Change in centripetal acceleration from change in linear velocity and radius: Worked examples

WEBVTT
Kind: captions
Language: en

00:00:00.613 --> 00:00:02.498
- [Instructor] We are told
that a van drives around

00:00:02.498 --> 00:00:06.676
a circular curve of radius
r with linear speed v.

00:00:06.676 --> 00:00:09.542
On a second curve of the same radius,

00:00:09.542 --> 00:00:13.197
the van has linear speed 1/3 v.

00:00:13.197 --> 00:00:15.078
And you could view linear
speed as the magnitude

00:00:15.078 --> 00:00:17.179
of your linear velocity.

00:00:17.179 --> 00:00:21.228
How does the magnitude of the
van's centripetal acceleration

00:00:21.228 --> 00:00:24.699
change after the linear speed decreases?

00:00:24.699 --> 00:00:27.485
So pause this video and see
if you can figure it out

00:00:27.485 --> 00:00:29.679
on your own, and I'll give
you a little bit of a hint.

00:00:29.679 --> 00:00:31.734
We know that the magnitude
of centripetal acceleration

00:00:31.734 --> 00:00:35.173
in general is equal to
linear speed squared

00:00:35.173 --> 00:00:37.255
divided by radius.

00:00:37.255 --> 00:00:39.255
The radius of the curve.

00:00:40.381 --> 00:00:42.242
Alright, now let's work
through this together.

00:00:42.242 --> 00:00:44.889
So let's first think
about the first curve.

00:00:44.889 --> 00:00:48.174
So the first curve the
magnitude of our centripetal

00:00:48.174 --> 00:00:51.333
acceleration for curve one,
I have another subscript

00:00:51.333 --> 00:00:54.269
one here, this is around the first curve.

00:00:54.269 --> 00:00:55.978
They tell us that our linear speed is v.

00:00:55.978 --> 00:01:00.064
So we have v squared over and
the radius of that curve is r.

00:01:00.064 --> 00:01:02.391
This is going to be a
straight up v squared over r

00:01:02.391 --> 00:01:03.480
for that first curve.

00:01:03.480 --> 00:01:05.780
The magnitude of our
centripetal acceleration.

00:01:05.780 --> 00:01:07.679
Now what about the second curve?

00:01:07.679 --> 00:01:09.825
So the magnitude of our
centripetal acceleration

00:01:09.825 --> 00:01:13.585
around second curve,
that's what that two is,

00:01:13.585 --> 00:01:15.292
is going to be equal to,

00:01:15.292 --> 00:01:18.068
they tell us we now have
a linear speed of 1/3 v.

00:01:18.068 --> 00:01:20.426
So in our numerator
we're gonna square that,

00:01:20.426 --> 00:01:23.665
1/3 v squared all of that over

00:01:23.665 --> 00:01:25.311
the curve of the same radius.

00:01:25.311 --> 00:01:27.254
So our radius is still r.

00:01:27.254 --> 00:01:29.764
So let's just do a little
algebraic simplification.

00:01:29.764 --> 00:01:32.419
1/3 v times 1/3 v is just going to be

00:01:32.419 --> 00:01:34.886
1/9 v squared.

00:01:34.886 --> 00:01:39.053
So it's going to be 1/9 v squared over r.

00:01:40.122 --> 00:01:41.777
All I did is square this numerator here.

00:01:41.777 --> 00:01:45.944
Or I could write this as
1/9 times v squared over r.

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The reason why I wrote
this in green is because

00:01:51.047 --> 00:01:52.843
this is the exact same thing as this.

00:01:52.843 --> 00:01:55.544
And so this is going to be equal to,

00:01:55.544 --> 00:01:58.784
this is equal to 1/9
times, instead of writing

00:01:58.784 --> 00:02:01.508
v squared over r, I could say hey that's r

00:02:01.508 --> 00:02:03.633
the magnitude of our
centripetal acceleration

00:02:03.633 --> 00:02:05.331
around the first curve.

00:02:05.331 --> 00:02:07.501
The magnitude of our
centripetal acceleration

00:02:07.501 --> 00:02:08.979
around the first curve.

00:02:08.979 --> 00:02:11.156
So how does the magnitude
of the van's centripetal

00:02:11.156 --> 00:02:14.417
acceleration change after
the linear speed decreases?

00:02:14.417 --> 00:02:15.696
Well around the second curve,

00:02:15.696 --> 00:02:19.692
we have 1/9 the magnitude
of centripetal acceleration.

00:02:19.692 --> 00:02:22.192
So we could say the magnitude,

00:02:23.869 --> 00:02:25.577
or I could just say, they already asked us

00:02:25.577 --> 00:02:27.325
how does the magnitude change

00:02:27.325 --> 00:02:29.717
so we could say decreases,

00:02:29.717 --> 00:02:31.467
decreases by a factor

00:02:35.369 --> 00:02:36.619
factor of nine.

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And I wrote it in this language.

00:02:39.196 --> 00:02:42.765
You could say it got
multiplied by a factor of 1/9

00:02:42.765 --> 00:02:45.123
or you could say decreases
by a factor of nine

00:02:45.123 --> 00:02:47.157
because on the Khan Academy exercises

00:02:47.157 --> 00:02:49.805
that deal with this, they
use language like that.

00:02:49.805 --> 00:02:51.888
Let's do another example.

00:02:54.730 --> 00:02:57.468
Here we are told a
father spins his daughter

00:02:57.468 --> 00:03:01.323
in a circle of radius r
at angular speed omega.

00:03:01.323 --> 00:03:04.390
Then the father extends
his arms and spins her

00:03:04.390 --> 00:03:08.557
in a circle of radius two r
with the same angular speed.

00:03:09.745 --> 00:03:12.617
How does the magnitude of
the child's centripetal

00:03:12.617 --> 00:03:17.194
acceleration change when
the father extends his arms?

00:03:17.194 --> 00:03:19.460
Once again pause this
video and see if you can

00:03:19.460 --> 00:03:21.235
figure it out.

00:03:21.235 --> 00:03:25.318
Well the key realization
here and we derived this

00:03:26.389 --> 00:03:29.108
at a previous video, is to
realize that the magnitude

00:03:29.108 --> 00:03:33.025
of centripetal acceleration
is equal to r times

00:03:34.382 --> 00:03:36.549
our angular speed squared.

00:03:38.316 --> 00:03:41.545
And so initially so the
magnitude of our centripetal

00:03:41.545 --> 00:03:45.270
acceleration initially,
I'll do that with a sub i.

00:03:45.270 --> 00:03:48.437
That is going to be equal
to, well they're using

00:03:48.437 --> 00:03:49.952
the same notation.

00:03:49.952 --> 00:03:51.575
We have omega as our angular speed.

00:03:51.575 --> 00:03:52.498
And our radius is r.

00:03:52.498 --> 00:03:55.510
So it's just going to be r omega squared.

00:03:55.510 --> 00:03:58.738
And then when we think about the father,

00:03:58.738 --> 00:04:00.224
he extends his arms.

00:04:00.224 --> 00:04:03.125
So then you have the
magnitude of your centripetal

00:04:03.125 --> 00:04:06.389
acceleration, I could
say final or extended or

00:04:06.389 --> 00:04:07.934
I'll just say final sub f.

00:04:07.934 --> 00:04:10.103
What does that going to be equal to?

00:04:10.103 --> 00:04:14.270
Well now our radius, the
radius of our circle is two r.

00:04:15.633 --> 00:04:17.712
So it's going to be two r

00:04:17.712 --> 00:04:19.794
and they say the same angular speed.

00:04:19.794 --> 00:04:22.040
So our angular speed is still omega.

00:04:22.040 --> 00:04:23.707
Two r omega squared.

00:04:25.421 --> 00:04:29.828
Well this part right over
here r omega squared,

00:04:29.828 --> 00:04:32.043
that was just the magnitude of our initial

00:04:32.043 --> 00:04:33.660
centripetal acceleration.

00:04:33.660 --> 00:04:35.666
That was the magnitude of our initial

00:04:35.666 --> 00:04:37.391
centripetal acceleration.

00:04:37.391 --> 00:04:40.854
And so you see that the magnitude of our

00:04:40.854 --> 00:04:45.728
centripetal acceleration, has
increased by a factor of two.

00:04:45.728 --> 00:04:48.395
Increased, increased by a factor

00:04:52.577 --> 00:04:53.410
of two.

00:04:54.364 --> 00:04:55.697
And we are done.


Deriving formula for centripetal acceleration from angular velocity

WEBVTT
Kind: captions
Language: en

00:00:00.940 --> 00:00:03.228
- [Instructor] In multiple
videos we have already talked

00:00:03.228 --> 00:00:07.799
about if something is
moving in a circular motion

00:00:07.799 --> 00:00:12.508
at a fixed speed, its velocity
is constantly changing.

00:00:12.508 --> 00:00:13.369
Why is that?

00:00:13.369 --> 00:00:15.823
Because velocity is a vector

00:00:15.823 --> 00:00:18.317
and a vector has not just a magnitude

00:00:18.317 --> 00:00:19.620
which would be its speed

00:00:19.620 --> 00:00:21.279
but also its direction,

00:00:21.279 --> 00:00:23.873
so even if I have the
same speed at this point

00:00:23.873 --> 00:00:28.064
as I have at say this
point right over here,

00:00:28.064 --> 00:00:31.859
my velocity will be pointing
in a different direction

00:00:31.859 --> 00:00:34.456
so now the magnitude might be the same,

00:00:34.456 --> 00:00:37.014
I'll try to draw it roughly the same,

00:00:37.014 --> 00:00:39.270
the length of this arrow
should be the same,

00:00:39.270 --> 00:00:40.741
so the magnitude,

00:00:40.741 --> 00:00:42.388
V without an arrow on top,

00:00:42.388 --> 00:00:44.306
you could view that as the linear speed,

00:00:44.306 --> 00:00:45.256
that will be the same

00:00:45.256 --> 00:00:46.915
but now the direction has changed

00:00:46.915 --> 00:00:48.820
and in order to change the direction,

00:00:48.820 --> 00:00:51.568
you must have this ball

00:00:51.568 --> 00:00:53.074
that's moving in this circular motion

00:00:53.074 --> 00:00:54.978
must be accelerated.

00:00:54.978 --> 00:00:55.813
That's the only way.

00:00:55.813 --> 00:00:57.802
If you have a change in velocity,

00:00:57.802 --> 00:00:59.952
then you must have acceleration

00:00:59.952 --> 00:01:02.488
and it's a little
counterintuitive at first

00:01:02.488 --> 00:01:04.302
because you're saying well,
my magnitude didn't change,

00:01:04.302 --> 00:01:05.708
only my direction did

00:01:05.708 --> 00:01:08.734
but any change in your velocity
implies that acceleration

00:01:08.734 --> 00:01:10.884
and in previous videos we saw

00:01:10.884 --> 00:01:14.387
that that acceleration is
constantly going to be inward

00:01:14.387 --> 00:01:17.265
if you have your uniform circular motion

00:01:17.265 --> 00:01:19.773
and we call that inward acceleration,

00:01:19.773 --> 00:01:23.199
we call that centripetal acceleration

00:01:23.199 --> 00:01:26.679
and though if I wrote
this A sub C like this,

00:01:26.679 --> 00:01:29.185
this means the magnitude of
my centripetal acceleration.

00:01:29.185 --> 00:01:31.570
If I'm talking about the
magnitude and the direction

00:01:31.570 --> 00:01:35.161
I would put an arrow on it just like that.

