49.350
Well in this case there
are a lot of them.

00:01:49.350 --> 00:01:53.040
So this 0 is less than this 1,
this 0 is less than this 2,

00:01:53.040 --> 00:01:54.410
this 3 is less than this 8.

00:01:54.410 --> 00:01:56.780
So we're going to
have to borrow.

00:01:56.780 --> 00:01:59.990
Some people will like to
do their borrowing and

00:01:59.990 --> 00:02:02.250
subtracting, they kind of
alternate between the two.

00:02:02.250 --> 00:02:04.510
I like to do all of my
borrowing ahead of time.

00:02:04.510 --> 00:02:06.540
So what I do is I start in
the top right and I say

00:02:06.540 --> 00:02:09.990
OK, 0 is less than 1.

00:02:09.990 --> 00:02:13.670
So that 0 becomes a 10.

00:02:13.670 --> 00:02:15.710
But in order to become a
10 I would have had to

00:02:15.710 --> 00:02:17.620
borrow 1 from some place.

00:02:17.620 --> 00:02:19.580
I look to the left of that
0 and I say well, can

00:02:19.580 --> 00:02:21.500
I borrow the 1 from 0?

00:02:21.500 --> 00:02:22.570
Well, no.

00:02:22.570 --> 00:02:23.440
This is just the way I do it.

00:02:23.440 --> 00:02:25.610
There are people who would
actually let you borrow the 1

00:02:25.610 --> 00:02:27.630
from the 0, but I say no,
instead of borrowing the 1

00:02:27.630 --> 00:02:30.790
from the 0, I borrow the
1 from this entire 30.

00:02:30.790 --> 00:02:34.470
So this 30 -- see, there's a
30 so I'm going to borrow 1

00:02:34.470 --> 00:02:36.650
from it and it becomes 29.

00:02:39.590 --> 00:02:45.710
So we borrowed 1 from this 30
to get a 10 here, and now let's

00:02:45.710 --> 00:02:49.700
check again to see if all of
our numbers on top are larger

00:02:49.700 --> 00:02:50.900
than all the numbers
on the bottom.

00:02:50.900 --> 00:02:56.550
Well 10 is larger than 1,
9 is larger than 2, 2

00:02:56.550 --> 00:02:58.090
is not larger than 8.

00:02:58.090 --> 00:02:59.230
So we have to borrow again.

00:02:59.230 --> 00:03:05.040
So if we're going to borrow,
the 2 becomes a 12, and the 7

00:03:05.040 --> 00:03:08.100
-- we borrowed 1 from
that -- becomes a 6.

00:03:08.100 --> 00:03:08.970
So let's check again.

00:03:08.970 --> 00:03:13.500
10 is larger than 1, 9 is
larger than 2, 12 is larger

00:03:13.500 --> 00:03:17.280
than 8, 6 is larger than 0,
and 5 is larger than 0.

00:03:17.280 --> 00:03:19.330
So now we've done all of our
borrowing and we're ready to

00:03:19.330 --> 00:03:22.010
do some subtraction, and
this is the easy part.

00:03:22.010 --> 00:03:25.630
10 minus 1 is 9.

00:03:25.630 --> 00:03:29.460
9 minus 2 is 7.

00:03:29.460 --> 00:03:33.370
12 minus 8 is 4.

00:03:33.370 --> 00:03:36.420
6 minus 0 is 6.

00:03:36.420 --> 00:03:39.920
5 minus 0 is 5.

00:03:39.920 --> 00:03:42.640
And we just bring down
the decimal point.

00:03:42.640 --> 00:03:43.850
So there's our answer.

00:03:43.850 --> 00:03:55.130
5.73 minus 0.0821 is
equal to 5.6479.

00:03:55.130 --> 00:03:56.530
There you go.

00:03:56.530 --> 00:03:58.655
I probably confused you, so
let's do some more problems.

00:04:01.880 --> 00:04:05.260
Here's another one.

00:04:05.260 --> 00:04:09.795
8 -- let me leave some space
on top to do the borrowing

00:04:09.795 --> 00:04:25.880
-- 8.25 minus 0.0105.

00:04:25.880 --> 00:04:29.100
So what was that first step
that I always have to do?

00:04:29.100 --> 00:04:30.030
Right.

00:04:30.030 --> 00:04:31.190
To line up the decimals.

00:04:31.190 --> 00:04:32.720
So let me do that.

00:04:32.720 --> 00:04:46.080
So it's 8.25 and 0.0105.

00:04:46.080 --> 00:04:50.300
Notice I lined up this decimal
right below this decimal.

00:04:50.300 --> 00:04:54.330
Now I add the 0s, just because
this 0 and this 5 need to be

00:04:54.330 --> 00:04:56.040
subtracted from something.

00:04:56.040 --> 00:04:57.630
Now let me do my borrowing.

00:04:57.630 --> 00:05:00.190
So once again, all I do is
check to see whether the

00:05:00.190 --> 00:05:03.180
top number is larger than
the number below it.

00:05:03.180 --> 00:05:08.550
Well, this 0 is smaller than 5,
so I'm going to have to borrow.

00:05:08.550 --> 00:05:09.350
So I'm going to borrow.

00:05:09.350 --> 00:05:11.400
I can't borrow from this
0, I have to borrow

00:05:11.400 --> 00:05:12.990
from this entire 50.

00:05:12.990 --> 00:05:16.900
So this 50, if I borrow
1 from 50 I get 49.

00:05:19.820 --> 00:05:22.840
And this 0 will then
become a 10, right?

00:05:22.840 --> 00:05:26.060
I borrowed 1 from
50 to get a 10.

00:05:26.060 --> 00:05:27.560
Now, am I done?

00:05:27.560 --> 00:05:29.440
10 is larger than 5.

00:05:29.440 --> 00:05:31.550
9 is larger than 0.

00:05:31.550 --> 00:05:33.970
4 is larger than 1.

00:05:33.970 --> 00:05:35.910
2 is larger than 0.

00:05:35.910 --> 00:05:37.530
8 is larger than 0.

00:05:37.530 --> 00:05:39.610
So I think I'm
ready to subtract.

00:05:39.610 --> 00:05:44.130
10 minus 5, well that's 5.

00:05:44.130 --> 00:05:47.240
9 minus 0 is 9.

00:05:47.240 --> 00:05:51.140
4 minus 1 is 3.

00:05:51.140 --> 00:05:54.710
2 minus 0 is 2.

00:05:54.710 --> 00:05:57.960
8 minus 0 is 8.

00:05:57.960 --> 00:06:01.730
And I bring down
the decimal point.

00:06:01.730 --> 00:06:05.690
So if you mastered level four
subtraction, the decimal

00:06:05.690 --> 00:06:08.450
problems really are just about
lining up the decimal point,

00:06:08.450 --> 00:06:10.060
adding the 0s and then
just doing a normal

00:06:10.060 --> 00:06:11.440
subtraction problem.

00:06:11.440 --> 00:06:14.400
In general with subtraction I
think most people have the most

00:06:14.400 --> 00:06:17.260
trouble with the borrowing.

00:06:17.260 --> 00:06:19.230
The way I do it I think is a
little bit different than is

00:06:19.230 --> 00:06:20.100
taught in a lot of schools.

00:06:20.100 --> 00:06:22.140
A lot of schools they'll do
the subtraction and they'll

00:06:22.140 --> 00:06:23.670
borrow alternatively.

00:06:23.670 --> 00:06:27.140
But I find this easier when I
just borrow ahead of time, and

00:06:27.140 --> 00:06:30.320
I also, like for example in
this problem, when I had to

00:06:30.320 --> 00:06:32.930
make this 0 into a 10, instead
of borrowing from the 0, which

00:06:32.930 --> 00:06:35.820
is not intuitive because I
can't really borrow from the 0,

00:06:35.820 --> 00:06:40.720
I borrowed from this entire 50
and I made that into a 49.

00:06:40.720 --> 00:06:42.040
Let's do one more problem.

00:06:47.000 --> 00:06:58.500
If I have 2.64 minus 0.0486.

00:06:58.500 --> 00:07:01.670
So once again, let's line
up the decimal points.

00:07:01.670 --> 00:07:11.720
2.64 and it's .0486.

00:07:11.720 --> 00:07:16.620
Lined up the decimal
points, the 0s on top.

00:07:16.620 --> 00:07:19.380
You're going to have a 0
here, so I have to borrow.

00:07:19.380 --> 00:07:20.490
Becomes a 10.

00:07:20.490 --> 00:07:22.260
Can't borrow from the
0, so I have to borrow

00:07:22.260 --> 00:07:23.920
from this entire 40.

00:07:23.920 --> 00:07:27.166
So this 40 becomes a 39.

00:07:27.166 --> 00:07:29.440
I think I'm running
out of space.

00:07:29.440 --> 00:07:31.450
So 10 is larger than 6.

00:07:31.450 --> 00:07:33.020
9 is larger than 8.

00:07:33.020 --> 00:07:34.420
3 is not larger than 4.

00:07:34.420 --> 00:07:36.750
So this 3 I'm going
to have to borrow.

00:07:36.750 --> 00:07:38.480
So 3 becomes a 13.

00:07:38.480 --> 00:07:41.000
I apologize, I'm
becoming scrunched.

00:07:41.000 --> 00:07:43.396
And this 6 becomes a 5.

00:07:43.396 --> 00:07:46.040
This is really bad, I
shouldn't do it so messy.

00:07:46.040 --> 00:07:49.040
But now we say the 10 is larger
than 6, the 9 is larger than

00:07:49.040 --> 00:07:52.560
the 8, this 13, this 13
should be on top of that 3.

00:07:52.560 --> 00:07:55.550
The 13 is larger than 4,
and 5 is larger than 0.

00:07:55.550 --> 00:07:57.120
So we're ready to subtract.

00:07:57.120 --> 00:07:59.270
10 minus 6 is 4.

00:07:59.270 --> 00:08:01.590
9 minus 8 is 1.

00:08:01.590 --> 00:08:04.320
13 minus 4 is 9.

00:08:04.320 --> 00:08:07.800
5 minus 0 is 5.

00:08:07.800 --> 00:08:09.880
2 minus nothing is 2.

00:08:09.880 --> 00:08:12.180
Bring down the decimal point.

00:08:12.180 --> 00:08:21.530
So 2.64 minus 0.0486
is equal to 2.5914.

00:08:21.530 --> 00:08:23.200
Hope I didn't confuse
you too much.

00:08:23.200 --> 00:08:26.740
But I think you're
ready now to try the

00:08:26.740 --> 00:08:28.680
subtraction of decimals.

00:08:28.680 --> 00:08:30.250
Have fun.


Unit conversion within the metric system

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:03.310
Welcome to the
presentation on units.

00:00:03.310 --> 00:00:05.390
Let's get started.

00:00:05.390 --> 00:00:12.770
So if I were to ask you, or if
I were to say, I have traveled

00:00:12.770 --> 00:00:20.910
0.05 kilometers-- some
people say KIL-ometers

00:00:20.910 --> 00:00:22.020
or kil-O-meters.

00:00:24.850 --> 00:00:28.250
If I have traveled 0.05
kilometers, how many

00:00:28.250 --> 00:00:30.865
centimeters have I traveled?

00:00:30.865 --> 00:00:32.590
That's question
mark centimeters.

00:00:35.880 --> 00:00:38.390
So before we break into the
math, it's important to just

00:00:38.390 --> 00:00:41.730
know what these prefixes
centi and kilo mean.

00:00:41.730 --> 00:00:44.650
And it's good to memorize this,
or when you're first starting

00:00:44.650 --> 00:00:46.130
to do these problems, you can
just write them down on a piece

00:00:46.130 --> 00:00:48.080
of paper, just so you
have a reference.