00:01:35.161 --> 00:01:37.326
Now, we have also, in previous videos

00:01:37.326 --> 00:01:39.648
have been able to connect
what is the magnitude

00:01:39.648 --> 00:01:41.151
of centripetal acceleration,

00:01:41.151 --> 00:01:42.382
how can we figure that out

00:01:42.382 --> 00:01:45.617
from our linear speed and the radius

00:01:45.617 --> 00:01:47.000
and we had the formula,

00:01:47.000 --> 00:01:49.226
the magnitude of centripetal acceleration

00:01:49.226 --> 00:01:52.643
is equal to the magnitude of our velocity

00:01:53.577 --> 00:01:56.283
or our linear speed squared

00:01:56.283 --> 00:01:58.116
divided by our radius.

00:02:00.655 --> 00:02:02.377
Now, what I wanna do in this video

00:02:02.377 --> 00:02:06.041
is see if I can connect our
centripetal acceleration

00:02:06.041 --> 00:02:08.405
to angular velocity,

00:02:08.405 --> 00:02:11.731
our nice variable omega right over here

00:02:11.731 --> 00:02:12.564
and omega right over here

00:02:12.564 --> 00:02:13.950
you could use angular speed.

00:02:13.950 --> 00:02:17.229
It's the magnitude, I
could say our magnitude

00:02:17.229 --> 00:02:19.387
of our angular velocity,

00:02:19.387 --> 00:02:20.624
so our angular speed here.

00:02:20.624 --> 00:02:24.334
So, how can we make this connection?

00:02:24.334 --> 00:02:27.165
Well, the key realization
is to be able to connect

00:02:27.165 --> 00:02:30.670
your linear speed with your angular speed.

00:02:30.670 --> 00:02:33.853
So, in previous videos,
I think it was the second

00:02:33.853 --> 00:02:35.831
or third when we introduced ourselves

00:02:35.831 --> 00:02:38.067
to angular velocity or the magnitude of it

00:02:38.067 --> 00:02:39.400
which would be angular speed,

00:02:39.400 --> 00:02:43.521
we saw that our linear
speed is going to be equal

00:02:43.521 --> 00:02:47.688
to our radius, the radius of
our uniform circular motion

00:02:48.636 --> 00:02:52.219
times the magnitude of
our angular velocity

00:02:54.494 --> 00:02:57.089
and I don't like to
just memorize formulas.

00:02:57.089 --> 00:02:58.555
It's always good to have an intuition

00:02:58.555 --> 00:03:00.288
of why this makes sense.

00:03:00.288 --> 00:03:03.027
Remember, angular velocity

00:03:03.027 --> 00:03:04.673
or the magnitude of angular velocity

00:03:04.673 --> 00:03:07.423
is measured in radians per second

00:03:10.128 --> 00:03:12.920
and we typically view radians as an angle

00:03:12.920 --> 00:03:14.892
but if you think of it as an arc length,

00:03:14.892 --> 00:03:16.840
a radian you could view it

00:03:16.840 --> 00:03:20.124
as how many radii in length
am I completing per second?

00:03:20.124 --> 00:03:22.992
And so, if I multiply that
times the actual length

00:03:22.992 --> 00:03:25.470
of the radii, then you can get a sense

00:03:25.470 --> 00:03:29.295
of well, how much distance
am I covering per second?

00:03:29.295 --> 00:03:30.263
Hopefully that makes some sense

00:03:30.263 --> 00:03:32.754
and we actually prove this formula,

00:03:32.754 --> 00:03:35.768
we get an intuition for this
formula in previous videos

00:03:35.768 --> 00:03:38.534
but from this formula it's
easy to make a substitution

00:03:38.534 --> 00:03:39.954
back into our original one

00:03:39.954 --> 00:03:43.190
to have en expression for
centripetal acceleration,

00:03:43.190 --> 00:03:45.266
the magnitude of centripetal acceleration

00:03:45.266 --> 00:03:48.741
in terms of radius and the magnitude

00:03:48.741 --> 00:03:51.085
of angular velocity

00:03:51.085 --> 00:03:52.441
and I encourage you, pause this video

00:03:52.441 --> 00:03:55.498
and see if you can drive that on your own.

00:03:55.498 --> 00:03:56.331
All right, let's do this together.

00:03:56.331 --> 00:03:58.498
So, if we start with this,

00:03:59.769 --> 00:04:03.052
we have the magnitude of
our centripetal acceleration

00:04:03.052 --> 00:04:04.671
is going to be equal to,

00:04:04.671 --> 00:04:07.504
instead of putting V squared here,

00:04:10.854 --> 00:04:13.687
instead of V, I can write R omega,

00:04:14.759 --> 00:04:16.348
so let me do that,

00:04:16.348 --> 00:04:17.765
R and then omega.

00:04:20.809 --> 00:04:22.479
There you go and all I did I said look,

00:04:22.479 --> 00:04:25.063
our linear speed right over here

00:04:25.063 --> 00:04:26.931
is equal to our radius

00:04:26.931 --> 00:04:29.864
times the magnitude of
our angular velocity

00:04:29.864 --> 00:04:30.828
or angular speed,

00:04:30.828 --> 00:04:32.802
so everywhere I saw a V here,

00:04:32.802 --> 00:04:35.934
I'm just replacing it with an R omega

00:04:35.934 --> 00:04:37.805
and so, I have R omega,

00:04:37.805 --> 00:04:40.638
the entire quantity squared over R

00:04:42.220 --> 00:04:44.231
and then we can simplify this.

00:04:44.231 --> 00:04:46.359
This is going to be equal to,

00:04:46.359 --> 00:04:48.078
I just use my exponent properties here,

00:04:48.078 --> 00:04:49.926
R omega times R omega

00:04:49.926 --> 00:04:51.676
is gonna be R squared

00:04:53.393 --> 00:04:55.060
times omega squared,

00:04:57.203 --> 00:04:58.786
all of that over R.

00:05:00.187 --> 00:05:01.849
If I have it R squared over R,

00:05:01.849 --> 00:05:05.575
well, that's just going
to simplify to an R,

00:05:05.575 --> 00:05:06.646
so there we have it,

00:05:06.646 --> 00:05:08.473
we have our formula for the magnitude

00:05:08.473 --> 00:05:09.910
of our centripetal acceleration

00:05:09.910 --> 00:05:13.041
in terms of the magnitude
of our angular velocity.

00:05:13.041 --> 00:05:14.624
It is going to be R

00:05:16.144 --> 00:05:20.311
times the magnitude of our
angular velocity omega squared.

00:05:22.161 --> 00:05:23.022
There you have it

00:05:23.022 --> 00:05:23.900
and in future videos

00:05:23.900 --> 00:05:24.794
we'll do worked examples

00:05:24.794 --> 00:05:27.877
where we actually apply this formula.


Change in period and frequency from change in angular velocity: Worked examples

WEBVTT
Kind: captions
Language: en

00:00:00.941 --> 00:00:03.254
- [Instructor] We're told
that a large tire spins

00:00:03.254 --> 00:00:06.808
with angular velocity four omega.

00:00:06.808 --> 00:00:11.239
A smaller tire spins with
half the angular velocity.

00:00:11.239 --> 00:00:13.157
I'm assuming half the angular velocity

00:00:13.157 --> 00:00:14.835
of the large tire.

00:00:14.835 --> 00:00:19.740
How does the period T
sub-large of the large tire

00:00:19.740 --> 00:00:24.201
compare with the period T
sub-small of the small tire?

00:00:24.201 --> 00:00:25.623
So pause this video and see

00:00:25.623 --> 00:00:27.229
if you can figure that out and figure out

00:00:27.229 --> 00:00:29.980
which choice you would pick.

00:00:29.980 --> 00:00:32.836
Okay, so the key here is
to realize the connection

00:00:32.836 --> 00:00:35.620
between angular velocity and period.

00:00:35.620 --> 00:00:37.970
And instead of just blindly
memorizing the formula,

00:00:37.970 --> 00:00:40.394
I always like to reason
it through a little bit.

00:00:40.394 --> 00:00:43.037
We know that period is equal to,

00:00:43.037 --> 00:00:44.080
well think about it.

00:00:44.080 --> 00:00:45.961
In order to complete
one cycle, if I'm doing

00:00:45.961 --> 00:00:48.144
uniform circular motion, if I am going

00:00:48.144 --> 00:00:50.573
in a circle around like this,

00:00:50.573 --> 00:00:54.260
in order to do one complete
lap around the circle,

00:00:54.260 --> 00:00:57.734
or to complete one cycle, I
have to cover two pi radians.

00:00:57.734 --> 00:01:00.251
So two pi radians is what I need to cover,

00:01:00.251 --> 00:01:02.872
and then I divide that
by my angular velocity,

00:01:02.872 --> 00:01:05.714
how fast am I going through the radians.

00:01:05.714 --> 00:01:08.859
So that's how I like to
reason through this formula

00:01:08.859 --> 00:01:11.683
that connects angular
velocity, or the magnitude

00:01:11.683 --> 00:01:14.507
of angular velocity, and the period.

00:01:14.507 --> 00:01:16.787
And so, we can say T sub-large,

00:01:16.787 --> 00:01:18.550
I'll do some two different colors,

00:01:18.550 --> 00:01:20.467
we can say T sub-large,

00:01:22.501 --> 00:01:26.860
T sub-large is going to
be equal to two pi over,

00:01:26.860 --> 00:01:28.161
it says a large tire spins

00:01:28.161 --> 00:01:30.811
with angular velocity four omega,

00:01:30.811 --> 00:01:32.447
so it's going to be two pi,

00:01:32.447 --> 00:01:35.484
its angular velocity is four omega,

00:01:35.484 --> 00:01:37.493
so two pi over four omega.

00:01:37.493 --> 00:01:40.136
And then T sub-small, a smaller tire spins

00:01:40.136 --> 00:01:44.053
with half the angular
velocity, so T sub-small,

00:01:45.471 --> 00:01:48.248
T sub-small is going
to be equal to two pi,

00:01:48.248 --> 00:01:50.681
and it's going to have
half the angular velocity

00:01:50.681 --> 00:01:53.439
as the large tire, so
that half of four omega

00:01:53.439 --> 00:01:54.522
is two omega.

00:01:55.534 --> 00:01:57.601
So how do these two things compare?

00:01:57.601 --> 00:01:59.680
Well it might be helpful to just simplify

00:01:59.680 --> 00:02:01.835
these expressions a little bit.

00:02:01.835 --> 00:02:05.389
So T sub-large, the
period of the large tire,

00:02:05.389 --> 00:02:08.139
is going to be pi over two omega,

00:02:09.588 --> 00:02:12.968
and T sub-small, the
period of the smaller tire,

00:02:12.968 --> 00:02:16.135
that's just going to be pi over omega.

00:02:17.288 --> 00:02:20.072
And so, the red expression right over here

00:02:20.072 --> 00:02:21.959
is half of this blue expression.

00:02:21.959 --> 00:02:26.907
I can rewrite this as being
equal to one half times,

00:02:26.907 --> 00:02:29.324
one half times pi over omega.

00:02:30.413 --> 00:02:31.702
Or another way of writing this,

00:02:31.702 --> 00:02:34.607
I could write T sub-large,

00:02:34.607 --> 00:02:37.992
T sub-large is equal to one half times,

00:02:37.992 --> 00:02:39.373
this expression right over here

00:02:39.373 --> 00:02:43.540
is the period of the smaller
tire, so T sub-small.

00:02:45.030 --> 00:02:47.175
Now which of these
choices match up to that?

00:02:47.175 --> 00:02:49.199
Well it is this one right over here:

00:02:49.199 --> 00:02:51.108
the period of the larger
tire is going to be

00:02:51.108 --> 00:02:54.441
one half the period of the smaller tire.

00:02:55.433 --> 00:02:57.537
Now, it's always nice
if you have the time,

00:02:57.537 --> 00:02:59.512
if you aren't under time pressure,

00:02:59.512 --> 00:03:01.987
to just think about
whether that makes sense.

00:03:01.987 --> 00:03:05.368
So a large tire spins
with an angular velocity

00:03:05.368 --> 00:03:07.680
of four omega; the smaller tire spins

00:03:07.680 --> 00:03:10.444
with half the angular velocity.

00:03:10.444 --> 00:03:13.167
So, if it has half the angular velocity,

00:03:13.167 --> 00:03:16.168
it's rotating half as fast.

00:03:16.168 --> 00:03:18.098
If it's rotating half as fast,

00:03:18.098 --> 00:03:21.999
it would take twice as
long to complete one cycle.

00:03:21.999 --> 00:03:24.040
So the smaller tire is
gonna take twice as long,

00:03:24.040 --> 00:03:26.027
or you could view it as the large tire

00:03:26.027 --> 00:03:28.539
takes half as long as the small tire.

00:03:28.539 --> 00:03:30.094
So that makes sense.

00:03:30.094 --> 00:03:32.177
Let's do another example.

00:03:34.483 --> 00:03:38.039
An ice skater spins with
angular velocity two omega.

00:03:38.039 --> 00:03:40.273
She brings her arms away from her body,

00:03:40.273 --> 00:03:43.922
decreasing her angular velocity to omega.

00:03:43.922 --> 00:03:47.254
How does the frequency of her spin change?

00:03:47.254 --> 00:03:49.182
Once again pause this video and see

00:03:49.182 --> 00:03:52.133
if you can figure that out on your own.

00:03:52.133 --> 00:03:54.182
Well let's just think about how frequency

00:03:54.182 --> 00:03:57.218
is connected to angular velocity.

00:03:57.218 --> 00:03:59.627
We already know that period
from the last question

00:03:59.627 --> 00:04:02.547
is equal to, or you'll have
to complete two pi radians

00:04:02.547 --> 00:04:04.715
to complete a cycle, and
then we can divide that

00:04:04.715 --> 00:04:07.965
by how quickly our angle is increasing.

00:04:09.728 --> 00:04:11.013
And so, there you have it.

00:04:11.013 --> 00:04:11.846
That's our period.

00:04:11.846 --> 00:04:14.215
It's two pi over our angular velocity.

00:04:14.215 --> 00:04:15.982
And if we want frequency, frequency

00:04:15.982 --> 00:04:17.954
is just the reciprocal of period.

00:04:17.954 --> 00:04:22.121
So frequency is just going
to be omega over two pi.

00:04:24.005 --> 00:04:27.327
This is how many cycles we
can complete in a second.

00:04:27.327 --> 00:04:30.632
And so, at first the ice skater spins

00:04:30.632 --> 00:04:33.514
with an angular velocity of two omega.

00:04:33.514 --> 00:04:36.994
So let's say frequency, let's call it

00:04:36.994 --> 00:04:39.244
frequency initial, initial,

00:04:41.175 --> 00:04:42.958
is going to be equal to,

00:04:42.958 --> 00:04:45.177
her angular velocity is two omega,

00:04:45.177 --> 00:04:48.273
it's going to be two omega over two pi.

00:04:48.273 --> 00:04:52.250
And then her frequency final I'll say,

00:04:52.250 --> 00:04:53.583
frequency final,

00:04:55.094 --> 00:04:57.451
so after she puts her
arms away from her body

00:04:57.451 --> 00:04:59.085
decreasing her angular velocity,

00:04:59.085 --> 00:05:00.713
and we'll talk more about that phenomena

00:05:00.713 --> 00:05:03.280
in other videos, is going to be,

00:05:03.280 --> 00:05:06.025
so decreasing her angular
velocity to omega,

00:05:06.025 --> 00:05:08.971
so now her angular velocity is omega,

00:05:08.971 --> 00:05:10.991
and it's gonna be over two pi.

00:05:10.991 --> 00:05:12.658
So do these compare?

00:05:13.995 --> 00:05:15.493
Well if I write her initial frequency,

00:05:15.493 --> 00:05:19.016
I can rewrite it as initial frequency,

00:05:19.016 --> 00:05:22.933
initial frequency, is
equal to two times omega,

00:05:25.249 --> 00:05:26.511
I'm gonna do that in another color,

00:05:26.511 --> 00:05:28.844
two times omega over two pi.

00:05:31.519 --> 00:05:32.437
Right?

00:05:32.437 --> 00:05:35.392
Two times all of this
business right over here.

00:05:35.392 --> 00:05:37.504
Which is the exact same thing,

00:05:37.504 --> 00:05:41.337
which is equal to two
times our final frequency.

00:05:41.337 --> 00:05:43.837
Two times our frequency final.

00:05:45.885 --> 00:05:48.001
Or another way of thinking about it,

00:05:48.001 --> 00:05:51.418
her frequency final, her frequency final,

00:05:52.696 --> 00:05:55.821
if I divide this and this by two,

00:05:55.821 --> 00:05:58.229
is going to be equal to one half

00:05:58.229 --> 00:06:00.145
of her frequency initial,

00:06:00.145 --> 00:06:02.812
is going to be equal to one half

00:06:04.203 --> 00:06:07.870
of her initial frequency,
frequency initial.

00:06:09.586 --> 00:06:12.658
If your initial frequency
is twice your final,

00:06:12.658 --> 00:06:14.955
then your final is going to
be one half your initial.

00:06:14.955 --> 00:06:17.672
I could just divide both sides by two.

00:06:17.672 --> 00:06:20.455
So once again how does the
frequency of her spin change?

00:06:20.455 --> 00:06:22.521
Well it looks like her frequency goes

00:06:22.521 --> 00:06:24.092
by half, and that makes sense.

00:06:24.092 --> 00:06:27.202
If your angular velocity
is going down by half,

00:06:27.202 --> 00:06:29.376
you are rotating half as fast

00:06:29.376 --> 00:06:30.809
and so you're going to be able to complete

00:06:30.809 --> 00:06:33.024
half as many cycles per second.

00:06:33.024 --> 00:06:35.600
So it makes sense that we are decreasing,

00:06:35.600 --> 00:06:37.783
our frequency is decreasing

00:06:37.783 --> 00:06:40.080
by a factor of two, it is halving.

00:06:40.080 --> 00:06:42.448
Decreasing by a factor
of two is the same thing

00:06:42.448 --> 00:06:46.615
as saying your frequency
gets multiplied by one half.


Connecting period and frequency to angular velocity

WEBVTT
Kind: captions
Language: en

00:00:01.134 --> 00:00:01.992
- [Instructor] What we're going to do

00:00:01.992 --> 00:00:03.549
in this video is continue talking

00:00:03.549 --> 00:00:05.600
about uniform circular motion.

00:00:05.600 --> 00:00:07.249
And in that context, we're gonna talk

00:00:07.249 --> 00:00:09.840
about the idea of period,

00:00:09.840 --> 00:00:11.590
which we denote with a capital T,

00:00:11.590 --> 00:00:13.788
or we tend to denote with a capital T,

00:00:13.788 --> 00:00:16.251
and a very related idea.

00:00:16.251 --> 00:00:18.092
And that's of frequency,

00:00:18.092 --> 00:00:21.842
which we typically denote
with a lower case f.

00:00:21.842 --> 00:00:25.030
So you might have seen these
ideas in other contexts,

00:00:25.030 --> 00:00:26.830
but we'll just make sure we get them.

00:00:26.830 --> 00:00:28.682
And then we'll connect it to the idea

00:00:28.682 --> 00:00:31.659
of angular velocity, in
particular the magnitude

00:00:31.659 --> 00:00:34.080
of angular velocity,
which we've already seen

00:00:34.080 --> 00:00:36.742
we can denote with a lower case omega.

00:00:36.742 --> 00:00:38.792
Since I don't have a little arrow on top,

00:00:38.792 --> 00:00:40.908
you could view it, just
the lower case omega,

00:00:40.908 --> 00:00:43.551
as the magnitude of angular velocity.

00:00:43.551 --> 00:00:47.182
But first, what is period
and what is frequency?

00:00:47.182 --> 00:00:49.362
Well, period is how long does it take

00:00:49.362 --> 00:00:51.111
to complete a cycle?

00:00:51.111 --> 00:00:53.971
And if we're talking about
uniform circular motion,

00:00:53.971 --> 00:00:56.122
a cycle is how long does it take,

00:00:56.122 --> 00:00:58.000
if this is, say, some
type of a tennis ball

00:00:58.000 --> 00:01:00.592
that's tethered to a nail right over here

00:01:00.592 --> 00:01:04.260
and it's moving with some uniform speed,

00:01:04.260 --> 00:01:06.080
a period is, well, how long does it take

00:01:06.080 --> 00:01:08.127
to go all the way around once?

00:01:08.127 --> 00:01:11.392
So, for example, if you
have a period of one second,

00:01:11.392 --> 00:01:14.279
this ball would move like this,

00:01:14.279 --> 00:01:16.279
one second, two seconds,

00:01:17.549 --> 00:01:20.693
three seconds, four seconds.

00:01:20.693 --> 00:01:22.411
That would be a period of one second.

00:01:22.411 --> 00:01:24.173
If you had a period of two seconds,

00:01:24.173 --> 00:01:26.373
well, it would go half the speed.

00:01:26.373 --> 00:01:30.233
You would have one second, two seconds,

00:01:30.233 --> 00:01:32.566
three seconds, four seconds,

00:01:33.944 --> 00:01:36.111
five seconds, six seconds.

00:01:37.314 --> 00:01:38.733
And if you went the other way,

00:01:38.733 --> 00:01:40.852
if you had a period of half a second,

00:01:40.852 --> 00:01:44.451
well, then it would be one second,

00:01:44.451 --> 00:01:45.451
two seconds,

00:01:46.624 --> 00:01:50.257
and so your period would be half a second.

00:01:50.257 --> 00:01:54.882
It would take you half a
second to complete a cycle.

00:01:54.882 --> 00:01:58.967
The unit of period is
going to be the second,

00:01:58.967 --> 00:02:02.619
the unit of time and it's
typically given in seconds.

00:02:02.619 --> 00:02:04.308
Now, what about frequency?

00:02:04.308 --> 00:02:08.819
Well, frequency literally is
the reciprocal of the period.

00:02:08.819 --> 00:02:11.652
So frequency is equal to one over,

00:02:12.990 --> 00:02:14.867
let me write that one a little bit neater,

00:02:14.867 --> 00:02:17.137
one over the period.

00:02:17.137 --> 00:02:18.795
And one way to think about it is

00:02:18.795 --> 00:02:22.788
well, how many cycles can
you complete in a second?

00:02:22.788 --> 00:02:25.266
Period is how many seconds does it take

00:02:25.266 --> 00:02:27.846
to complete a cycle, while frequency is

00:02:27.846 --> 00:02:31.096
how many cycles can you do in a second?