00:00:48.080 --> 00:01:06.460
So kilo means 1,000, hecto
means 100, deca means 10.

00:01:06.460 --> 00:01:09.840
You might recognize that
from decade, 10 years.

00:01:09.840 --> 00:01:13.980
And then, of course, you
have no prefix, means 1.

00:01:13.980 --> 00:01:15.580
No prefix.

00:01:15.580 --> 00:01:18.710
No prefix equals 1.

00:01:18.710 --> 00:01:28.100
deci is equal to 0.1 or 1/10.

00:01:28.100 --> 00:01:32.510
centi-- I keep changing
between cases.

00:01:32.510 --> 00:01:38.590
centi is equal to
0.01, or 1/100.

00:01:38.590 --> 00:01:45.110
And then milli is equal
to 0.001, and that's the

00:01:45.110 --> 00:01:48.710
same thing as 1/1,000.

00:01:48.710 --> 00:01:52.810
And the way I remember, I mean,
centi, if you think of a

00:01:52.810 --> 00:01:54.740
centipede, it has a 100 feet.

00:01:54.740 --> 00:01:58.360
A millipede, I'm not sure if a
millipede has 1,000 feet, but

00:01:58.360 --> 00:02:00.850
that's the implication when
someone says a millipede

00:02:00.850 --> 00:02:02.550
because pede means feet.

00:02:02.550 --> 00:02:03.810
So let's go back
to the problem.

00:02:03.810 --> 00:02:08.490
If I have 0.05 kilometers, how
many centimeters do I have?

00:02:08.490 --> 00:02:10.350
Whenever I do a problem like
this, I like to actually

00:02:10.350 --> 00:02:12.930
convert my number to
meters because that's

00:02:12.930 --> 00:02:14.290
very easy for me.

00:02:14.290 --> 00:02:18.150
And actually, I'm going to
abbreviate this is km, and

00:02:18.150 --> 00:02:21.530
we can abbreviate this
as cm for centimeters.

00:02:21.530 --> 00:02:28.480
So let's say 0.05 km.

00:02:28.480 --> 00:02:32.500
Well, if I want to convert this
into meters, is it going to be

00:02:32.500 --> 00:02:37.150
more than 0.05 meters
or less than 0.05?

00:02:37.150 --> 00:02:40.820
Well, a kilometer is a very
large distance, so in terms

00:02:40.820 --> 00:02:43.430
of meters, it's going to
be a much bigger number.

00:02:43.430 --> 00:02:52.600
So we can multiply this times
1,000 meters, and I'll do

00:02:52.600 --> 00:02:53.880
it over 1, per kilometer.

00:02:56.490 --> 00:02:58.050
And what does that get?

00:02:58.050 --> 00:03:04.890
Well, 0.05 times 1,000
is equal to 50, right?

00:03:04.890 --> 00:03:07.670
I just multiplied
0.05 times 1,000.

00:03:07.670 --> 00:03:12.610
And with the units, I now
have kilometers times

00:03:12.610 --> 00:03:16.260
meters over kilometers.

00:03:16.260 --> 00:03:18.290
And the kilometers cancel out.

00:03:18.290 --> 00:03:22.260
And just so you're familiar
with this, you can treat units

00:03:22.260 --> 00:03:24.640
exactly the same way that you
would treat numbers

00:03:24.640 --> 00:03:25.670
or variables.

00:03:25.670 --> 00:03:28.990
As long as you have the same
unit in the numerator and the

00:03:28.990 --> 00:03:30.970
denominator, you can cancel
them out, assuming that you're

00:03:30.970 --> 00:03:33.490
not adding units, you're
multiplying units.

00:03:33.490 --> 00:03:36.540
So you have kilometers times
meters divided by kilometers,

00:03:36.540 --> 00:03:40.030
and that equals 50 meters.

00:03:40.030 --> 00:03:43.890
And it's good to always do a
reality check after every step.

00:03:43.890 --> 00:03:45.700
Usually when you do these types
of problems, you know, OK, if I

00:03:45.700 --> 00:03:48.780
want to go from kilometers to
meters, I'm going to use the

00:03:48.780 --> 00:03:51.010
number 1,000, because that's
the relationship between

00:03:51.010 --> 00:03:52.170
a kilometer and a meter.

00:03:52.170 --> 00:03:54.830
And you're always confused,
well, do I multiply by 1,000,

00:03:54.830 --> 00:03:56.250
or do I divide by 1,000?

00:03:56.250 --> 00:03:58.720
And you always have to say,
well, if I'm going from

00:03:58.720 --> 00:04:03.050
kilometers to meters, I'm
going-- 1 kilometer is

00:04:03.050 --> 00:04:05.550
1,000 meters, right?

00:04:05.550 --> 00:04:07.820
So I'm going to be
multiplying by 1,000.

00:04:07.820 --> 00:04:09.130
I'm going to get
a bigger number.

00:04:09.130 --> 00:04:12.490
So that's why I went from
0.05, and I multiplied it

00:04:12.490 --> 00:04:14.600
by 1,000, and I got 50.

00:04:14.600 --> 00:04:16.100
So let's get back
to the problem.

00:04:16.100 --> 00:04:19.400
0.05 kilometers is
equal to 50 meters.

00:04:19.400 --> 00:04:20.210
We're not done yet.

00:04:20.210 --> 00:04:23.280
Now, you need to convert those
50 meters into centimeters.

00:04:23.280 --> 00:04:25.540
Well, we do the same thing.

00:04:25.540 --> 00:04:32.730
50 meters times-- how many-- so
what's the relationship between

00:04:32.730 --> 00:04:33.740
meters and centimeters?

00:04:33.740 --> 00:04:36.320
Well, if we look at the
chart, we see it's 100.

00:04:36.320 --> 00:04:38.310
And the question I'm going to
ask you, am I going to multiply

00:04:38.310 --> 00:04:41.580
by 100, or am I going
to divide by 100?

00:04:41.580 --> 00:04:42.520
Well, it's the same thing.

00:04:42.520 --> 00:04:45.320
We're going from a bigger unit
to a smaller unit, so one of

00:04:45.320 --> 00:04:48.240
a bigger unit is equal to a
bunch of the smaller units.

00:04:48.240 --> 00:04:50.330
So we're going to multiply.

00:04:50.330 --> 00:04:56.590
So we say times 100
centimeters per meter, right?

00:04:56.590 --> 00:04:57.330
And that just makes sense.

00:04:57.330 --> 00:04:59.710
There's 100 centimeters
per meter.

00:04:59.710 --> 00:05:02.840
So 50 meters times 100
centimeters per meter is equal

00:05:02.840 --> 00:05:12.625
to 50 times 100 is 5,000, and
then the meters cancel out,

00:05:12.625 --> 00:05:15.260
and you get centimeters.

00:05:15.260 --> 00:05:21.330
So what we have here is that
0.05 kilometers is equal

00:05:21.330 --> 00:05:24.850
to 5,000 centimeters.

00:05:24.850 --> 00:05:26.090
Let's do another problem.

00:05:26.090 --> 00:05:28.580
I think the more examples
you see, it'll make them

00:05:28.580 --> 00:05:28.940
a little more sense.

00:05:28.940 --> 00:05:31.360
And always try to visualize
what we're doing, the scale.

00:05:31.360 --> 00:05:32.830
Otherwise, it's very
confusing whether you

00:05:32.830 --> 00:05:36.020
should multiply or divide.

00:05:36.020 --> 00:05:41.100
Let's say I have 422 decigrams.

00:05:47.110 --> 00:05:49.530
Grams are a measure of mass.

00:05:49.530 --> 00:05:51.470
One gram is actually
a very small amount.

00:05:51.470 --> 00:05:53.710
That's what you measure--
I guess in the metric

00:05:53.710 --> 00:05:56.770
system, they measure
gold in terms of grams.

00:05:56.770 --> 00:06:03.180
And I want to convert
this into milligrams.

00:06:03.180 --> 00:06:07.200
So before we start the problem,
let's just do a reality check.

00:06:07.200 --> 00:06:09.700
Am I going from a bigger unit
to a smaller unit, or a smaller

00:06:09.700 --> 00:06:10.870
unit to a bigger unit?

00:06:10.870 --> 00:06:18.870
Well, decigrams, that's 1/10
of a gram, and I'm going

00:06:18.870 --> 00:06:23.110
to 1/1,000 of a gram.

00:06:25.750 --> 00:06:26.960
So there's two ways
of doing this.

00:06:26.960 --> 00:06:29.940
We can convert to grams and
then convert to the other unit.

00:06:29.940 --> 00:06:32.350
That sometimes
make things easy.

00:06:32.350 --> 00:06:35.910
Or we could say, well,
how many milligrams is

00:06:35.910 --> 00:06:37.920
equal to one decigram?

00:06:37.920 --> 00:06:42.140
Well, a milligram, as
we see here, is 100

00:06:42.140 --> 00:06:43.270
times smaller, right?

00:06:43.270 --> 00:06:47.330
To go from 1/10 to 1/1,000,
you have to decrease

00:06:47.330 --> 00:06:48.780
in size by 100.

00:06:48.780 --> 00:07:02.570
So we could just say 422
decigrams times 100

00:07:02.570 --> 00:07:08.230
milligrams per decigram.

00:07:12.390 --> 00:07:17.620
And then the decigrams will
cancel out, and I'll get 422

00:07:17.620 --> 00:07:24.405
times 100, 42,200 milligrams.

00:07:29.950 --> 00:07:31.790
Now, another way you could have
done it is the way we just

00:07:31.790 --> 00:07:32.810
did that last problem.

00:07:32.810 --> 00:07:36.880
We could say 422 decigrams, we
could convert that to grams.

00:07:36.880 --> 00:07:41.620
We could say 422-- I'm
just going to say dg.

00:07:41.620 --> 00:07:45.500
That's not really a
familiar unit-- decigrams.

00:07:45.500 --> 00:07:48.150
And how many decigrams
are there per gram?

00:07:48.150 --> 00:07:51.260
If we're going to gram, 422
is going to be a smaller

00:07:51.260 --> 00:07:53.360
number of grams, right?

00:07:53.360 --> 00:07:59.110
So we could say times
1 decigram is equal

00:07:59.110 --> 00:08:01.230
to how many grams?

00:08:01.230 --> 00:08:07.200
Well, 1 decigram is
equal to-- no, sorry.

00:08:07.200 --> 00:08:09.800
1 gram is equal to
how many decigrams?

00:08:09.800 --> 00:08:13.650
Well, 1 gram is equal
to 10 decigrams.

00:08:13.650 --> 00:08:16.560
And the reason why this makes
sense is if we have a decigram

00:08:16.560 --> 00:08:18.290
in the numerator here, we want
a decigram in the

00:08:18.290 --> 00:08:19.790
denominator here.

00:08:19.790 --> 00:08:26.480
So if we have decigrams cancel
out, 422 decigrams will equal--

00:08:26.480 --> 00:08:33.240
that divided by 10 is
equal to 42.2 grams.

00:08:33.240 --> 00:08:35.770
And now we can just go
from grams to milligrams.

00:08:35.770 --> 00:08:37.120
Well, that's an easy one.

00:08:37.120 --> 00:08:41.150
1 gram is equal to 1,000
milligrams, so we could say

00:08:41.150 --> 00:08:48.620
times 1,000 milligram per gram.

00:08:48.620 --> 00:08:53.680
The grams cancel out, and
we're left with 42,200

00:08:53.680 --> 00:08:55.310
milligrams, right?