00:02:32.027 --> 00:02:36.349
So, for example, if I can
do two cycles in a second,

00:02:36.349 --> 00:02:38.349
one second, two seconds,

00:02:39.588 --> 00:02:41.186
three seconds,

00:02:41.186 --> 00:02:45.079
then my frequency is
two cycles per second.

00:02:45.079 --> 00:02:47.869
And the unit for frequency is,

00:02:47.869 --> 00:02:50.617
sometimes you'll hear
people say just per second,

00:02:50.617 --> 00:02:53.308
so the unit, sometimes
you'll see people just say

00:02:53.308 --> 00:02:55.479
an inverse second like that,

00:02:55.479 --> 00:02:58.667
or sometimes they'll use the shorthand Hz,

00:02:58.667 --> 00:03:00.584
which stands for Hertz.

00:03:01.978 --> 00:03:06.360
And Hertz is sometimes substituted
with cycles per second.

00:03:06.360 --> 00:03:08.628
So this you could view as seconds or even

00:03:08.628 --> 00:03:10.360
seconds per cycle.

00:03:10.360 --> 00:03:13.260
And this is cycles per second.

00:03:13.260 --> 00:03:15.008
Now with that out of the way,

00:03:15.008 --> 00:03:17.268
let's see if we can connect these ideas

00:03:17.268 --> 00:03:20.249
to the magnitude of angular velocity.

00:03:20.249 --> 00:03:22.519
So let's just think about
a couple of scenarios.

00:03:22.519 --> 00:03:26.109
Let's say that the magnitude
of our angular velocity,

00:03:26.109 --> 00:03:28.359
let's say it is pi radians,

00:03:30.629 --> 00:03:32.462
pi radians per second.

00:03:33.662 --> 00:03:37.452
So if we knew that, what
is the period going to be?

00:03:37.452 --> 00:03:40.949
Pause this video and see
if you can figure that out.

00:03:40.949 --> 00:03:42.261
So let's work through it together.

00:03:42.261 --> 00:03:44.361
So, this ball is going to move

00:03:44.361 --> 00:03:46.931
through pi radians every second.

00:03:46.931 --> 00:03:48.571
So how long is it going to take

00:03:48.571 --> 00:03:50.629
for it to complete two pi radians?

00:03:50.629 --> 00:03:54.796
'Cause remember, one complete
rotation is two pi radians.

00:03:55.770 --> 00:03:58.420
Well, if it's going pi radians per second,

00:03:58.420 --> 00:04:02.003
it's gonna take it two
seconds to go two pi radians.

00:04:02.003 --> 00:04:04.932
And so, the period here, let me write it,

00:04:04.932 --> 00:04:09.900
the period here is going
to be equal to two seconds.

00:04:09.900 --> 00:04:12.202
Now, I kind of did that intuitively,

00:04:12.202 --> 00:04:16.452
but how did I actually
manipulate the omega here?

00:04:16.452 --> 00:04:18.542
Well, one way to think
about it, the period,

00:04:18.542 --> 00:04:21.796
I said, look, in order to
complete one entire rotation,

00:04:21.796 --> 00:04:24.629
I have to complete two pi radians.

00:04:26.079 --> 00:04:29.803
So that is one entire cycle
is going to be two pi radians.

00:04:29.803 --> 00:04:32.641
And then I'm gonna divide it by how fast,

00:04:32.641 --> 00:04:35.213
what my angular velocity is going to be.

00:04:35.213 --> 00:04:36.493
So I'm gonna divide it by,

00:04:36.493 --> 00:04:40.612
in this case I'm gonna
divide it by pi radians,

00:04:40.612 --> 00:04:44.779
pi, and I could write it
out pi radians per second.

00:04:45.791 --> 00:04:47.494
I'm saying how far do I have to go

00:04:47.494 --> 00:04:49.293
to complete a cycle and
that I'm dividing it

00:04:49.293 --> 00:04:52.733
by how fast I am going through the angles.

00:04:52.733 --> 00:04:56.512
And that's where I got
the two seconds from.

00:04:56.512 --> 00:04:59.121
And so, already you can think of a formula

00:04:59.121 --> 00:05:02.621
that connects period and angular velocity.

00:05:04.516 --> 00:05:08.183
Period is equal to,
remember, two pi radians

00:05:09.145 --> 00:05:10.575
is an entire cycle.

00:05:10.575 --> 00:05:13.226
And so you just want to
divide that by how quickly

00:05:13.226 --> 00:05:15.417
you're going through the angles.

00:05:15.417 --> 00:05:17.896
And so that there will connect your period

00:05:17.896 --> 00:05:19.814
and angular velocity.

00:05:19.814 --> 00:05:21.065
Now if we know the period,

00:05:21.065 --> 00:05:24.026
it's quite straightforward
to figure out the frequency.

00:05:24.026 --> 00:05:27.515
So the frequency is just
one over the period.

00:05:27.515 --> 00:05:29.946
So the frequency is,
we've already said it's

00:05:29.946 --> 00:05:32.356
one over the period, and so the reciprocal

00:05:32.356 --> 00:05:35.237
of two pi over omega is going to be

00:05:35.237 --> 00:05:36.737
omega over two pi.

00:05:40.598 --> 00:05:43.799
And in this situation where
the period was two seconds,

00:05:43.799 --> 00:05:45.648
you don't even know what omega is,

00:05:45.648 --> 00:05:47.978
and someone says the
period is two seconds,

00:05:47.978 --> 00:05:50.645
then you know that the frequency

00:05:51.709 --> 00:05:54.709
is going to be one over two seconds.

00:05:59.209 --> 00:06:02.805
Or, you could view this
as being equal to 1/2.

00:06:02.805 --> 00:06:05.086
You could sometimes see units like that,

00:06:05.086 --> 00:06:07.095
which is kind of per second.

00:06:07.095 --> 00:06:10.428
But I like to use Hertz, and in my brain

00:06:11.294 --> 00:06:15.286
I say this means 1/2 cycles per second.

00:06:15.286 --> 00:06:16.586
So one way to think about it,

00:06:16.586 --> 00:06:18.626
it takes two seconds to complete.

00:06:18.626 --> 00:06:20.944
If I'm doing pi radians per second,

00:06:20.944 --> 00:06:24.237
my ball here is going to go one second,

00:06:24.237 --> 00:06:27.654
two seconds, three seconds, four seconds.

00:06:28.567 --> 00:06:30.013
And you see, just like that,

00:06:30.013 --> 00:06:32.236
my period is indeed two seconds.

00:06:32.236 --> 00:06:34.366
And you also see that in each second,

00:06:34.366 --> 00:06:36.697
remember in each second
I cover pi radians.

00:06:36.697 --> 00:06:39.826
Well, pi radians is half a cycle.

00:06:39.826 --> 00:06:42.743
I complete half a cycle per second.


Visually dividing unit fraction by a whole number

WEBVTT
Kind: captions
Language: en

00:00:00.608 --> 00:00:04.304
- We are asked to figure out
what is 1/7 divided by four,

00:00:04.304 --> 00:00:06.551
and they help us out with this diagram.

00:00:06.551 --> 00:00:09.893
We have a whole divided
into seven equal sections.

00:00:09.893 --> 00:00:11.485
Each of those is a seventh,

00:00:11.485 --> 00:00:13.806
and we have one of those
sevenths filled in,

00:00:13.806 --> 00:00:17.065
so this is 1/7 right over here,

00:00:17.065 --> 00:00:20.034
and then they divide it
into four equal sections.

00:00:20.034 --> 00:00:22.762
In fact, they divide
all of the sevenths into

00:00:22.762 --> 00:00:25.731
four equal sections, and so 1/7,

00:00:25.731 --> 00:00:28.957
which is this whole green
bar divided by four,

00:00:28.957 --> 00:00:32.833
well what would be this
fraction of the whole

00:00:32.833 --> 00:00:34.461
that is in a question mark.

00:00:34.461 --> 00:00:37.110
Can you pause this video
and figure out what fraction

00:00:37.110 --> 00:00:40.093
of the whole is this question mark?

00:00:40.093 --> 00:00:43.439
When we divided the first
seventh into four equal

00:00:43.439 --> 00:00:45.846
sections, we also divided
all of the sevenths into

00:00:45.846 --> 00:00:47.315
four equal sections,

00:00:47.315 --> 00:00:51.314
and so now the entire
whole is 28 equal sections

00:00:51.314 --> 00:00:54.893
because you have a four by seven grid.

00:00:54.893 --> 00:00:57.372
You have one, two, three, four rows

00:00:57.372 --> 00:00:58.918
and you still have your seven columns,

00:00:58.918 --> 00:01:02.418
and you can count them, seven, 14, 21, 28,

00:01:03.726 --> 00:01:07.227
and so 1/7 divided by four is going to be

00:01:07.227 --> 00:01:09.310
one of these 28 sections.

00:01:10.608 --> 00:01:13.608
This right over here is one over 28.

00:01:15.968 --> 00:01:18.743
This is 1/28.

00:01:18.743 --> 00:01:20.826
Let's do another example.

00:01:21.888 --> 00:01:25.265
We're told use the number
line below to help visualize

00:01:25.265 --> 00:01:28.432
1/5 being divided by three.

00:01:29.981 --> 00:01:32.627
As we go from zero to
one on the number line,

00:01:32.627 --> 00:01:34.709
you can divide it into five
equal sections where that's

00:01:34.709 --> 00:01:39.147
1/5, 2/5, 3/5, 4/5,

00:01:39.147 --> 00:01:42.112
and of course 5/5 is equal to one,

00:01:42.112 --> 00:01:44.997
but we want 1/5 divided by three,

00:01:44.997 --> 00:01:48.170
so we took the section from zero to 1/5

00:01:48.170 --> 00:01:50.613
and we divided it into
three equal sections,

00:01:50.613 --> 00:01:52.286
and so the first of those sections,

00:01:52.286 --> 00:01:53.595
this one right over here,

00:01:53.595 --> 00:01:56.537
that would be 1/5 divided by three.

00:01:56.537 --> 00:01:58.982
What is this going to be equal to?

00:01:58.982 --> 00:02:02.157
Pause this video again and see
if you can figure that out.

00:02:02.157 --> 00:02:06.311
The key realization is when
we divided each of the fifths

00:02:06.311 --> 00:02:08.557
into three more equal sections,

00:02:08.557 --> 00:02:12.173
we can now think of each of
these steps as a fifteenth

00:02:12.173 --> 00:02:15.189
because now we have one,
two, three, four, five,

00:02:15.189 --> 00:02:19.356
six, seven, eight, nine,
10, 11, 12, 13, 14, 15 equal

00:02:20.247 --> 00:02:23.178
sections between zero and one,

00:02:23.178 --> 00:02:24.755
and where did that 15 come from?

00:02:24.755 --> 00:02:27.319
We had five equal
sections and then we split

00:02:27.319 --> 00:02:30.325
each of those five into
three more equal sections

00:02:30.325 --> 00:02:32.477
so five times three is 15.

00:02:32.477 --> 00:02:35.249
This right over here is 1/15,

00:02:35.249 --> 00:02:37.492
this is 2/15,

00:02:37.492 --> 00:02:40.670
this is 3/15, which is equivalent to 1/5

00:02:40.670 --> 00:02:42.828
and we can keep going on and on and on,

00:02:42.828 --> 00:02:46.039
but the key realization
here is if I take that first

00:02:46.039 --> 00:02:49.741
1/5 and if I divide it
into three equal sections

00:02:49.741 --> 00:02:52.594
and I go only as far as that
first of the three equal

00:02:52.594 --> 00:02:56.761
sections, that is going to be
1/15, 1/15 and we are done.


Example diving a unit fraction by a whole number

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:02.057
- [Instructor] So let's
see if we can figure out

00:00:02.057 --> 00:00:05.402
what 1/3 divided by five is.

00:00:05.402 --> 00:00:06.877
And I'll give you a hint.

00:00:06.877 --> 00:00:09.470
Try to draw out 1/3 of a whole

00:00:09.470 --> 00:00:11.978
and then divide it into
five equal sections.

00:00:11.978 --> 00:00:14.634
Pause this video and try to do that.

00:00:14.634 --> 00:00:16.341
I always try to work through it together

00:00:16.341 --> 00:00:20.021
and to help us as I promised
or I suggested I guess.

00:00:20.021 --> 00:00:22.757
We said let's draw 1/3.

00:00:22.757 --> 00:00:27.267
I will represent a whole by
that square right over there.

00:00:27.267 --> 00:00:31.389
And now let me split up
into three equal sections.

00:00:31.389 --> 00:00:35.607
So this is all hand drawn
with the aid of a computer.

00:00:35.607 --> 00:00:37.176
So it's not going to be perfect.

00:00:37.176 --> 00:00:39.986
But let's say that that
is three equal sections.

00:00:39.986 --> 00:00:41.404
It's roughly three equal sections.

00:00:41.404 --> 00:00:42.364
I didn't do it perfectly

00:00:42.364 --> 00:00:44.013
but you hopefully get the idea.

00:00:44.013 --> 00:00:48.757
And so 1/3 would be one of
those three equal sections.

00:00:48.757 --> 00:00:50.632
So that's a third right over there

00:00:50.632 --> 00:00:52.538
that I have just shaded in

00:00:52.538 --> 00:00:54.977
and I want to divide it into five.

00:00:54.977 --> 00:00:57.812
I want to divide it by five, I should say.

00:00:57.812 --> 00:00:59.288
So let's do that.

00:00:59.288 --> 00:01:02.460
So to divide it by five, I'm
gonna divide it into five

00:01:02.460 --> 00:01:03.677
equal sections.

00:01:03.677 --> 00:01:06.570
And if I'm dividing that one
into five equal sections,

00:01:06.570 --> 00:01:07.919
let me just divide all of the thirds

00:01:07.919 --> 00:01:09.322
into five equal sections.

00:01:09.322 --> 00:01:12.001
I'm essentially just going
to make five rows here.

00:01:12.001 --> 00:01:13.588
One, and I'm gonna eyeball it,

00:01:13.588 --> 00:01:14.879
so it's going to be approximate.

00:01:14.879 --> 00:01:19.046
Two, three, and then four
and five equal sections.

00:01:23.559 --> 00:01:27.726
I now split this whole into
one, two, three, four, five rows

00:01:29.703 --> 00:01:31.258
of equal height.

00:01:31.258 --> 00:01:34.642
Now if I go to my original
third and I divide it by five,

00:01:34.642 --> 00:01:38.189
I would be left with this right over here.

00:01:38.189 --> 00:01:41.858
But what fraction is this of the whole?

00:01:41.858 --> 00:01:44.576
Well what I've done now
is I've split my whole

00:01:44.576 --> 00:01:46.729
into 15 equal sections.

00:01:46.729 --> 00:01:47.562
How do I know that?

00:01:47.562 --> 00:01:48.443
Well I could count them.

00:01:48.443 --> 00:01:50.855
One, two, three, four,
five, six, seven, eight,

00:01:50.855 --> 00:01:53.867
nine, 10, 11, 12, 13, 14, 15.

00:01:53.867 --> 00:01:58.256
Or you could just say look
I had one, two, three thirds

00:01:58.256 --> 00:02:00.610
and now each of those have been split into

00:02:00.610 --> 00:02:04.258
one, two, three, four,
five equal sections.

00:02:04.258 --> 00:02:06.891
So three times five is 15.

00:02:06.891 --> 00:02:08.683
So each of these is a 15th

00:02:08.683 --> 00:02:13.009
and so the 1/3 divided by five is just one

00:02:13.009 --> 00:02:14.873
of those 15ths.

00:02:14.873 --> 00:02:17.423
So that right over there
is one of those 15ths.

00:02:17.423 --> 00:02:21.045
So this is going to be equal to 1/15

00:02:21.045 --> 00:02:22.378
and we are done.


Visually dividing whole numbers by unit fractions

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.913
- [Narrator] If five is
divided into pieces that are

00:00:02.913 --> 00:00:07.080
each one half of a whole,
how many pieces are there?

00:00:08.352 --> 00:00:09.633
And this would be the
equivalent of saying,

00:00:09.633 --> 00:00:12.762
"What is five divided by 1/2?"

00:00:12.762 --> 00:00:14.668
And they help us out with this visual.

00:00:14.668 --> 00:00:16.930
So pause this video and see if

00:00:16.930 --> 00:00:18.827
you can figure out what that is.

00:00:18.827 --> 00:00:22.698
How many pieces would you
get if you divide five

00:00:22.698 --> 00:00:26.531
into pieces that are
each one half of a whole?

00:00:27.458 --> 00:00:30.005
All right, now let's work
through this together.

00:00:30.005 --> 00:00:33.556
So here on this number line
we go from zero to five,

00:00:33.556 --> 00:00:36.193
and then notice they've
divided into pieces

00:00:36.193 --> 00:00:38.233
that are each a half of a whole.

00:00:38.233 --> 00:00:40.920
This is one piece right over here.

00:00:40.920 --> 00:00:42.919
So how many of those halves,

00:00:42.919 --> 00:00:45.337
so this right over here is one half,

00:00:45.337 --> 00:00:49.091
how many of those halves
does it take to make five?

00:00:49.091 --> 00:00:53.133
Well, two halves make a whole,
and we have five wholes.

00:00:53.133 --> 00:00:57.060
So it's going to be five times two, or 10.

00:00:57.060 --> 00:00:58.065
And we see that right over here.

00:00:58.065 --> 00:01:02.179
One half, two, three, four,
five, six, seven, eight,

00:01:02.179 --> 00:01:04.929
nine, 10 halves make five wholes.

00:01:06.897 --> 00:01:09.210
So this is going to be equal to 10.

00:01:09.210 --> 00:01:13.740
So five divided into pieces
of one half, or five wholes

00:01:13.740 --> 00:01:17.907
divided into pieces of one half
would be equal to 10 pieces.

00:01:19.081 --> 00:01:21.528
Let's do another example.

00:01:21.528 --> 00:01:23.502
So we have a similar question here.

00:01:23.502 --> 00:01:24.455
Here we're asked,

00:01:24.455 --> 00:01:26.657
"If three wholes are divided into pieces

00:01:26.657 --> 00:01:30.824
that are each 1/6 of a whole,
how many pieces are there?"

00:01:31.683 --> 00:01:35.092
Once again, pause this
video and think about it.

00:01:35.092 --> 00:01:38.117
Well, they really help
us out with this visual

00:01:38.117 --> 00:01:39.220
because we have three wholes.

00:01:39.220 --> 00:01:42.483
This is one whole, two
whole, and then three wholes,

00:01:42.483 --> 00:01:45.706
and then we have divided them into pieces

00:01:45.706 --> 00:01:48.015
that are each 1/6 of a whole.

00:01:48.015 --> 00:01:50.098
This is a sixth right
over here, this is a sixth

00:01:50.098 --> 00:01:52.765
right over here, so each
of these are sixths.

00:01:52.765 --> 00:01:57.634
And so, if we look at this,
we have six sixths in a whole,

00:01:57.634 --> 00:02:01.801
and so in three wholes we're
gonna have six, 12, 18 pieces.

00:02:02.707 --> 00:02:04.931
And you could literally
just count these up.

00:02:04.931 --> 00:02:06.471
But it makes sense.

00:02:06.471 --> 00:02:10.053
If you take three wholes and
you divide it into sixths,

00:02:10.053 --> 00:02:12.806
so this is a sixth right over here,

00:02:12.806 --> 00:02:16.828
each of these wholes are
going to be six sixths,

00:02:16.828 --> 00:02:18.851
so three wholes are going to be

00:02:18.851 --> 00:02:21.934
three times six sixths, or 18 sixths.

00:02:23.843 --> 00:02:24.926
There you go.


Example dividing a whole by a unit fraction

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.865
- Lets think about what three divided

00:00:02.865 --> 00:00:05.032
by one fourth is equal to.

00:00:07.021 --> 00:00:08.253
Pause this video and see if you can

00:00:08.253 --> 00:00:09.962
figure it out on your own.

00:00:09.962 --> 00:00:12.721
And I'll give you a
hint, take three wholes

00:00:12.721 --> 00:00:16.401
and divide it into pieces
or sections that are each

00:00:16.401 --> 00:00:18.522
one fourth of a whole and then think

00:00:18.522 --> 00:00:21.609
about how many sections you have.

00:00:21.609 --> 00:00:23.511
Alright now lets work
through this together.

00:00:23.511 --> 00:00:25.681
Now let me draw three wholes.

00:00:25.681 --> 00:00:27.840
And so lets say this is...

00:00:27.840 --> 00:00:29.879
I'm gonna hand draw it, so
its not gonna be perfect.

00:00:29.879 --> 00:00:32.603
But lets say that is one
whole right over there.

00:00:32.603 --> 00:00:35.603
This is two wholes right over there.

00:00:37.485 --> 00:00:42.045
And then I have my third
whole just like that.

00:00:42.045 --> 00:00:44.395
And now I'm going to divide it into pieces

00:00:44.395 --> 00:00:48.295
or sections that are each
one fourth of a whole.

00:00:48.295 --> 00:00:51.128
So this first whole is four pieces

00:00:53.215 --> 00:00:56.345
that are each one fourth
each, of four fourths.

00:00:56.345 --> 00:01:00.495
And then I have another four
fourths right over here.

00:01:00.495 --> 00:01:04.355
And then I have another four
fourths right over here.

00:01:04.355 --> 00:01:06.655
Another way to think about
it, I took the three wholes,

00:01:06.655 --> 00:01:08.682
and I divided it into fourths.

00:01:08.682 --> 00:01:10.345
So if you divide it into fourths how

00:01:10.345 --> 00:01:12.630
many pieces am I going to have?

00:01:12.630 --> 00:01:15.244
Well this is pretty straight
forward, you have one

00:01:15.244 --> 00:01:19.411
two three four five six seven
eight nine ten 11 12 pieces.

00:01:24.477 --> 00:01:26.659
And I didn't even have to count that,

00:01:26.659 --> 00:01:28.982
each whole is gonna be four fourths.

00:01:28.982 --> 00:01:33.071
So three wholes is going to
be three times four fourths,

00:01:33.071 --> 00:01:35.404
or 12 pieces or 12 sections.

00:01:36.651 --> 00:01:39.632
So that is going to be equal to 12.

00:01:39.632 --> 00:01:42.589
Another way to think about
it is if you take three

00:01:42.589 --> 00:01:46.058
and you divide it into equal
sections that are each a fourth

00:01:46.058 --> 00:01:49.391
you are going to have 12 equal sections.


Visually dividing a fraction by a whole number

WEBVTT
Kind: captions
Language: en

00:00:00.279 --> 00:00:01.457
- [Instructor] Let's
see if we can figure out

00:00:01.457 --> 00:00:05.457
what 2/3 divided by five is equal to.