00:08:55.310 --> 00:08:59.380
42.2 times 1,000.

00:08:59.380 --> 00:09:01.530
Hopefully, that doesn't
confuse you too much.

00:09:01.530 --> 00:09:04.330
The important thing is to
always take a step back and

00:09:04.330 --> 00:09:06.960
really visualize and think
about, should I be getting a

00:09:06.960 --> 00:09:09.310
larger number or a smaller
number than the one

00:09:09.310 --> 00:09:10.920
I started off with?

00:09:10.920 --> 00:09:13.840
I think you're ready to
now try some problems.

00:09:13.840 --> 00:09:15.320
Have fun!


Level 1 Exponents

WEBVTT
Kind: captions
Language: en

00:00:01.200 --> 00:00:04.430
Welcome to the presentation on exponents.

00:00:04.696 --> 00:00:07.012
If I were to ask you what

00:00:07.012 --> 00:00:11.570
-- let me make sure this pen is the right width--

00:00:11.570 --> 00:00:14.860
if I were to ask you what two times three is,

00:00:14.860 --> 00:00:17.110
well I think at this point this should be pretty easy for you.

00:00:17.120 --> 00:00:25.190
That's the same thing as two plus two plus two, which equals six.

00:00:25.190 --> 00:00:26.840
You didn't have to do this.

00:00:26.850 --> 00:00:30.595
You all know that two times three is equal to six.

00:00:30.595 --> 00:00:32.465
So what we're going to do now is

00:00:32.465 --> 00:00:37.136
we're going to learn exponents which are the same thing

00:00:37.136 --> 00:00:41.730
that multiplication is to addition, exponents are to multiplication.

00:00:41.740 --> 00:00:43.163
I'll explain that in a second.

00:00:43.163 --> 00:00:44.740
I know that probably confused you.

00:00:44.740 --> 00:00:50.380
If I were to say what two to the third power is,

00:00:50.380 --> 00:00:52.709
instead of two plus two plus two,

00:00:52.709 --> 00:01:02.070
this is equal to two times two times two, which equals eight.

00:01:02.070 --> 00:01:07.640
Or if I were to say three the second power,

00:01:07.650 --> 00:01:13.160
that is equal to three times three.

00:01:13.344 --> 00:01:19.870
Remember, I said three times two would be three plus three.

00:01:19.870 --> 00:01:23.620
So the first three times three this equals nine,

00:01:23.620 --> 00:01:25.940
and three plus three equals six.

00:01:26.000 --> 00:01:28.165
The reason why I'm doing this is because there's always this temptation

00:01:28.165 --> 00:01:30.504
when you first learn exponents is to multiply.

00:01:30.504 --> 00:01:33.812
When I first learned it and I saw three to the second power or three squared,

00:01:33.812 --> 00:01:35.631
I'd always be like oh, that's six.

00:01:35.631 --> 00:01:37.898
But you also remember it's three times three,

00:01:37.898 --> 00:01:40.035
which equals nine.

00:01:40.035 --> 00:01:42.640
Let's do some more problems.

00:01:44.930 --> 00:01:52.710
If I were to tell you negative four squared.

00:01:52.730 --> 00:01:54.969
Once again, that's the same thing as

00:01:54.969 --> 00:02:01.250
negative four times negative four.

00:02:01.260 --> 00:02:04.540
Well, we all learn from multiplying negative numbers

00:02:04.540 --> 00:02:07.540
that a negative times a negative is a positive.

00:02:07.540 --> 00:02:08.861
And then four times four,

00:02:08.861 --> 00:02:10.029
well this equals positive sixteen.

00:02:10.029 --> 00:02:11.584
You don't have to write the positive,

00:02:11.584 --> 00:02:13.369
I'm just doing that for emphasis.

00:02:13.369 --> 00:02:19.584
If I would ask you what negative four to the third power is,

00:02:19.584 --> 00:02:26.440
well that equals negative four times negative four

00:02:26.440 --> 00:02:29.620
times negative four.

00:02:29.620 --> 00:02:32.340
Well we know already that negative four times negative four,

00:02:32.350 --> 00:02:35.130
that this equals sixteen, positive sixteen,

00:02:35.130 --> 00:02:38.470
and then we multiply that times negative four.

00:02:38.470 --> 00:02:42.490
And then that equals minus sixty-four.

00:02:42.500 --> 00:02:44.720
So something very interesting here to observe.

00:02:44.720 --> 00:02:46.267
When I took a negative number,

00:02:46.267 --> 00:02:47.968
and we call this the base,

00:02:47.968 --> 00:02:49.134
when the base is negative,

00:02:49.134 --> 00:02:50.454
in this case, negative four,

00:02:50.681 --> 00:02:53.248
and I raise it to an even power,

00:02:53.248 --> 00:02:55.552
I got a positive number, right?

00:02:55.552 --> 00:02:58.553
Negative four to an even power is positive sixteen,

00:02:58.553 --> 00:03:02.600
and when I took a negative number to an odd power, to three,

00:03:02.610 --> 00:03:03.984
I got a negative number.

00:03:03.984 --> 00:03:06.332
And that makes sense because every time

00:03:06.332 --> 00:03:08.298
you multiply by a negative number again,

00:03:08.298 --> 00:03:09.749
it switches signs.

00:03:09.749 --> 00:03:12.189
I'll show you the, I guess you'd call it

00:03:12.189 --> 00:03:13.939
the simplest example.

00:03:13.939 --> 00:03:18.580
Negative one to the one power is equal to negative one, right?

00:03:18.590 --> 00:03:21.930
Because that's just negative one times itself one time.

00:03:21.930 --> 00:03:26.877
And if I said negative one squared,

00:03:26.877 --> 00:03:29.435
well that's negative one times negative one,

00:03:29.435 --> 00:03:32.418
well that equals positive one.

00:03:32.418 --> 00:03:39.584
But if I said negative one to the third power,

00:03:39.584 --> 00:03:44.750
once again that's negative one times negative one times negative one.

00:03:44.750 --> 00:03:47.850
Well now this equals negative one times negative one is positive one

00:03:47.860 --> 00:03:50.340
times negative one equals negative one again.

00:03:50.340 --> 00:03:51.803
So I could tell you

00:03:51.803 --> 00:03:56.070
what negative one to the fifty-first power is.

00:03:56.080 --> 00:03:57.892
Because fifty-one is odd,

00:03:57.892 --> 00:04:01.322
we know that that is equal to negative one.

00:04:01.322 --> 00:04:03.250
If it was a fifty then it would be a positive one.

00:04:03.468 --> 00:04:05.510
Hope I didn't confuse you too much.

00:04:05.520 --> 00:04:08.932
Let's do a couple more problems.

00:04:08.932 --> 00:04:13.059
If I asked you what five to the third power is,

00:04:13.059 --> 00:04:19.207
well that will go into five times five times five,

00:04:19.207 --> 00:04:22.601
which equals one hundred and twenty-five.

00:04:22.601 --> 00:04:23.935
So really if I were ask to ask

00:04:23.935 --> 00:04:27.703
what negative five to the third power is,

00:04:27.703 --> 00:04:36.326
that would be negative five times negative five times negative five,

00:04:36.326 --> 00:04:39.910
which would be negative one hundred and twenty-five.

00:04:39.920 --> 00:04:41.686
Now one principle of exponents

00:04:41.686 --> 00:04:45.060
that might not seem completely intuitive to you at first

00:04:45.060 --> 00:04:46.869
is when I raise something to the zero power.

00:04:46.869 --> 00:04:51.111
So let's say I had two to the zero power.

00:04:51.111 --> 00:04:54.200
It turns out that anything to the zero power is equal to one.

00:04:54.200 --> 00:04:56.442
So two to the zero power is one,

00:04:56.442 --> 00:05:00.384
three to the zero power is equal to one,

00:05:00.384 --> 00:05:07.471
negative nine hundred to the zero power is equal to one.

00:05:07.471 --> 00:05:09.571
And let me see if I can give you a little bit of intuition of

00:05:09.571 --> 00:05:13.009
why that is actually the case.

00:05:15.916 --> 00:05:17.981
So if I were to ask you,

00:05:17.981 --> 00:05:23.231
let's do three to the fourth power.

00:05:23.231 --> 00:05:29.265
That equals three times three times three times three,

00:05:29.265 --> 00:05:32.249
which equals eighty-one.

00:05:32.249 --> 00:05:38.200
three to the third power is equal to twenty-seven,

00:05:38.200 --> 00:05:41.132
this three times three times three.

00:05:41.132 --> 00:05:46.930
three to second power is equal to nine.

00:05:46.930 --> 00:05:52.790
three to the first power is equal to three.

00:05:52.800 --> 00:05:54.753
Now we're going to say what's three to zero power?

00:05:54.753 --> 00:05:56.357
Well we already know, I already told you the rule,

00:05:56.357 --> 00:05:58.103
anything with the zero power is equal to one,

00:05:58.103 --> 00:06:00.040
but this will hopefully give you some intuition.

00:06:00.050 --> 00:06:03.490
When we went from the fourth power to the third power,

00:06:03.500 --> 00:06:05.640
we divided by three, right?

00:06:05.640 --> 00:06:08.240
eighty-one divided by three is twenty-seven.

00:06:08.250 --> 00:06:11.048
When you went from the third power to the second power,

00:06:11.048 --> 00:06:13.490
we divided by three.

00:06:13.490 --> 00:06:15.230
When we went from the second power from nine to three,

00:06:15.230 --> 00:06:17.350
we divided by three.

00:06:17.360 --> 00:06:19.744
So it kind of makes logical sense that

00:06:19.744 --> 00:06:22.478
when we go from the first power to the zero power,

00:06:22.478 --> 00:06:25.911
we'll just divide by three again.

00:06:25.911 --> 00:06:27.890
So three divided by three is one.

00:06:27.890 --> 00:06:30.032
Hopefully that gives you a little bit of an intuitive sense.

00:06:30.032 --> 00:06:33.480
You might want to replay that and think about why that is.

00:06:33.480 --> 00:06:36.270
And there's actually other aspects of exponents that

00:06:36.270 --> 00:06:37.719
why this also makes sense,

00:06:37.719 --> 00:06:39.683
why something to the zero power

00:06:39.683 --> 00:06:41.202
is equal to one.

00:06:41.202 --> 00:06:42.960
But let's just do some more problems in the time we have.

00:06:42.970 --> 00:06:46.750
I don't want to get you too confused.

00:06:46.750 --> 00:06:49.600
So if I were to ask you seven squared,

00:06:49.610 --> 00:06:53.720
well that's seven times seven, that's forty-nine.

00:06:53.797 --> 00:07:00.789
If I asked you negative six to the third power

00:07:00.820 --> 00:07:04.419
-- parentheses around here so you know it's a whole negative six to the third power--

00:07:04.419 --> 00:07:09.086
that equals negative six times negative six times negative six.

00:07:09.117 --> 00:07:15.319
Negative six times negative six is positive thirty-six times negative six,

00:07:15.381 --> 00:07:16.730
and that equals what?

00:07:16.730 --> 00:07:20.127
It's one hundred and eighty and thirty-six,

00:07:20.127 --> 00:07:22.164
that's minus two hundred and sixteen,

00:07:22.164 --> 00:07:24.296
if my mental math is correct.

00:07:24.296 --> 00:07:27.332
You could have actually multiplied it out.

00:07:27.594 --> 00:07:31.200
I think you're getting the point at this point.

00:07:31.200 --> 00:07:32.344
Oh, and another thing,

00:07:32.344 --> 00:07:35.176
if I told you zero to the hundredth power,

00:07:35.176 --> 00:07:36.290
well, that's pretty easy.