00:00:09.309 --> 00:00:12.731
Pause this video and see
if you can figure this out.

00:00:12.731 --> 00:00:14.776
Well, there is a couple of
ways that we can approach it.

00:00:14.776 --> 00:00:16.889
We can first do it in a conceptual way,

00:00:16.889 --> 00:00:17.966
think about it visually.

00:00:17.966 --> 00:00:20.578
To do that, let me represent 2/3,

00:00:20.578 --> 00:00:23.483
so let's say that what I'm drawing

00:00:23.483 --> 00:00:25.251
right over here is a whole.

00:00:25.251 --> 00:00:29.071
This is a square, and
it represents a whole.

00:00:29.071 --> 00:00:31.305
Now, I can divide into
three equal sections.

00:00:31.305 --> 00:00:33.012
I'm gonna try to hand-draw that.

00:00:33.012 --> 00:00:36.262
So, this is, that looks pretty good,

00:00:36.262 --> 00:00:38.626
three equal sections here.

00:00:38.626 --> 00:00:42.054
Each is 1/3, and we have 2/3,

00:00:42.054 --> 00:00:45.027
so we are really representing

00:00:45.027 --> 00:00:47.893
all of this stuff right over here.

00:00:47.893 --> 00:00:49.976
That is two of my thirds.

00:00:51.867 --> 00:00:55.727
Now, I wanna divide those 2/3 by five.

00:00:55.727 --> 00:00:57.271
Well, the way I could do this is

00:00:57.271 --> 00:00:59.831
I could divide it into
five equal sections.

00:00:59.831 --> 00:01:01.029
But, if I'm doing it, I might as well

00:01:01.029 --> 00:01:02.928
just divide everything, all the thirds,

00:01:02.928 --> 00:01:05.743
into five equal sections,
so let me do that.

00:01:05.743 --> 00:01:09.910
So, one, two, three, and then
four, and five equal sections.

00:01:15.604 --> 00:01:19.487
So, what is one of those
five equal sections

00:01:19.487 --> 00:01:22.311
of my original 2/3?

00:01:22.311 --> 00:01:23.704
Well, this right over here

00:01:23.704 --> 00:01:28.560
is one of those five equal
sections of my original 2/3.

00:01:28.560 --> 00:01:31.452
Notice I could draw that, I
could draw another one here,

00:01:31.452 --> 00:01:34.763
another one here, another one
there, and another one there,

00:01:34.763 --> 00:01:39.247
and I would have five equal
sections that make up those 2/3.

00:01:39.247 --> 00:01:41.909
But, what does just one of them represent?

00:01:41.909 --> 00:01:45.017
And if we figure out what
this represents of the whole,

00:01:45.017 --> 00:01:49.090
then we know what 2/3 divided by five is.

00:01:49.090 --> 00:01:51.742
Well, when I took my
thirds and I divided them

00:01:51.742 --> 00:01:55.847
into five equal sections, I
essentially constructed 15ths.

00:01:55.847 --> 00:01:57.074
How do I know that?

00:01:57.074 --> 00:02:00.142
Well, I had one, two, three thirds,

00:02:00.142 --> 00:02:02.308
and then I divided it into

00:02:02.308 --> 00:02:05.628
one, two, three, four, five sections,

00:02:05.628 --> 00:02:09.631
so each of these squares
right over here is a 15th.

00:02:09.631 --> 00:02:11.469
You have three times five,

00:02:11.469 --> 00:02:13.206
and you could count 'em if you like.

00:02:13.206 --> 00:02:17.647
And, what we have circled off
in red is two of these 15ths.

00:02:17.647 --> 00:02:20.379
We have 1/15 right over here,

00:02:20.379 --> 00:02:23.444
and then 2/15 right over there.

00:02:23.444 --> 00:02:27.887
So, this is going to be equal to 2/15.

00:02:27.887 --> 00:02:30.232
Now, another way that
you could think about,

00:02:30.232 --> 00:02:32.276
and over time this is the
way you will approach it,

00:02:32.276 --> 00:02:34.814
but it's nice to think
about it conceptually,

00:02:34.814 --> 00:02:37.071
when you divide by any number,

00:02:37.071 --> 00:02:40.307
it's the same thing as
multiplying by the reciprocal.

00:02:40.307 --> 00:02:44.474
So, five is the same thing
as five wholes, or 5/1.

00:02:47.624 --> 00:02:50.582
And so, 2/3 divided by
five is the same thing

00:02:50.582 --> 00:02:55.121
as 2/3 times the reciprocal of five,

00:02:55.121 --> 00:02:58.383
or the reciprocal of 5/1,
which is you just swap

00:02:58.383 --> 00:03:02.679
the numerator and the
denominator, which is 1/5.

00:03:02.679 --> 00:03:05.054
And so, another way of
thinking about this is

00:03:05.054 --> 00:03:08.368
this is 1/5 of 2/3, which
it once again will be

00:03:08.368 --> 00:03:11.610
this section right over here.

00:03:11.610 --> 00:03:13.063
The way you could compute this,

00:03:13.063 --> 00:03:15.390
conceptually, you see that this is 2/15,

00:03:15.390 --> 00:03:17.563
but you could also say well,
I could just multiply it.

00:03:17.563 --> 00:03:19.068
When I multiply fractions

00:03:19.068 --> 00:03:20.409
I can just multiply the numerators.

00:03:20.409 --> 00:03:22.664
Two times one is two.

00:03:22.664 --> 00:03:24.635
We do that same red color.

00:03:24.635 --> 00:03:27.032
Two times one is two,

00:03:27.032 --> 00:03:28.820
and then I could multiply
the denominators.

00:03:28.820 --> 00:03:31.543
Three times five is 15.

00:03:31.543 --> 00:03:33.635
And, hopefully, what we just drew out

00:03:33.635 --> 00:03:37.635
may help make sense of
why dividing by something

00:03:38.654 --> 00:03:41.478
is the same thing as
multiplying by the reciprocal.

00:03:41.478 --> 00:03:43.435
Then, when you multiply fractions,

00:03:43.435 --> 00:03:46.119
it's the same thing as
multiplying the numerators

00:03:46.119 --> 00:03:48.244
to get our new numerator,
and then multiplying

00:03:48.244 --> 00:03:51.911
the denominators to get
our new denominator.


Dividing mixed numbers example

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.079
- [Tutor] Let's see if we can figure out

00:00:01.079 --> 00:00:04.941
what four and four fifths
divided by one and one half is

00:00:04.941 --> 00:00:06.632
and I encourage you to pause the video

00:00:06.632 --> 00:00:08.048
and see if you can
figure it out on your own

00:00:08.048 --> 00:00:09.373
and I'll give you a hint,

00:00:09.373 --> 00:00:12.227
see if you can rewrite these mixed numbers

00:00:12.227 --> 00:00:16.685
as what is sometimes
called improper fractions.

00:00:16.685 --> 00:00:19.110
Alright, now let's do this together,

00:00:19.110 --> 00:00:22.231
so how can we rewrite
four and four fifths?

00:00:22.231 --> 00:00:25.186
Well, four and four
fifths is the same thing

00:00:25.186 --> 00:00:26.762
as, if we take the four,
that's the same thing

00:00:26.762 --> 00:00:30.679
as four plus four fifths,
four plus four fifths

00:00:31.732 --> 00:00:35.065
and four is the same thing as 20 fifths,

00:00:36.331 --> 00:00:39.168
so 4 is the same thing as 20 fifths

00:00:39.168 --> 00:00:42.539
and then plus four fifths,
well, what are you going to get?

00:00:42.539 --> 00:00:44.257
Your going to get, you add the numerators,

00:00:44.257 --> 00:00:46.674
you get 24 fifths, 24 fifths,

00:00:47.872 --> 00:00:50.378
another way to think about
it is take this denominator,

00:00:50.378 --> 00:00:54.663
take the fifths, multiply
by four, you get 20 fifths,

00:00:54.663 --> 00:00:59.102
plus the four fifths that
you already have is 24 fifths

00:00:59.102 --> 00:01:02.372
and so this is the same
things as 24 fifths

00:01:02.372 --> 00:01:04.854
divided by, use the same idea,

00:01:04.854 --> 00:01:08.437
one and a half is the same
thing as one plus one half,

00:01:08.437 --> 00:01:11.751
one is the same thing as two halves

00:01:11.751 --> 00:01:14.986
plus one half and so that's
going to be, add the numerators,

00:01:14.986 --> 00:01:18.435
that's going to be three
halves, so just like this,

00:01:18.435 --> 00:01:21.150
we're able to rewrite our expression

00:01:21.150 --> 00:01:23.948
as 24 fifths divided by three halves

00:01:23.948 --> 00:01:26.115
and now the key realization is

00:01:26.115 --> 00:01:28.503
is that that is the same thing

00:01:28.503 --> 00:01:29.503
as 24 fifths

00:01:31.186 --> 00:01:34.773
times the reciprocal of three halves,

00:01:34.773 --> 00:01:36.512
so times, pause the video,

00:01:36.512 --> 00:01:39.043
what's the reciprocal of three halves?

00:01:39.043 --> 00:01:40.485
Well, the reciprocal of three halves,

00:01:40.485 --> 00:01:42.339
you just swap the numerator
and the denominator,

00:01:42.339 --> 00:01:45.762
it's going to be two over three,

00:01:45.762 --> 00:01:47.649
now what is this going to be?

00:01:47.649 --> 00:01:49.660
Well, there's a couple of ways to do it,

00:01:49.660 --> 00:01:51.718
you could just straight
up multiply the numerators

00:01:51.718 --> 00:01:54.729
and you would get 48 and then
multiply the denominators

00:01:54.729 --> 00:01:58.272
and you would get 15,
so you get 48 over 15,

00:01:58.272 --> 00:02:00.487
but you might be able to rewrite that

00:02:00.487 --> 00:02:04.728
in sometimes what's called
a more simplified way,

00:02:04.728 --> 00:02:06.220
but another way of thinking about this

00:02:06.220 --> 00:02:09.413
is you could just say,
well this is the same thing

00:02:09.413 --> 00:02:11.580
as 24 times two, times two

00:02:13.812 --> 00:02:16.758
over five times three and simplify things,

00:02:16.758 --> 00:02:19.415
before you even multiply them out,

00:02:19.415 --> 00:02:21.665
five times five times three

00:02:24.185 --> 00:02:25.730
and you realize that look,

00:02:25.730 --> 00:02:28.670
24 and three are both divisible by three,

00:02:28.670 --> 00:02:31.371
so let me divide them both by three,

00:02:31.371 --> 00:02:33.816
so 24 divided by three is eight

00:02:33.816 --> 00:02:37.316
and three divided by three is equal to one

00:02:38.287 --> 00:02:40.150
and then you could multiply the numerators

00:02:40.150 --> 00:02:42.483
and the denominators and so,

00:02:43.828 --> 00:02:47.141
and so you get in the numerator,

00:02:47.141 --> 00:02:48.831
eight times two is 16,

00:02:48.831 --> 00:02:51.599
in the denominator, you get a five,

00:02:51.599 --> 00:02:52.940
so you get 16 fifths

00:02:52.940 --> 00:02:56.177
and then if you wanna express
that as a mixed number,

00:02:56.177 --> 00:02:59.951
16 over five, well, five goes into 16

00:02:59.951 --> 00:03:01.808
three times with one left over,

00:03:01.808 --> 00:03:04.308
so this is three and one fifth

00:03:05.296 --> 00:03:07.909
and one thing to
appreciate, right over here,

00:03:07.909 --> 00:03:10.365
I simplified that the 24
and the three of this step,

00:03:10.365 --> 00:03:13.454
sometimes you'll see people
simplifying at this step,

00:03:13.454 --> 00:03:16.175
so they'll say, hey look,
eventually I'm gonna have

00:03:16.175 --> 00:03:17.887
a 24 in the numerator and
a three in the denominator,

00:03:17.887 --> 00:03:20.664
so let me divide both of those by three,

00:03:20.664 --> 00:03:23.131
so they'll say 24
divided by three is eight

00:03:23.131 --> 00:03:25.283
and then three divided by three is one

00:03:25.283 --> 00:03:27.849
and this is sometimes
called a cross reduction,

00:03:27.849 --> 00:03:31.849
but this is all that's
going on right over here.