00:07:36.290 --> 00:07:39.021
That's zero times itself one hundred times,

00:07:39.021 --> 00:07:41.940
which is still equal to zero.

00:07:41.940 --> 00:07:49.340
If I were to ask you one to the thousandth power,

00:07:49.350 --> 00:07:50.370
well that's just equal to one, right?

00:07:50.370 --> 00:07:52.930
You can multiply one by itself as many times as you want

00:07:52.940 --> 00:07:54.750
and you're still going to get one.

00:07:54.760 --> 00:07:59.430
And remember, if I had negative one to the one thousandth power,

00:07:59.430 --> 00:08:02.910
well, this is an even exponent so you're still going to get one.

00:08:02.920 --> 00:08:06.960
If it was negative one to the one thousand and one,

00:08:06.960 --> 00:08:07.870
then it would be negative one.

00:08:07.870 --> 00:08:09.672
I think you remember why this is,

00:08:09.672 --> 00:08:12.540
because when you multiply a negative times itself an even number of times,

00:08:12.550 --> 00:08:13.550
the negatives cancel out.

00:08:13.550 --> 00:08:16.445
And then if you multiply it by a negative one more time

00:08:16.445 --> 00:08:17.970
it becomes a negative number again.

00:08:17.980 --> 00:08:19.430
Well let's just do some normal problems.

00:08:19.430 --> 00:08:22.699
I just want to make sure you get the basics of exponents down.

00:08:23.010 --> 00:08:25.682
If I were to tell you

00:08:25.682 --> 00:08:28.132
-- let me think of a good one--

00:08:28.132 --> 00:08:30.330
eight squared,

00:08:30.330 --> 00:08:34.947
that equals eight times eight equals sixty-four.

00:08:34.947 --> 00:08:39.442
If I were to tell you twenty-five squared,

00:08:39.442 --> 00:08:46.880
that's twenty-five times twenty-five, which equals six hundred and twenty-five.

00:08:46.880 --> 00:08:49.870
Powers of two is always very interesting.

00:08:49.880 --> 00:08:51.639
It's especially interesting

00:08:51.639 --> 00:08:53.860
if you one day go into computer science.

00:08:53.870 --> 00:08:55.496
So two to the fourth power,

00:08:55.496 --> 00:08:59.820
that's two times two times two times two.

00:08:59.820 --> 00:09:06.750
So two times two is four so this equals sixteen.

00:09:06.750 --> 00:09:08.350
And I did something very interesting here

00:09:08.360 --> 00:09:09.720
kind of on purpose.

00:09:09.720 --> 00:09:18.210
Notice that two to the fourth is equal to four times four, right?

00:09:18.210 --> 00:09:20.370
Because we did four times four here.

00:09:20.370 --> 00:09:22.678
I'm going to detail this more later on,

00:09:22.678 --> 00:09:24.020
but I want to think about what that means.

00:09:24.020 --> 00:09:27.070
Because four itself is the same thing as two squared.

00:09:27.080 --> 00:09:29.094
So we learned, just real fast,

00:09:29.094 --> 00:09:30.381
two to the fourth is the same thing

00:09:30.381 --> 00:09:34.130
as two squared times two squared.

00:09:34.130 --> 00:09:35.742
So I'll let you sit and think about that,

00:09:35.742 --> 00:09:37.727
but other than that I think you have the general idea

00:09:37.727 --> 00:09:40.430
of how basic exponents work,

00:09:40.468 --> 00:09:43.891
and I think you're ready to try the level one exponent module.

00:09:43.891 --> 99:59:59.999
have fun.


Speed translation

WEBVTT
Kind: captions
Language: en

00:00:01.280 --> 00:00:04.450
Welcome to the
presentation on units.

00:00:04.450 --> 00:00:05.570
Let's get started.

00:00:05.570 --> 00:00:09.180
So if I were to tell you -- let
me make sure my pen is set up

00:00:09.180 --> 00:00:13.700
right -- if I were to tell you
that someone is, let's say

00:00:13.700 --> 00:00:16.893
they're driving at a speed
of -- let's say it's Zack.

00:00:21.750 --> 00:00:24.170
So let's say I have Zack.

00:00:24.170 --> 00:00:29.100
And they're driving at a
speed of, let me say,

00:00:29.100 --> 00:00:38.680
28 feet per minute.

00:00:38.680 --> 00:00:44.430
So what I'm going to ask you is
if he's going 28 feet in every

00:00:44.430 --> 00:00:55.890
minute, how many inches will
Zack travel in 1 second?

00:00:55.890 --> 00:00:59.460
So how many inches per second
is he going to be going?

00:00:59.460 --> 00:01:01.570
Let's try to figure
this one out.

00:01:01.570 --> 00:01:06.700
So let's say if I had 28, and
I'll write ft short for feet,

00:01:06.700 --> 00:01:11.210
feet per minute, and I'll
write min short for a minute.

00:01:11.210 --> 00:01:14.330
So 28 feet per minute, let's
first figure out how many

00:01:14.330 --> 00:01:16.770
inches per minute that is.

00:01:16.770 --> 00:01:28.010
Well, we know that there are
12 inches per foot, right?

00:01:28.010 --> 00:01:29.790
If you didn't know
that you do now.

00:01:29.790 --> 00:01:32.590
So we know that there
are 12 inches per foot.

00:01:32.590 --> 00:01:35.800
So if you're going 28 feet per
minute, he's going to be going

00:01:35.800 --> 00:01:40.800
12 times that many
inches per minute.

00:01:40.800 --> 00:01:47.120
So, 12 times 28 -- let me do
the little work down here -- 28

00:01:47.120 --> 00:01:54.920
times 12 is 16, 56 into 280.

00:01:54.920 --> 00:01:56.930
I probably shouldn't be
doing it this messy.

00:01:56.930 --> 00:01:59.620
And this kind of stuff it would
be OK to use a calculator,

00:01:59.620 --> 00:02:01.260
although it's always good to
do the math yourself,

00:02:01.260 --> 00:02:02.510
it's good practice.

00:02:02.510 --> 00:02:07.870
So that's 6, 5 plus 8 is 13.

00:02:07.870 --> 00:02:09.350
336.

00:02:09.350 --> 00:02:21.280
So that equals 336
inches per minute.

00:02:21.280 --> 00:02:23.420
And something interesting
happened here is that you

00:02:23.420 --> 00:02:26.080
noticed that I had a foot in
the numerator here, and I had a

00:02:26.080 --> 00:02:27.730
foot in the denominator here.

00:02:27.730 --> 00:02:29.890
So you can actually treat
units just the same way

00:02:29.890 --> 00:02:32.730
that you would treat actual
numbers or variables.

00:02:32.730 --> 00:02:34.550
You have the same number in the
numerator and you have the same

00:02:34.550 --> 00:02:37.080
number in the denominator, and
your multiplying not adding,

00:02:37.080 --> 00:02:38.150
you can cancel them out.

00:02:38.150 --> 00:02:41.010
So the feet and the feet
canceled out and that's

00:02:41.010 --> 00:02:44.020
why we were left with
inches per minute.

00:02:44.020 --> 00:02:53.730
I could have also written
this as 336 foot per minute

00:02:53.730 --> 00:03:00.380
times inches per foot.

00:03:00.380 --> 00:03:03.180
Because the foot per minute
came from here, and the inches

00:03:03.180 --> 00:03:05.280
per foot came from here.

00:03:05.280 --> 00:03:07.580
Then I'll just cancel this
out and I would have

00:03:07.580 --> 00:03:09.000
gotten inches per minute.

00:03:09.000 --> 00:03:10.620
So anyway, I don't want
to confuse you too much

00:03:10.620 --> 00:03:12.710
with all of that unit
cancellation stuff.

00:03:12.710 --> 00:03:15.650
The bottom line is you just
remember, well if I'm going 28

00:03:15.650 --> 00:03:18.930
feet per minute, I'm going to
go 12 times that many inches

00:03:18.930 --> 00:03:22.580
per minute, right, because
there are 12 inches per foot.

00:03:22.580 --> 00:03:28.380
So I'm going 336
inches per minute.

00:03:28.380 --> 00:03:30.940
So now I have the question, but
we're not done, because the

00:03:30.940 --> 00:03:33.230
question is how many inches am
I going to be traveling

00:03:33.230 --> 00:03:35.030
in 1 second.

00:03:35.030 --> 00:03:37.320
So let me erase some of the
stuff here at the bottom.

00:03:57.170 --> 00:04:09.820
So 336 inches -- let's write it
like that -- inches per minute,

00:04:09.820 --> 00:04:13.390
and I want to know how
many inches per second.

00:04:13.390 --> 00:04:15.010
Well what do we know?

00:04:15.010 --> 00:04:21.730
We know that 1 minute -- and
notice, I write it in the

00:04:21.730 --> 00:04:23.790
numerator here because I want
to cancel it out with

00:04:23.790 --> 00:04:24.970
this minute here.

00:04:24.970 --> 00:04:27.760
1 minute is equal to
how many seconds?

00:04:27.760 --> 00:04:30.370
It equals 60 seconds.

00:04:32.890 --> 00:04:37.640
And this part can be confusing,
but it's always good to just

00:04:37.640 --> 00:04:39.360
take a step back and think
about what I'm doing.

00:04:39.360 --> 00:04:45.370
If I'm going to be going 336
inches per minute, how many

00:04:45.370 --> 00:04:47.380
inches am I going to
travel in 1 second?

00:04:47.380 --> 00:04:50.235
Am I going to travel more
than 336 or am I going

00:04:50.235 --> 00:04:53.640
to travel less than
336 inches per second.

00:04:53.640 --> 00:04:55.650
Well obviously less,
because a second is a much

00:04:55.650 --> 00:04:57.400
shorter period of time.

00:04:57.400 --> 00:04:59.475
So if I'm in a much shorter
period of time, I'm going

00:04:59.475 --> 00:05:02.140
to be traveling a much
shorter distance, if I'm

00:05:02.140 --> 00:05:03.360
going the same speed.

00:05:03.360 --> 00:05:05.910
So I should be dividing by a
number, which makes sense.

00:05:05.910 --> 00:05:08.090
I'm going to be dividing by 60.

00:05:08.090 --> 00:05:10.020
I know this can be very
confusing at the beginning, but

00:05:10.020 --> 00:05:12.840
that's why I always want you to
think about should I be getting

00:05:12.840 --> 00:05:15.370
a larger number or should I be
getting a smaller number and

00:05:15.370 --> 00:05:17.800
that will always give you
a good reality check.

00:05:17.800 --> 00:05:19.830
And if you just want to look at
how it turns out in terms of

00:05:19.830 --> 00:05:23.390
units, we know from the problem
that we want this minutes to

00:05:23.390 --> 00:05:25.630
cancel out with something
and get into seconds.

00:05:25.630 --> 00:05:28.890
So if we have minutes in the
denominator in the units here,

00:05:28.890 --> 00:05:31.910
we want the minutes in the
numerator here, and the seconds

00:05:31.910 --> 00:05:33.250
in the denominator here.

00:05:33.250 --> 00:05:36.950
And 1 minute is equal
to 60 seconds.

00:05:36.950 --> 00:05:39.540
So here, once again,
the minutes and the

00:05:39.540 --> 00:05:41.090
minutes cancel out.

00:05:41.090 --> 00:05:51.275
And we get 336 over 60
inches per second.

00:05:57.020 --> 00:06:01.440
Now if I were to actually
divide this out, actually we

00:06:01.440 --> 00:06:04.910
could just divide the numerator
and the denominator by 6.

00:06:04.910 --> 00:06:08.850
6 goes into 336,
what, 56 times?

00:06:08.850 --> 00:06:14.330
56 over 10, and then we can
divide that again by 2.

00:06:14.330 --> 00:06:18.290
So then that gets us 28 over 5.

00:06:18.290 --> 00:06:30.150
And 28 over 5 -- let's see, 5
goes into 28 five times, 25.

00:06:34.170 --> 00:06:37.680
3, 5.6.

00:06:37.680 --> 00:06:38.775
So this equals 5.6.

00:06:41.890 --> 00:06:45.530
So I think we now just
solved the problem.

00:06:45.530 --> 00:06:49.580
If Zack is going 28 feet in
every minute, that's his

00:06:49.580 --> 00:06:56.240
speed, he's actually going
5.6 inches per second.

00:06:56.240 --> 00:06:58.850
Hopefully that kind
of made sense.

00:06:58.850 --> 00:07:00.400
Let's try to see if we
could do another one.

00:07:05.200 --> 00:07:16.750
If I'm going 91 feet per
second, how many miles

00:07:16.750 --> 00:07:17.730
per hour is that?

00:07:22.860 --> 00:07:27.210
Well, 91 feet per second.

00:07:29.990 --> 00:07:32.080
If we want to say how many
miles that is, should we be

00:07:32.080 --> 00:07:34.480
dividing or should
we be multiplying?

00:07:34.480 --> 00:07:35.640
We should be dividing
because it's going to be a

00:07:35.640 --> 00:07:37.560
smaller number of miles.

00:07:37.560 --> 00:07:44.820
We know that 1 mile is equal to
-- and you might want to just

00:07:44.820 --> 00:07:47.950
memorize this -- 5,280 feet.

00:07:47.950 --> 00:07:50.430
It's actually a pretty
useful number to know.

00:07:50.430 --> 00:07:54.520
And then that will actually
cancel out the feet.

00:07:54.520 --> 00:07:59.150
Then we want to go from
seconds to hours, right?

00:07:59.150 --> 00:08:02.760
So, if we go from seconds to
hours, if I can travel 91 feet

00:08:02.760 --> 00:08:05.190
per second, how many will I
travel in an hour, I'm going to

00:08:05.190 --> 00:08:07.950
be getting a larger number
because an hour's a much larger

00:08:07.950 --> 00:08:09.530
period of time than a second.

00:08:09.530 --> 00:08:11.500
And how many seconds
are there in an hour?

00:08:11.500 --> 00:08:14.080
Well, there are 3,600
seconds in an hour.

00:08:14.080 --> 00:08:18.870
60 seconds per minute and
60 minutes per hour.

00:08:18.870 --> 00:08:26.090
So 3,600 over 1
seconds per hour.

00:08:26.090 --> 00:08:28.880
And these seconds
will cancel out.

00:08:28.880 --> 00:08:32.960
Then we're just left with, we
just multiply everything out.

00:08:32.960 --> 00:08:42.030
We get in the numerator,
91 times 3,600, right?

00:08:42.030 --> 00:08:44.950
91 times 1 times 3,600.

00:08:44.950 --> 00:08:50.880
In the denominator
we just have 5,280.

00:08:50.880 --> 00:08:53.750
This time around I'm actually
going to use a calculator --

00:08:53.750 --> 00:08:55.860
let me bring up the calculator
just to show you that I'm

00:08:55.860 --> 00:08:57.960
using the calculator.

00:08:57.960 --> 00:09:13.180
Let's see, so if I say 91 times
3,600, that equals a huge

00:09:13.180 --> 00:09:31.160
number divided by 5,280.

00:09:31.160 --> 00:09:32.730
Let me see if I can type it.

00:09:32.730 --> 00:09:44.520
91 times 3,600 divided
by 5,280 -- 62.05.

00:09:44.520 --> 00:09:52.380
So that equals 62.05
miles per hour.


Dividing fractions

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:03.110
Welcome to the presentation
on dividing fractions.

00:00:03.110 --> 00:00:04.490
Let's get started.

00:00:04.490 --> 00:00:06.640
So before I give you the
intuition-- actually, I might

00:00:06.640 --> 00:00:09.340
do that in a different module--
I'm just going to show you the

00:00:09.340 --> 00:00:11.740
mechanics of how you
divide a fraction.

00:00:11.740 --> 00:00:13.740
And it turns out that
it's actually not much

00:00:13.740 --> 00:00:16.030
more difficult than
multiplying fractions.

00:00:16.030 --> 00:00:21.410
If I were to ask you, 1/2
divided by 1/2, whenever you

00:00:21.410 --> 00:00:25.110
divide by a fraction, or
actually, when you divide by

00:00:25.110 --> 00:00:29.960
any number, it's the same thing
as multiplying by its inverse.

00:00:29.960 --> 00:00:36.670
So 1/2 divided by 1/2 is
equal to 1/2 times 2/1.

00:00:36.670 --> 00:00:44.990
We just inverted--
inverse-- the second 1/2.

00:00:44.990 --> 00:00:47.630
And we know from the
multiplication module, 1/2

00:00:47.630 --> 00:00:51.110
times 2/1, well, that's
just equal to 2/2,

00:00:51.110 --> 00:00:53.560
or it's equal to 1.

00:00:53.560 --> 00:00:56.020
And that makes sense because,
actually, any number divided

00:00:56.020 --> 00:00:58.750
by itself is equal to 1.

00:00:58.750 --> 00:01:03.220
1/2 divided by 1/2 is 1, just
like 5 divided by 5 is 1, just

00:01:03.220 --> 00:01:05.240
like 100 divided by 100 is 1.

00:01:05.240 --> 00:01:06.850
And this isn't a new principal.

00:01:06.850 --> 00:01:08.970
Actually, you were
always doing it.

00:01:08.970 --> 00:01:15.000
Think about it this way:
What is 2 divided by 2?

00:01:15.000 --> 00:01:16.290
Well, you know that's 1.

00:01:16.290 --> 00:01:20.560
But isn't this also the
same thing as 2 times the

00:01:20.560 --> 00:01:24.210
inverse of 2, which is 1?

00:01:24.210 --> 00:01:24.950
I'll show it to you.

00:01:24.950 --> 00:01:26.990
Actually, let me give you a
couple more examples to show

00:01:26.990 --> 00:01:31.340
that dividing fractions really
isn't a new concept, this whole

00:01:31.340 --> 00:01:34.840
notion of multiplying
by the inverse.

00:01:34.840 --> 00:01:40.540
If I were to tell you
what is 12 divided by 4?

00:01:40.540 --> 00:01:42.650
Well, we know the answer to
this, but I'm going to show

00:01:42.650 --> 00:01:50.640
you that this is the same
thing as 12 times 1/4.

00:01:50.640 --> 00:01:56.230
12/1 times 1/4 4 is
12/4, which is 3.

00:01:56.230 --> 00:01:59.480
And 12/4 is really just another
way of writing 12 divided by 4,

00:01:59.480 --> 00:02:02.535
so it's kind of a long way of
getting to the same point.

00:02:02.535 --> 00:02:04.990
But I just wanted to show you
that what we're doing in this

00:02:04.990 --> 00:02:07.970
module is nothing new than
what we've always been doing

00:02:07.970 --> 00:02:09.320
when we divide by a number.

00:02:09.320 --> 00:02:11.360
Division is the same thing.

00:02:11.360 --> 00:02:14.310
Dividing by a number is the
same thing as multiplying by

00:02:14.310 --> 00:02:15.960
the inverse of that number.

00:02:15.960 --> 00:02:19.880
And just as a review, an
inverse, if I have a number

00:02:19.880 --> 00:02:28.070
A, the inverse-- inv, short
for inverse-- is 1 over A.

00:02:28.070 --> 00:02:36.290
So the inverse of 2/3 is 3/2,
or the inverse of 5, because 5

00:02:36.290 --> 00:02:39.670
is the same thing as 5/1,
so the inverse is 1/5.

00:02:39.670 --> 00:02:41.000
We're just flipping it.

00:02:41.000 --> 00:02:43.320
We're switching the
numerator and denominator.

00:02:43.320 --> 00:02:46.475
So let's do some fraction
division problems.

00:02:49.270 --> 00:02:56.340
What is 2/3 divided by 5/6?

00:02:56.340 --> 00:03:05.970
Well, we know that this is the
same thing as 2/3 times 6/5,

00:03:05.970 --> 00:03:09.230
and that's equal to 12/15.

00:03:09.230 --> 00:03:14.570
We can divide the numerator and
denominator by 3, that's 4/5.

00:03:14.570 --> 00:03:22.900
What is 7/8 divided by 1/4?

00:03:22.900 --> 00:03:30.520
Well, that's the same
thing as 7/8 times 4/1.

00:03:30.520 --> 00:03:32.820
Remember, I just
flipped this 1/4.

00:03:32.820 --> 00:03:36.840
Divide by 1/4 is the same
thing as multiplying by 4/1.

00:03:36.840 --> 00:03:38.230
That's all you've got to do.

00:03:38.230 --> 00:03:39.990
And then we could use a little
shortcut we learned in the

00:03:39.990 --> 00:03:41.480
multiplication module.

00:03:41.480 --> 00:03:42.950
8 divided by 4 is 2.

00:03:42.950 --> 00:03:44.800
4 divided by 4 is 1.

00:03:44.800 --> 00:03:47.450
So that equals 7/2.

00:03:47.450 --> 00:03:49.900
Or if you wanted to write that
as a mixed number, this is, of

00:03:49.900 --> 00:03:51.200
course, an improper fraction.

00:03:51.200 --> 00:03:53.440
Improper fractions have
a numerator larger

00:03:53.440 --> 00:03:54.830
than the denominator.

00:03:54.830 --> 00:03:58.670
If you wanted to write that as
a mixed number, 2 goes into 7

00:03:58.670 --> 00:04:03.680
three times with a remainder
of 1, so that's 3 and a half.

00:04:03.680 --> 00:04:04.440
You can write it either way.

00:04:04.440 --> 00:04:05.990
I tend to keep it this
way because it's

00:04:05.990 --> 00:04:07.800
easier to deal with.

00:04:07.800 --> 00:04:10.130
Let's do a ton of more
problems, or at least as many

00:04:10.130 --> 00:04:13.830
more as we can do in the
next four or five minutes.

00:04:13.830 --> 00:04:23.850
What's negative 2/3
divided by 5/2?

00:04:23.850 --> 00:04:29.110
Once again, that's the same
thing as minus 2/3-- whoops--

00:04:29.110 --> 00:04:34.850
as minus 2/3 times what?

00:04:34.850 --> 00:04:40.110
It's times the inverse of
5/2, which is 2/5, and

00:04:40.110 --> 00:04:45.630
that equals minus 4/15.

00:04:45.630 --> 00:04:52.300
What is 3/2 divided by 1/6?

00:04:52.300 --> 00:04:59.850
Well, that's just the same
thing as 3/2 times 6/1,

00:04:59.850 --> 00:05:03.000
which equals 3 and 1.

00:05:03.000 --> 00:05:09.610
We just divided the 6 by 2 and
the 2 by 2, so that equals 9.

00:05:09.610 --> 00:05:11.280
I think you might
be getting it now.

00:05:11.280 --> 00:05:12.950
Let's see, let's
do a couple more.

00:05:12.950 --> 00:05:16.290
And, of course, you can always
pause, and look at this whole

00:05:16.290 --> 00:05:19.420
presentation again, so you can
get confused all over again.

00:05:19.420 --> 00:05:27.240
Let's see, let's do minus
5/7 divided by 10/3.