Whole number divided by a fraction example

WEBVTT
Kind: captions
Language: en

00:00:00.164 --> 00:00:03.299
- Let's see if we can figure
out what 4 divided by 2/3 is,

00:00:03.299 --> 00:00:05.197
and like always, pause this video,

00:00:05.197 --> 00:00:08.124
and see if you can figure
it out on your own.

00:00:08.124 --> 00:00:10.180
Well, one way to approach it is to realize

00:00:10.180 --> 00:00:12.603
that this is the same thing

00:00:12.603 --> 00:00:16.270
as 4 times the reciprocal of 2/3.

00:00:17.324 --> 00:00:18.991
So it'd be 4 times 3

00:00:20.483 --> 00:00:23.647
over 2, and what is this
going to be equal to?

00:00:23.647 --> 00:00:27.002
You could pause the video
again if you're so inspired.

00:00:27.002 --> 00:00:29.741
Well, what you need to realize
is this is the same thing.

00:00:29.741 --> 00:00:34.197
4 could be written as
a fraction as 4 over 1.

00:00:34.197 --> 00:00:36.947
So 4 over 1 times 3/2,

00:00:38.869 --> 00:00:41.286
and we've multiplied fractions before.

00:00:41.286 --> 00:00:43.527
To do that, you just
multiply the numerators.

00:00:43.527 --> 00:00:46.436
4 times 3 is equal to 12,

00:00:46.436 --> 00:00:47.856
and you multiply the denominators.

00:00:47.856 --> 00:00:49.971
1 times 2 is equal to 2.

00:00:49.971 --> 00:00:54.243
12/2, well that's the same thing as 6.

00:00:54.243 --> 00:00:56.706
This is the same thing as 12 divided by 2,

00:00:56.706 --> 00:00:59.455
but a key question is
why does this make sense.

00:00:59.455 --> 00:01:01.454
You know, I said dividing by something is

00:01:01.454 --> 00:01:03.945
the same thing as multiplying
by the reciprocal.

00:01:03.945 --> 00:01:07.414
And to think about that,
let's draw four wholes.

00:01:07.414 --> 00:01:11.242
So let me draw it in the same red color.

00:01:11.242 --> 00:01:15.629
So let's say that this is
one whole right over here.

00:01:15.629 --> 00:01:17.212
This is two wholes.

00:01:18.670 --> 00:01:20.420
This is three wholes.

00:01:21.872 --> 00:01:23.760
And then this is four wholes.

00:01:23.760 --> 00:01:25.287
So I have four wholes there.

00:01:25.287 --> 00:01:28.139
And imagine splitting it up into groups

00:01:28.139 --> 00:01:31.411
that are each 2/3 of a whole.

00:01:31.411 --> 00:01:33.484
So actually, let me just divide everything

00:01:33.484 --> 00:01:35.581
into 1/3s real fast.

00:01:35.581 --> 00:01:39.748
So I'm gonna divide everything
into 1/3s, into 1/3s.

00:01:41.101 --> 00:01:43.365
So I'm gonna make each
group a different color.

00:01:43.365 --> 00:01:46.948
So here's one group that is 2/3.

00:01:48.117 --> 00:01:51.784
Here's another group that is 2/3.

00:01:52.782 --> 00:01:57.496
Here is another group that
is, or another section,

00:01:57.496 --> 00:02:00.647
that I could say, that represents 2/3.

00:02:00.647 --> 00:02:04.814
Here is another section
that represents 2/3.

00:02:05.899 --> 00:02:08.497
Here is another section.

00:02:08.497 --> 00:02:10.411
Folks, let me do that
in a different color.

00:02:10.411 --> 00:02:11.244
Here is,

00:02:12.710 --> 00:02:14.627
here is another section

00:02:16.727 --> 00:02:17.924
that can represent 2/3

00:02:17.924 --> 00:02:20.707
if I take those two blue 1/3s together.

00:02:20.707 --> 00:02:24.874
That's 2/3, and then last but
not least, I have another 2/3.

00:02:25.976 --> 00:02:29.862
So how many sections that
are each 2/3 large do I have?

00:02:29.862 --> 00:02:31.279
Well, I have one,

00:02:32.488 --> 00:02:34.321
and then these is two,

00:02:35.258 --> 00:02:38.502
and then I have three,
and then I have four,

00:02:38.502 --> 00:02:41.250
and then these two combined make a,

00:02:41.250 --> 00:02:44.792
make my fifth section that is 2/3 large,

00:02:44.792 --> 00:02:47.170
and then finally, I have six.

00:02:47.170 --> 00:02:48.396
So I have six.

00:02:48.396 --> 00:02:51.020
I can take four wholes and split it

00:02:51.020 --> 00:02:55.187
into six equal sections that
are each 2/3 of a whole.

00:02:56.254 --> 00:03:00.253
So 4 divided into sections
that are 2/3 of a whole,

00:03:00.253 --> 00:03:02.253
you will get 6 sections.


Adding tenths to hundredths

WEBVTT
Kind: captions
Language: en

00:00:01.073 --> 00:00:02.819
- [Instructor] So what we're
going to try to do in this

00:00:02.819 --> 00:00:07.712
video is add 7/10 to 13/100,
pause this video and see

00:00:07.712 --> 00:00:10.295
if you can figure what that is.

00:00:11.758 --> 00:00:14.993
Alright so despite being a
little bit intimidating at first

00:00:14.993 --> 00:00:18.964
because we're adding tenths
here, 7/10 and we're adding

00:00:18.964 --> 00:00:22.625
hundredths here 13/100, how
do I add a certain number

00:00:22.625 --> 00:00:25.597
of tenths to a certain
number of hundredths?

00:00:25.597 --> 00:00:29.764
The key idea is to re-express
7/10 as a certain number of

00:00:30.741 --> 00:00:33.383
hundredths, so how do we do that?

00:00:33.383 --> 00:00:36.970
Well let's just first visualize
each of these fractions.

00:00:36.970 --> 00:00:41.137
So 7/10 if we imagine this
square is a whole and that we've

00:00:41.976 --> 00:00:44.661
divided it into ten equal
sections I tried to hand draw it

00:00:44.661 --> 00:00:48.051
as best as I can, notice I
have filled in seven of those

00:00:48.051 --> 00:00:51.557
ten equal sections that we
have split the whole into.

00:00:51.557 --> 00:00:55.724
So this represents 7/10 and
13/100 you could split the whole

00:00:57.129 --> 00:01:00.376
into 100 equal sections and
I tried to hand draw it,

00:01:00.376 --> 00:01:02.029
let's assume that these
are 100 equal sections.

00:01:02.029 --> 00:01:06.196
And so notice this is a ten
by ten square, it's got 100

00:01:07.543 --> 00:01:11.910
of these squares and notice we
have ten plus three 13 filled

00:01:11.910 --> 00:01:16.089
in so we want to add these
7/10 to these 13/100.

00:01:16.089 --> 00:01:18.700
Now how do we express 7/10
in terms of hundredths?

00:01:18.700 --> 00:01:23.240
Well visually you could take
each of your tenths and split

00:01:23.240 --> 00:01:26.875
them into ten equal sections,
now you have your whole split

00:01:26.875 --> 00:01:31.042
into hundredths and each of
your tenths is now a hundredth

00:01:33.411 --> 00:01:37.684
so you have ten times as many
things in the denominator

00:01:37.684 --> 00:01:42.235
and you also have ten times as
many things in the numerator.

00:01:42.235 --> 00:01:47.052
Before you had seven of the
tenths filled in, now you have

00:01:47.052 --> 00:01:49.635
70 of the hundredths filled in.

00:01:50.710 --> 00:01:52.962
Or another way to think
about it, we multiplied

00:01:52.962 --> 00:01:56.724
the numerator by ten and the
we multiplied the denominator

00:01:56.724 --> 00:01:57.767
by ten.

00:01:57.767 --> 00:01:59.994
And if you do the same
thing to the numerator

00:01:59.994 --> 00:02:02.535
and the denominator, you
multiply or divide it by the same

00:02:02.535 --> 00:02:05.304
number, you're not changing
the value, think about it,

00:02:05.304 --> 00:02:08.392
ten over ten is the same thing
as one, so we're just taking

00:02:08.392 --> 00:02:11.640
one and multiplying it by
7/10 isn't going to change

00:02:11.640 --> 00:02:13.080
the value.

00:02:13.080 --> 00:02:16.167
But this is as we already
talked about this is equivalent

00:02:16.167 --> 00:02:20.334
to 70/100, so this is equal
to 70/100 which is this

00:02:23.995 --> 00:02:28.162
right over here plus 13/100, plus 13/100.

00:02:29.871 --> 00:02:32.576
Well now we're adding 100s
in both cases, if I have 70

00:02:32.576 --> 00:02:36.289
of something and I add to
that 13 of the same somethings

00:02:36.289 --> 00:02:39.472
in this case the something
is hundredths, I'm going to

00:02:39.472 --> 00:02:42.660
have 83 of that so this is
going to be 70, get the same

00:02:42.660 --> 00:02:46.827
color, this is going to be equal
to 70 plus 13, plus 13/100.

00:02:54.412 --> 00:02:58.733
70 plus 13/100 and what's 70
plus 13, well that of course

00:02:58.733 --> 00:03:01.483
is going to be 83/100.

00:03:03.388 --> 00:03:04.721
And we are done.


Word problem subtracting fractions with like denominators

WEBVTT
Kind: captions
Language: en

00:00:00.996 --> 00:00:03.527
- [Instructor] After a rain
storm, Lily measures the depth

00:00:03.527 --> 00:00:06.302
of several puddles in her backyard.

00:00:06.302 --> 00:00:09.088
She records her results in a table.

00:00:09.088 --> 00:00:11.987
So here are three different
puddles and she measures

00:00:11.987 --> 00:00:14.162
the depth in inches.

00:00:14.162 --> 00:00:18.547
We're asked how much deeper
was the puddle under the swing

00:00:18.547 --> 00:00:21.220
than the puddle on the sidewalk.

00:00:21.220 --> 00:00:25.604
So pause this video and see
if you can figure that out.

00:00:25.604 --> 00:00:29.278
So they say how much deeper
was the puddle under the swing,

00:00:29.278 --> 00:00:31.955
so that's this one right over
here it's one and 1/4 inches

00:00:31.955 --> 00:00:33.954
deep it's under the swing.

00:00:33.954 --> 00:00:38.121
How much deeper was that than
the puddle on the sidewalk?

00:00:38.981 --> 00:00:41.828
Do that in a different color,
the puddle on the sidewalk.