00:05:27.240 --> 00:05:33.880
Well, this is the same thing
as minus 5/7 times 3/10.

00:05:33.880 --> 00:05:35.420
I just multiplied
by the inverse.

00:05:35.420 --> 00:05:38.120
That's all I keep doing
over and over again.

00:05:38.120 --> 00:05:40.180
Minus 5 times 3.

00:05:40.180 --> 00:05:42.610
Minus 15.

00:05:42.610 --> 00:05:47.350
7 times 10 is 70.

00:05:47.350 --> 00:05:49.900
If we divide the numerator
and the denominator by

00:05:49.900 --> 00:05:56.050
5, we get minus 3/14.

00:05:56.050 --> 00:05:57.500
We could have also
just done it here.

00:05:57.500 --> 00:05:59.890
We could have done 5, 2,
and we would have gotten

00:05:59.890 --> 00:06:02.510
minus 3/14 as well.

00:06:02.510 --> 00:06:05.420
Let's do one or two
more problems.

00:06:05.420 --> 00:06:06.630
I think you kind of
get it, though.

00:06:09.600 --> 00:06:14.500
Let's say 1/2
divided by minus 3.

00:06:14.500 --> 00:06:14.965
Ah-ha!

00:06:14.965 --> 00:06:17.940
So what happens when you take a
fraction and you divide it by

00:06:17.940 --> 00:06:19.730
a whole number or an integer?

00:06:19.730 --> 00:06:22.970
Well, we know any whole number
can be written as a fraction.

00:06:22.970 --> 00:06:29.010
This is the same thing as
1/2 divided by minus 3/1.

00:06:29.010 --> 00:06:33.870
And dividing by a fraction is
the same thing as multiplying

00:06:33.870 --> 00:06:37.430
by it's inverse.

00:06:37.430 --> 00:06:42.150
So the inverse of negative 3/1
is negative 1/3, and this

00:06:42.150 --> 00:06:45.200
equals negative 1/6.

00:06:45.200 --> 00:06:46.040
Let's do it the other way.

00:06:46.040 --> 00:06:51.880
What if I had negative
3 divided by 1/2?

00:06:51.880 --> 00:06:52.500
Same thing.

00:06:52.500 --> 00:07:00.370
Negative 3 is the same thing as
minus 3/1 divided by 1/2, which

00:07:00.370 --> 00:07:07.940
is the same thing as minus 3/1
times 2/1, which is equal to

00:07:07.940 --> 00:07:12.010
minus 6/1, which
equals minus 6.

00:07:15.810 --> 00:07:17.350
Now, let me give you a
little bit of intuition

00:07:17.350 --> 00:07:19.730
of why this works.

00:07:19.730 --> 00:07:24.240
Let's say I said 2
divided by 1/3.

00:07:24.240 --> 00:07:27.650
Well, we know that this
is equal to 2/1 times

00:07:27.650 --> 00:07:30.120
3/1, which equals 6.

00:07:30.120 --> 00:07:32.700
So how does 2, 1/3,
and 6 relate?

00:07:32.700 --> 00:07:33.690
Well, let's look
at it this way.

00:07:33.690 --> 00:07:36.930
If I had two pieces of pizza.

00:07:36.930 --> 00:07:38.660
I have two pieces of pizza.

00:07:38.660 --> 00:07:41.520
Here's my two pieces
of pizza right.

00:07:41.520 --> 00:07:42.530
Two right here.

00:07:42.530 --> 00:07:45.050
So I have two pieces of pizza,
and I'm going to divide

00:07:45.050 --> 00:07:48.080
them into thirds.

00:07:48.080 --> 00:07:50.600
So I'm going to divide
each pizza into a third.

00:07:50.600 --> 00:07:52.860
I'll draw the little
Mercedes sign.

00:07:52.860 --> 00:07:57.050
So I'm dividing each pizza
into a third, right?

00:07:57.050 --> 00:07:58.210
How many pieces do I have?

00:07:58.210 --> 00:08:02.925
Let's see, 1, 2, 3, 4, 5, 6.

00:08:02.925 --> 00:08:04.800
I have 6 pieces.

00:08:04.800 --> 00:08:08.140
So you might want to sit and
ponder that for a little bit,

00:08:08.140 --> 00:08:12.850
but I think it might make a
little bit of sense to you.

00:08:12.850 --> 00:08:17.190
Let's do one more just
to tire your brain.

00:08:17.190 --> 00:08:25.750
If I had negative 7/2 divided
by 4/9-- let's pick a negative

00:08:25.750 --> 00:08:30.580
4/9-- well, that's the same
thing as minus 7/2 times

00:08:30.580 --> 00:08:33.720
minus 9/4, right?

00:08:33.720 --> 00:08:37.950
I just multiplied by the
inverse of negative 4/9.

00:08:37.950 --> 00:08:41.220
9 times 7 is equal to--
negative 7 times negative

00:08:41.220 --> 00:08:47.800
9 is positive 63,
and 2 times 4 is 8.

00:08:47.800 --> 00:08:51.460
Hopefully, I think you have a
good idea of how to divide by

00:08:51.460 --> 00:08:55.960
a fraction now, and you can
try out the dividing

00:08:55.960 --> 00:08:57.310
fractions modules.

00:08:57.310 --> 00:08:58.890
Have fun!


Multiplying fractions (old)

WEBVTT
Kind: captions
Language: en

00:00:01.180 --> 00:00:04.170
Welcome to the presentation
on multiplying fractions.

00:00:04.170 --> 00:00:08.120
Well, I think today you'll be
very happy because you'll find

00:00:08.120 --> 00:00:10.860
out that this is one of the few
times where multiplying

00:00:10.860 --> 00:00:14.730
something is easier than adding
it, I think, or subtracting

00:00:14.730 --> 00:00:15.630
it for that matter.

00:00:15.630 --> 00:00:17.455
And if you don't believe me,
let's do some problems.

00:00:20.430 --> 00:00:29.020
Let's start with 1/2 times 1/2.

00:00:29.020 --> 00:00:32.200
So when you multiply fractions
it's very straightforward.

00:00:32.200 --> 00:00:34.450
It's essentially just two
separate multiplication

00:00:34.450 --> 00:00:35.360
problems.

00:00:35.360 --> 00:00:39.100
You multiply the numerators,
so you get 1 times 1.

00:00:39.100 --> 00:00:43.080
And you multiply the
denominators, 2 times 2.

00:00:43.080 --> 00:00:45.330
1 times 1 is 1.

00:00:45.330 --> 00:00:47.870
2 times 2 is 4.

00:00:47.870 --> 00:00:51.360
So 1/2 times 1/2
is equal to 1/4.

00:00:51.360 --> 00:00:52.140
That makes sense.

00:00:52.140 --> 00:00:58.810
That's like saying 1/2 of 1/2
is 1/4, which makes sense.

00:00:58.810 --> 00:01:00.340
What if we had
negative numbers?

00:01:00.340 --> 00:01:06.880
Well, if I had 1/2 times
negative 1/2 -- and when you

00:01:06.880 --> 00:01:09.280
have a negative fraction it's
good ascribe the

00:01:09.280 --> 00:01:09.970
negative number.

00:01:09.970 --> 00:01:11.540
I tend to ascribe the negative
number to the numerator

00:01:11.540 --> 00:01:15.110
-- negative 1 over 2.

00:01:15.110 --> 00:01:17.740
You realize that negative
1/2 is the same thing

00:01:17.740 --> 00:01:19.820
as negative 1 over 2.

00:01:19.820 --> 00:01:21.950
Hopefully that make sense.

00:01:21.950 --> 00:01:25.670
So 1/2 times negative 1/2,
that's just the same thing as 1

00:01:25.670 --> 00:01:34.070
times negative 1 over 2 times
2, which equals negative 1

00:01:34.070 --> 00:01:40.300
over 4, which is the same
thing as negative 1/4.

00:01:40.300 --> 00:01:42.710
What if I had different
denominators, and when you're

00:01:42.710 --> 00:01:44.430
adding and subtracting
fractions that tends to

00:01:44.430 --> 00:01:45.320
make things difficult.

00:01:45.320 --> 00:01:47.360
Well, it's not necessarily
the case here.

00:01:47.360 --> 00:01:55.590
If I had 2/3 times 1/2, just
multiply the numerators, 2

00:01:55.590 --> 00:01:59.800
times 1, and you multiply
that denominators 3 times 2.

00:01:59.800 --> 00:02:04.690
So you get 2 times 1
is 2, 3 times 2 is 6.

00:02:04.690 --> 00:02:06.920
And 2 over 6 we know from
equivalent fractions is

00:02:06.920 --> 00:02:09.420
the same thing as 1/3.

00:02:09.420 --> 00:02:10.510
That was an
interesting problem.

00:02:10.510 --> 00:02:12.590
Let's do it again and I want to
show you a little trick here.

00:02:15.360 --> 00:02:22.090
So, 2 over 3 times 1/2 -- as
we said, any multiplication

00:02:22.090 --> 00:02:23.890
problem you just multiply the
numerators, multiply the

00:02:23.890 --> 00:02:25.620
denominators and you
have your answer.

00:02:25.620 --> 00:02:31.540
But sometimes there's a little
trick here where you can divide

00:02:31.540 --> 00:02:33.660
the numerators and the
denominators by a number,

00:02:33.660 --> 00:02:35.500
because you know that this is
going to be the same thing as 2

00:02:35.500 --> 00:02:39.360
times 1 over 3 times 2.

00:02:39.360 --> 00:02:41.415
Which is the same thing -- I'm
just switching the order on top

00:02:41.415 --> 00:02:45.910
-- as 1 times 2 over 3 times 2.

00:02:45.910 --> 00:02:48.250
All I did is I switched the
order on top, because you can

00:02:48.250 --> 00:02:49.990
multiply in either direction.

00:02:49.990 --> 00:02:55.170
And that's the same thing
as 1/3 times 2 over 2.

00:02:55.170 --> 00:03:00.430
Well that's just is 1/3 times
1, which is equal to 1/3.

00:03:00.430 --> 00:03:01.700
And why did I do that?

00:03:01.700 --> 00:03:05.110
Well I want to show you that
these 2s, these 2s, all I did

00:03:05.110 --> 00:03:08.040
is switch the order, but at
all times we had 1, 2 in the

00:03:08.040 --> 00:03:10.210
numerator raider and I had
1, 2 in the denominator.

00:03:10.210 --> 00:03:13.320
If I wanted to, and this is
kind of a trick for doing

00:03:13.320 --> 00:03:16.700
multiplication really fast so
you don't have to reduce the

00:03:16.700 --> 00:03:22.650
final fraction too much,
you get 2/3 times 1/3 --

00:03:22.650 --> 00:03:24.650
2/3 times 1/2, sorry.

00:03:24.650 --> 00:03:26.190
You say I have a 2 in the
numerator, 2 in the

00:03:26.190 --> 00:03:30.740
denominator, let me divide them
both by 2, that equals 1/3.

00:03:30.740 --> 00:03:31.680
Just a fast trick.

00:03:31.680 --> 00:03:33.380
I hope I didn't confuse you.

00:03:33.380 --> 00:03:35.190
Let's do a couple of more
problems, and I'll do

00:03:35.190 --> 00:03:38.230
it both with the trick
and without the trick.

00:03:38.230 --> 00:03:44.460
What if I had 3/7
times 2 over 5.

00:03:44.460 --> 00:03:49.640
Well, multiply the
numerators, 3 times 2 is 6.

00:03:49.640 --> 00:03:51.820
7 times 5 is 35.

00:03:51.820 --> 00:03:53.610
That's it.

00:03:53.610 --> 00:03:55.440
Let's do some negative numbers.

00:03:55.440 --> 00:04:05.420
If I had negative 3 over 4
times 2 over 11, well, that's

00:04:05.420 --> 00:04:11.630
negative 6 over 44, which
is the same thing as

00:04:11.630 --> 00:04:14.920
negative 3 over 22.

00:04:14.920 --> 00:04:16.980
And we could have done that
cross-dividing trick here.

00:04:16.980 --> 00:04:18.380
Let's do it again
with the cross--.

00:04:21.240 --> 00:04:23.080
Times 2 over 11.

00:04:23.080 --> 00:04:25.653
We say oh, well 2 and 4,
they're both divisible by 2, so

00:04:25.653 --> 00:04:27.770
let's divide them both by 2.

00:04:27.770 --> 00:04:31.190
So 2 becomes 1, 4 becomes
2, and then our answer

00:04:31.190 --> 00:04:34.280
becomes minus 3 over 22.

00:04:34.280 --> 00:04:37.170
Negative 3 times
1 is minus is 3.

00:04:37.170 --> 00:04:40.560
2 times 11 is 22.

00:04:40.560 --> 00:04:41.470
Do another one right here.

00:04:41.470 --> 00:04:49.760
If I had negative 2/5 times
minus 2/5, well, that just is

00:04:49.760 --> 00:04:54.860
equal to negative 2 times
negative 2 is positive 4.

00:04:54.860 --> 00:04:56.690
It's 5 times 5 is 25.

00:04:56.690 --> 00:04:57.550
4 over 25.

00:04:57.550 --> 00:04:59.450
And that's, just remember, a
negative times a negative is a

00:04:59.450 --> 00:05:02.050
positive, which makes sense.

00:05:02.050 --> 00:05:03.630
Let's just do a couple
more problems since we

00:05:03.630 --> 00:05:04.330
have a lot of time.

00:05:04.330 --> 00:05:06.070
But I think you probably
got this by now.

00:05:06.070 --> 00:05:08.410
You're probably realizing that
multiplying fractions is a

00:05:08.410 --> 00:05:11.890
lot easier than adding or
subtracting them, hopefully.

00:05:11.890 --> 00:05:13.910
I guess it's not a bad thing if
you find adding or subtracting

00:05:13.910 --> 00:05:16.320
fractions easy as well.

00:05:16.320 --> 00:05:18.453
Let's do -- I'm just
making up numbers now

00:05:18.453 --> 00:05:25.910
-- 2/9 times 18 over 2.

00:05:25.910 --> 00:05:28.710
Well here we could, well, we
have a 2 in the numerator

00:05:28.710 --> 00:05:30.310
and a 2 in the denominator.

00:05:30.310 --> 00:05:34.530
Let's divide them both by
2, so they both become 1.

00:05:34.530 --> 00:05:37.460
And we have an 18 in
the numerator and a 9

00:05:37.460 --> 00:05:38.290
in the denominator.

00:05:38.290 --> 00:05:41.090
Well they both are divisible
by 9, so let's divide

00:05:41.090 --> 00:05:41.840
them both by 9.

00:05:41.840 --> 00:05:47.890
So 9 becomes a 1, and
the 18 becomes a 2.

00:05:47.890 --> 00:05:52.600
So you have 1 times 2 over 1
times 1, well, that just equals

00:05:52.600 --> 00:05:54.840
2 over 1 which equals 2.

00:05:54.840 --> 00:05:56.100
That was pretty
straightforward.

00:05:56.100 --> 00:05:58.650
We could have done it, I guess
you could call it the hard

00:05:58.650 --> 00:06:05.940
way, if we said 2 times 2
over 9 times 18 over 2.

00:06:05.940 --> 00:06:08.680
2 times 18 is 36.

00:06:08.680 --> 00:06:11.360
9 times 2 is 18.

00:06:11.360 --> 00:06:15.090
And 36 divided by 18, and we
can see 18 goes into 36 two

00:06:15.090 --> 00:06:17.190
times, that also equals 2.

00:06:17.190 --> 00:06:18.280
Either way is fine.

00:06:18.280 --> 00:06:20.650
If you don't feel comfortable
doing this trick right

00:06:20.650 --> 00:06:21.780
now, you don't have to.

00:06:21.780 --> 00:06:27.710
All that does is you won't end
up with huge numbers in your

00:06:27.710 --> 00:06:29.880
product that you'll have to
figure out if they can

00:06:29.880 --> 00:06:32.700
be reduced further.

00:06:32.700 --> 00:06:34.585
Let's do two more problems.

00:06:38.320 --> 00:06:46.120
Minus 5 over 7 times 1 over 3.

00:06:46.120 --> 00:06:49.950
Minus 5 times 1 is minus 5.

00:06:49.950 --> 00:06:52.750
7 over 3 is 21.

00:06:52.750 --> 00:06:54.270
That's it.

00:06:54.270 --> 00:06:59.650
Let me do one with the
little trick I showed you.

00:06:59.650 --> 00:07:04.660
Say I had 15, and here I think
you'll see why that trick

00:07:04.660 --> 00:07:11.580
is useful, over 21
times 14 over 5.

00:07:11.580 --> 00:07:13.880
Well clearly, if we multiply
this out we end up with

00:07:13.880 --> 00:07:15.090
pretty big numbers.

00:07:15.090 --> 00:07:18.840
I think 220 one 105 and
you have to reduce those.

00:07:18.840 --> 00:07:19.960
It becomes a big mess.

00:07:19.960 --> 00:07:22.770
But we can see that 15 and
5 are both divisible by 5.

00:07:22.770 --> 00:07:24.560
So let's divide them both by 5.

00:07:24.560 --> 00:07:27.260
So 15 divided by 5 is 3.

00:07:27.260 --> 00:07:30.180
5 divided by 5 is 1.

00:07:30.180 --> 00:07:33.240
14 and 21, they're
both divisible by 7.

00:07:33.240 --> 00:07:36.960
14 divided by 7 is 2.

00:07:36.960 --> 00:07:40.770
21 divided by 7 is 3.

00:07:40.770 --> 00:07:48.640
So we got 3 times 2 is 6 over 3
times 1 is 3, which equals 2.

00:07:48.640 --> 00:07:50.090
That's the same thing
as what I said before.

00:07:50.090 --> 00:07:52.510
If we had multiplied 15
times 14 that would

00:07:52.510 --> 00:07:56.200
have been 210 I think.

00:07:56.200 --> 00:07:58.430
Yeah, 15 times 14 is 210.

00:07:58.430 --> 00:08:02.100
And 21 times 5 would have been
105, and you would have to say,

00:08:02.100 --> 00:08:05.320
I guess in this case it's kind
of obvious, that 210 is 2 times

00:08:05.320 --> 00:08:07.840
105 and you would have
gotten 2 as well.

00:08:07.840 --> 00:08:09.530
So hopefully I didn't
confuse you too much

00:08:09.530 --> 00:08:10.670
with that last problem.

00:08:10.670 --> 00:08:12.900
But I hope you realize
multiplication's pretty

00:08:12.900 --> 00:08:13.110
straightforward.

00:08:13.110 --> 00:08:15.340
You just multiply the
numerators, you multiply the

00:08:15.340 --> 00:08:18.400
denominators, and then if you
have to reduce you reduce, but

00:08:18.400 --> 00:08:19.500
you're pretty much done.

00:08:19.500 --> 00:08:22.050
I think you're ready now to try
the multiplication module,

00:08:22.050 --> 00:08:25.120
and I hope you have fun.


Adding and subtracting fractions

WEBVTT
Kind: captions
Language: en

00:00:01.610 --> 00:00:03.710
Welcome to the presentation
on adding and

00:00:03.710 --> 00:00:05.300
subtracting fractions.

00:00:05.300 --> 00:00:08.450
Let's get started.

00:00:08.450 --> 00:00:12.120
Let's start with what I hope
shouldn't confuse you too much.

00:00:12.120 --> 00:00:15.130
This should hopefully be a
relatively easy question.

00:00:15.130 --> 00:00:23.940
If I were to ask you
what 1/4 plus 1/4 is.

00:00:23.940 --> 00:00:25.270
Let's think about
what that means.

00:00:25.270 --> 00:00:32.420
Let's say we had a pie and it
was divided into four pieces.

00:00:32.420 --> 00:00:35.180
So this is like saying this
first 1/4 right here -- let me

00:00:35.180 --> 00:00:37.730
do it in a different color.

00:00:37.730 --> 00:00:39.530
This 1/4 right here,
let's say it's this

00:00:39.530 --> 00:00:42.400
1/4 of the pie, right?

00:00:42.400 --> 00:00:45.570
And we're going to add it
to another 1/4 of the pie.

00:00:45.570 --> 00:00:51.610
Let's make it this one -- let
me change the color -- pink.

00:00:51.610 --> 00:00:57.070
This 1/4, this pink 1/4
is this 1/4 of the pie.

00:00:57.070 --> 00:01:02.270
So if I were to eat both 1/4s
or 1/4 and then I eat another

00:01:02.270 --> 00:01:04.570
1/4, how much have I eaten?

00:01:04.570 --> 00:01:08.260
Well, you could look from just
the picture, I have now eaten 2

00:01:08.260 --> 00:01:10.280
out of the 4 pieces of the pie.

00:01:10.280 --> 00:01:15.480
So if I eat 1/4 of a piece of
pie or 1/4 of a pie, and then I

00:01:15.480 --> 00:01:21.550
eat another 1/4 of a pie, I
will have eaten 2/4 of the pie.

00:01:21.550 --> 00:01:23.940
And we know from the equivalent
fractions module that this is

00:01:23.940 --> 00:01:27.490
the same thing as I've
eaten 1/2 of the pie,

00:01:27.490 --> 00:01:28.330
which makes sense.

00:01:28.330 --> 00:01:32.150
If I eat 2 out of 4 pieces of a
pie, then I've eaten 1/2 of it.

00:01:32.150 --> 00:01:33.700
And if we look at it
mathematically, what

00:01:33.700 --> 00:01:34.960
happened here?

00:01:34.960 --> 00:01:39.230
Well the denominators or the
bottom numbers, the bottom

00:01:39.230 --> 00:01:41.270
numbers in the fraction
stayed the same.

00:01:41.270 --> 00:01:42.730
Because that's just the
total number of pieces

00:01:42.730 --> 00:01:44.360
I have in this example.

00:01:44.360 --> 00:01:47.930
Well, I added the numerators,
which makes sense.

00:01:47.930 --> 00:01:51.490
I had 1 out of the 4 pieces of
pie, then I ate another 1 out

00:01:51.490 --> 00:01:54.600
of the 4 pieces of pie, so I
ate 2 out of the 4 pieces

00:01:54.600 --> 00:01:56.510
of pie, which is 1/2.

00:01:56.510 --> 00:01:57.840
Let me do a couple
more examples.

00:02:01.820 --> 00:02:09.240
What is 2/5 plus 1/5?

00:02:09.240 --> 00:02:11.760
Well we do the same thing here.

00:02:11.760 --> 00:02:14.230
We first check to make sure the
denominators are the same --

00:02:14.230 --> 00:02:16.130
we'll learn in a second what we
do when the denominators

00:02:16.130 --> 00:02:16.900
are different.

00:02:16.900 --> 00:02:19.340
If the denominators are the
same, the denominator of the

00:02:19.340 --> 00:02:21.030
answer will be the same.

00:02:21.030 --> 00:02:22.480
And we just add the numerators.

00:02:22.480 --> 00:02:27.580
2/5 plus 1/5 is just 2
plus 1 over 5, which

00:02:27.580 --> 00:02:31.100
is equal to 3 over 5.

00:02:31.100 --> 00:02:33.390
And it works the same
way with subtraction.

00:02:33.390 --> 00:02:42.430
If I had 3 over 7 minus 2 over
7, that just equals 1 over 7.

00:02:42.430 --> 00:02:45.480
I just subtracted the 3, I
subtracted the 2 from the 3

00:02:45.480 --> 00:02:48.150
to get 1 and I kept the
denominator the same.

00:02:48.150 --> 00:02:48.860
Which makes sense.

00:02:48.860 --> 00:02:53.040
If I have 3 out of the 7 pieces
of a pie and I were to give

00:02:53.040 --> 00:02:57.800
away 2 out of the 7 pieces of a
pie, I'd be left with 1 of

00:02:57.800 --> 00:03:00.180
the 7 pieces of a pie.

00:03:00.180 --> 00:03:02.370
So now let's tackle -- I
think it should be pretty

00:03:02.370 --> 00:03:04.330
straightforward when we
have the same denominator.

00:03:04.330 --> 00:03:05.870
Remember, the denominator
is just the bottom

00:03:05.870 --> 00:03:06.880
number in a fraction.

00:03:06.880 --> 00:03:08.410
Numerator is the top number.

00:03:08.410 --> 00:03:11.430
What happens when we have
different denominators?

00:03:11.430 --> 00:03:15.100
Well, hopefully it won't
be too difficult.

00:03:15.100 --> 00:03:24.330
Let's say I have 1/4 plus 1/2.

00:03:24.330 --> 00:03:27.190
Let's go back to that
original pie example.

00:03:27.190 --> 00:03:28.260
Let me draw that pie.

00:03:33.900 --> 00:03:37.250
So this first 1/4 right here,
let's just color it in,

00:03:37.250 --> 00:03:40.470
that's this 1/4 of the pie.

00:03:40.470 --> 00:03:44.550
And now I'm going to eat
another 1/2 of the pie.

00:03:44.550 --> 00:03:46.460
So I'm going to eat
1/2 of the pie.

00:03:46.460 --> 00:03:49.100
So this 1/2.

00:03:49.100 --> 00:03:51.400
I'll eat this whole
1/2 of the pie.

00:03:54.290 --> 00:03:55.230
So what does that equal?

00:03:55.230 --> 00:03:57.190
Well, there's a couple of ways
we could think about it.

00:03:57.190 --> 00:03:59.210
First we could just
re-write 1/2.

00:03:59.210 --> 00:04:04.135
1/2 of the pie, that's actually
the same thing as 2/4, right?

00:04:06.950 --> 00:04:12.290
There's 1/4 here and
then another 1/4 here.

00:04:12.290 --> 00:04:15.470
So 1/2 is the same thing as
2/4, and we know that from the

00:04:15.470 --> 00:04:17.520
equivalent fractions module.

00:04:17.520 --> 00:04:21.460
So we know that 1/4 plus 1/2,
this is the same thing as

00:04:21.460 --> 00:04:27.110
saying 1/4 plus 2/4, right?

00:04:27.110 --> 00:04:36.160
And all I did here is I changed
the 1/2 to a 2/4 by essentially

00:04:36.160 --> 00:04:38.820
multiplying the numerator
and the denominator of

00:04:38.820 --> 00:04:40.410
this fraction by 2.

00:04:40.410 --> 00:04:41.740
And you can do that
to any fraction.

00:04:41.740 --> 00:04:43.890
As long as you multiply the
numerator and the denominator

00:04:43.890 --> 00:04:47.620
by the same number, you
can multiply by anything.

00:04:47.620 --> 00:04:52.880
That makes sense because
1/2 times 1 is equal

00:04:52.880 --> 00:04:54.440
to 1/2, you know that.

00:04:54.440 --> 00:05:00.080
Well another way of writing
1 is 1/2 times 2/2.

00:05:00.080 --> 00:05:04.490
2 over 2 is the same thing as
1, and that equals 2 over 4.

00:05:04.490 --> 00:05:07.380
The reason why I picked 2
is because I wanted to get

00:05:07.380 --> 00:05:08.580
the same denominator here.

00:05:11.230 --> 00:05:13.530
I hope I'm not completely
confusing you.

00:05:13.530 --> 00:05:15.220
Well, let's just finish
up this problem.

00:05:15.220 --> 00:05:19.220
So we have 1/4 plus 2/4, so we
know that we just add the

00:05:19.220 --> 00:05:22.570
numerators, 3, and the
denominators are the same, 3/4.

00:05:22.570 --> 00:05:25.190
And if we look at the picture,
true enough, we have

00:05:25.190 --> 00:05:29.380
eaten 3/4 of this pie.

00:05:29.380 --> 00:05:30.320
Let's do another one.

00:05:34.030 --> 00:05:44.730
Let's do 1/2 plus 1/3.

00:05:44.730 --> 00:05:47.570
Well once again, we want to get
both denominators to be the

00:05:47.570 --> 00:05:51.370
same, but you can't just
multiply one of them to get --

00:05:51.370 --> 00:05:53.850
there's nothing I can multiply
3 by to get 2, or there's no,

00:05:53.850 --> 00:05:56.510
at least, integer I can
multiply 3 by to get 2.

00:05:56.510 --> 00:05:58.900
And there's nothing I can
multiply 2 by to get 3.

00:05:58.900 --> 00:06:01.870
So I have to multiply both of
them so they equal each other.

00:06:01.870 --> 00:06:06.230
It turns out that what we want
for, what we'll call the common

00:06:06.230 --> 00:06:08.850
denominator, it turns out
to be the least common

00:06:08.850 --> 00:06:11.130
multiple of 2 and 3.

00:06:11.130 --> 00:06:13.400
Well what's the least common
multiple of 2 and 3?

00:06:13.400 --> 00:06:16.750
Well that's the smallest
number that's a multiple

00:06:16.750 --> 00:06:17.860
of both 2 and 3.

00:06:17.860 --> 00:06:19.390
Well the smallest number
that's a multiple of

00:06:19.390 --> 00:06:23.480
both 2 and 3 is 6.

00:06:23.480 --> 00:06:27.890
So let's convert both of these
fractions to something over 6.

00:06:27.890 --> 00:06:30.340
So 1/2 is equal to what over 6.

00:06:30.340 --> 00:06:33.320
You should know this from the
equivalent fractions module.

00:06:33.320 --> 00:06:37.500
Well if I eat 1/2 of a pizza
with 6 pieces, I would have

00:06:37.500 --> 00:06:40.270
eaten 3 pieces, right?

00:06:40.270 --> 00:06:40.820
That make sense.

00:06:40.820 --> 00:06:43.950
1 is 1/2 of 2, 3 is 1/2 of 6.

00:06:43.950 --> 00:06:47.650
Similarly, if I eat 1/3 of a
pizza with 6 pieces, it's

00:06:47.650 --> 00:06:50.730
the same thing as 2 over 6.

00:06:50.730 --> 00:06:57.700
So 1/2 plus 1/3 is the same
thing as 3/6 plus 2/6.

00:06:57.700 --> 00:06:58.980
Notice I didn't do
anything crazy.

00:06:58.980 --> 00:07:02.030
All I did is I re-wrote both
of these fractions with

00:07:02.030 --> 00:07:03.220
different denominators.

00:07:03.220 --> 00:07:06.050
I essentially changed the
number of pieces in the

00:07:06.050 --> 00:07:08.830
pie, if that helps at all.

00:07:08.830 --> 00:07:10.410
Now that we're at this
point then the problem

00:07:10.410 --> 00:07:11.190
becomes very easy.

00:07:11.190 --> 00:07:14.830
We just add the numerators,
3 plus 2 is 5, and we keep

00:07:14.830 --> 00:07:16.430
the denominators the same.

00:07:16.430 --> 00:07:18.970
3 over 6 plus 2
over 6 equals 5/6.

00:07:22.240 --> 00:07:24.740
And subtraction is
the same thing.

00:07:24.740 --> 00:07:31.460
1/2 minus 1/3, well that's
the same thing as 3

00:07:31.460 --> 00:07:35.110
over 6 minus 2 over 6.

00:07:35.110 --> 00:07:39.530
Well that equals 1 over 6.

00:07:39.530 --> 00:07:43.060
Let's do a bunch more problems
and hopefully you'll

00:07:43.060 --> 00:07:43.990
start to get it.

00:07:43.990 --> 00:07:47.120
And always remember you can
re-watch the presentation, or

00:07:47.120 --> 00:07:49.630
you can pause it and try to do
the problems yourself, because

00:07:49.630 --> 00:07:53.250
I think sometimes I talk fast.

00:07:53.250 --> 00:07:55.100
Let me throw you a curve ball.

00:07:55.100 --> 00:07:59.330
What's 1/10 minus 1?

00:07:59.330 --> 00:08:01.630
Well, one doesn't even
look like a fraction.

00:08:01.630 --> 00:08:04.150
But you can write
it as a fraction.

00:08:04.150 --> 00:08:07.970
Well that's the same thing as
1/10 minus -- how could we

00:08:07.970 --> 00:08:11.020
write 1 so it has the
denominator of 10?

00:08:11.020 --> 00:08:11.590
Right.

00:08:11.590 --> 00:08:14.830
It's the same thing as
10 over 10, right?

00:08:14.830 --> 00:08:16.330
10 over 10 is 1.

00:08:16.330 --> 00:08:20.900
So 1/10 minus 10 over 10 is the
same thing as 1 minus 10 --

00:08:20.900 --> 00:08:24.950
remember, we only subtract the
numerators and we keep the

00:08:24.950 --> 00:08:31.170
denominator 10, and that
equals negative 9 over 10.

00:08:31.170 --> 00:08:34.380
1/10 minus 1 is equal
to negative 9 over 10.

00:08:34.380 --> 00:08:35.920
Let's do another one.

00:08:35.920 --> 00:08:36.470
Let's do one more.

00:08:36.470 --> 00:08:38.680
I think that's all
I have time for.

00:08:38.680 --> 00:08:47.320
Let's do minus 1/9
minus 1 over 4.

00:08:47.320 --> 00:08:53.770
Well the least common
multiple of 0 and 4 is 36.

00:08:53.770 --> 00:08:55.590
So that's equal to 36.

00:08:55.590 --> 00:09:00.070
So what's negative 1/9 where
we change the denominator

00:09:00.070 --> 00:09:02.010
from 9 to 36?

00:09:02.010 --> 00:09:05.030
Well, we multiply 9
times 4 to get 36.

00:09:05.030 --> 00:09:07.230
We have to multiply the
numerator times 4 as well.

00:09:07.230 --> 00:09:11.860
So we have negative 1, so
it becomes negative 4.

00:09:11.860 --> 00:09:16.860
Then minus over 36.

00:09:16.860 --> 00:09:20.110
Well to go from 4 to 36, we
have to multiply this fraction

00:09:20.110 --> 00:09:23.070
by 9, or we have to multiply
the denominator by 9, so you

00:09:23.070 --> 00:09:25.190
also have to multiply
the numerator by 9.

00:09:25.190 --> 00:09:28.370
1 times 9 is 9.

00:09:28.370 --> 00:09:35.770
So this equals minus 4 minus
9 over 36, which equals

00:09:35.770 --> 00:09:39.580
minus 13 over 36.

00:09:39.580 --> 00:09:41.850
I think that's all I have time
for right now, and I'll

00:09:41.850 --> 00:09:44.000
probably add a couple more
modules, but I think you might

00:09:44.000 --> 00:09:47.520
be ready now to do the adding
and subtracting module.

00:09:47.520 --> 00:09:49.020
Have fun.