00:00:41.828 --> 00:00:44.887
And we see here the puddle
on the sidewalk is 2/4 inches

00:00:44.887 --> 00:00:47.476
deep, so what we could
do is subtract the 2/4

00:00:47.476 --> 00:00:49.717
from the one and 1/4.

00:00:49.717 --> 00:00:53.884
So we could write one and
1/4 minus 2/4 could write it

00:00:54.762 --> 00:00:57.912
like that, and we could try
to subtract the fraction part

00:00:57.912 --> 00:01:01.452
2/4 from the fraction part
of this mixed number up here

00:01:01.452 --> 00:01:04.598
from 1/4 but we immediately
have a problem 'cause 2/4

00:01:04.598 --> 00:01:08.804
is a larger fraction than 1/4,
so how do we deal with that?

00:01:08.804 --> 00:01:12.809
Well the key is to realize
that one can be rewritten

00:01:12.809 --> 00:01:16.976
as a fraction, one and 1/4
is the same thing as one plus

00:01:18.597 --> 00:01:23.172
1/4 which is the same thing
as another way to write one

00:01:23.172 --> 00:01:27.339
in terms of fourths is 4/4
so this is 4/4 plus 1/4

00:01:30.405 --> 00:01:34.213
which is going to be equal to 5/4.

00:01:34.213 --> 00:01:38.006
So now you can do this as
5/4, this number is the same

00:01:38.006 --> 00:01:42.173
thing as 5/4 minus 2/4, let
me rewrite it, minus 2/4,

00:01:45.671 --> 00:01:47.942
minus two over four.

00:01:47.942 --> 00:01:50.246
And that's pretty
straightforward if I have five of

00:01:50.246 --> 00:01:52.485
something and I subtract
two of it, I'm going to have

00:01:52.485 --> 00:01:56.550
three of that something in this
case I'm talking about 3/4.

00:01:56.550 --> 00:02:00.265
So this is going to be 3/4 so
how much deeper was the puddle

00:02:00.265 --> 00:02:02.805
under the swing than the
puddle on the sidewalk?

00:02:02.805 --> 00:02:05.382
Well 3/4 of an inch.

00:02:05.382 --> 00:02:07.625
And just another way that you
could have visualized this

00:02:07.625 --> 00:02:11.861
is look I'm going to subtract
2/4 from one and 1/4.

00:02:11.861 --> 00:02:15.881
At first we could've thought
of one and 1/4 as a whole like

00:02:15.881 --> 00:02:19.875
this and then it's all let me
shade it, the whole and that's

00:02:19.875 --> 00:02:24.136
one and then I would have a
fourth of a whole so let me

00:02:24.136 --> 00:02:28.520
divide this into four sections,
so this is one and 1/4.

00:02:28.520 --> 00:02:31.479
And at first we say well how
do we take away 2/4 from just

00:02:31.479 --> 00:02:35.591
that I only have 1/4 right over
here and our key realization

00:02:35.591 --> 00:02:39.600
is well look, I actually
this whole right over here

00:02:39.600 --> 00:02:43.767
is actually 4/4 I can think of
it as 4/4, so I can think of

00:02:46.020 --> 00:02:50.187
it like this and now I have
5/4, one, two, three, four, five

00:02:51.222 --> 00:02:54.328
fourths and now I can take
away two of the fourths,

00:02:54.328 --> 00:02:58.803
so I can take away one of the
fourths and two of the fourths

00:02:58.803 --> 00:03:00.649
and what am I left with?

00:03:00.649 --> 00:03:04.840
Well then I am going to
be left with these 3/4

00:03:04.840 --> 00:03:06.257
right over there.


Opportunity cost and comparative advantage using an output table

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.038
- [Tutor] What we're
going to do in this video

00:00:02.038 --> 00:00:04.866
is draw a connection between the idea

00:00:04.866 --> 00:00:07.114
of opportunity cost of producing a good

00:00:07.114 --> 00:00:10.338
in a certain country and
comparative advantage

00:00:10.338 --> 00:00:13.255
between countries in a certain good

00:00:14.697 --> 00:00:18.149
and below, right over
here we have a chart,

00:00:18.149 --> 00:00:20.784
that shows production possibility curves

00:00:20.784 --> 00:00:22.593
for two different countries

00:00:22.593 --> 00:00:24.577
and as we see in many economic models,

00:00:24.577 --> 00:00:26.812
this is a, I would argue
oversimplified model,

00:00:26.812 --> 00:00:28.729
but it helps us get some insights,

00:00:28.729 --> 00:00:31.544
where in each country
workers can only produce

00:00:31.544 --> 00:00:35.211
some combination of
sneakers and basketballs

00:00:36.132 --> 00:00:38.715
and to help us understand
this and to appreciate

00:00:38.715 --> 00:00:42.046
that you can see this
information in multiple ways,

00:00:42.046 --> 00:00:46.381
let's present this also
as an output table,

00:00:46.381 --> 00:00:49.403
output table, which you will sometimes see

00:00:49.403 --> 00:00:52.260
and from either the
production possibility curves

00:00:52.260 --> 00:00:53.621
or from the output table,

00:00:53.621 --> 00:00:56.539
we can calculate the
opportunity costs of shoes

00:00:56.539 --> 00:00:59.024
and the opportunity costs of basketballs

00:00:59.024 --> 00:01:00.853
and then try to deduce some things

00:01:00.853 --> 00:01:02.957
about comparative advantage.

00:01:02.957 --> 00:01:07.097
So in an output table, we
would look at country A

00:01:07.097 --> 00:01:11.181
and we would look at country
B and we would think about,

00:01:11.181 --> 00:01:14.706
well, what is the max,
and I'll just draw it,

00:01:14.706 --> 00:01:17.239
what is the max basketballs

00:01:17.239 --> 00:01:19.854
and this is all per worker per day

00:01:19.854 --> 00:01:23.687
and we would also think
what is the max shoes,

00:01:24.894 --> 00:01:27.333
shoes, those look like
socks, but you get the idea,

00:01:27.333 --> 00:01:30.260
once again, per worker per day

00:01:30.260 --> 00:01:34.150
and so let me draw a little chart here,

00:01:34.150 --> 00:01:36.329
so we can do that

00:01:36.329 --> 00:01:39.600
and so what I'd like you
to do is pause this video

00:01:39.600 --> 00:01:42.041
and see if you can fill in this chart,

00:01:42.041 --> 00:01:44.429
what is the maximum
basketballs per worker per day

00:01:44.429 --> 00:01:46.113
in country A and then in country B

00:01:46.113 --> 00:01:49.362
and then do the same thing for shoes.

00:01:49.362 --> 00:01:51.951
Alright, now let's work this together,

00:01:51.951 --> 00:01:53.623
so first in country A,

00:01:53.623 --> 00:01:55.914
what is the maximum number of basketballs?

00:01:55.914 --> 00:01:56.785
Well, if in country A,

00:01:56.785 --> 00:01:58.813
they put all of their
energy into basketballs,

00:01:58.813 --> 00:01:59.646
we are right over here

00:01:59.646 --> 00:02:01.444
on the production possibilities curve,

00:02:01.444 --> 00:02:04.495
they can produce eight basketballs

00:02:04.495 --> 00:02:06.258
and if on the other end of the curve,

00:02:06.258 --> 00:02:08.221
they put all of their energy into shoes,

00:02:08.221 --> 00:02:11.494
they would produce no basketballs
and six pairs of shoes,

00:02:11.494 --> 00:02:13.041
we're assuming that
these are pairs of shoes,

00:02:13.041 --> 00:02:17.403
that we're talking
about, six pairs of shoes

00:02:17.403 --> 00:02:19.303
and similarly if we go
to company, (laughs)

00:02:19.303 --> 00:02:20.842
if we go to country B,

00:02:20.842 --> 00:02:23.215
I keep saying company, instead of country,

00:02:23.215 --> 00:02:25.604
if we go to country B,

00:02:25.604 --> 00:02:29.458
if we say what's the maximum
number of basketballs,

00:02:29.458 --> 00:02:31.830
well, if they put all their
energy into basketballs,

00:02:31.830 --> 00:02:34.986
we get four basketballs
and no pairs of shoes,

00:02:34.986 --> 00:02:37.711
so that's four basketballs,
but then if they put

00:02:37.711 --> 00:02:40.008
all of their energy into pairs of shoes,

00:02:40.008 --> 00:02:41.256
they produce no basketballs,

00:02:41.256 --> 00:02:43.731
they could produce four pairs of shoes

00:02:43.731 --> 00:02:44.933
and so it's as simple as that,

00:02:44.933 --> 00:02:47.759
this output table is
just showing the extremes

00:02:47.759 --> 00:02:51.475
from the production possibility
curves for these countries.

00:02:51.475 --> 00:02:54.298
Now with the information
about the output table

00:02:54.298 --> 00:02:56.272
and these production possibility curves,

00:02:56.272 --> 00:02:58.461
let's calculate the opportunity cost,

00:02:58.461 --> 00:03:00.506
so let me set up another
table and let me just say

00:03:00.506 --> 00:03:05.093
this is going to be our
opportunity cost table, OC,

00:03:05.093 --> 00:03:08.493
not Orange County, opportunity
costs and once again,

00:03:08.493 --> 00:03:12.952
it's going to be for
country A and country B

00:03:12.952 --> 00:03:15.801
and we're gonna think
about the opportunity costs

00:03:15.801 --> 00:03:17.521
of producing basketballs

00:03:17.521 --> 00:03:20.298
and that's gonna be in
terms of pairs of shoes

00:03:20.298 --> 00:03:23.774
and then the opportunity costs
for producing pairs of shoes

00:03:23.774 --> 00:03:27.046
and that's going to be
in terms of basketballs

00:03:27.046 --> 00:03:30.611
and so let me set up another table

00:03:30.611 --> 00:03:35.316
and so I encourage you once
again, pause this video

00:03:35.316 --> 00:03:37.639
and see if you can fill in this table,

00:03:37.639 --> 00:03:39.250
what is the opportunity costs?

00:03:39.250 --> 00:03:40.556
We'll start with what's
the opportunity costs

00:03:40.556 --> 00:03:44.723
for producing basketballs in
terms of shoes in country A?

00:03:46.065 --> 00:03:48.970
Alright, well there's a couple
of ways to think about it,

00:03:48.970 --> 00:03:49.965
imagine a world in country A,

00:03:49.965 --> 00:03:52.221
where you're producing no basketballs

00:03:52.221 --> 00:03:54.276
and you're producing six pairs of shoes,

00:03:54.276 --> 00:03:56.247
but then if you were
to increase the number

00:03:56.247 --> 00:03:59.235
of basketballs you produce by eight,

00:03:59.235 --> 00:04:00.710
so if you add eight basketballs,

00:04:00.710 --> 00:04:04.137
well, you're gonna give
up six pairs of shoes,

00:04:04.137 --> 00:04:05.813
you see that right over here,

00:04:05.813 --> 00:04:10.062
you give up six pairs of
shoes and so in country A

00:04:10.062 --> 00:04:15.031
eight basketballs cost six
shoes, let me write that down,

00:04:15.031 --> 00:04:17.864
so in country A, eight basketballs

00:04:18.816 --> 00:04:20.841
and I'll just say B for short,

00:04:20.841 --> 00:04:22.174
cost six, six S,

00:04:25.822 --> 00:04:27.800
S is shoes for short,

00:04:27.800 --> 00:04:29.011
or another way to think about it,

00:04:29.011 --> 00:04:31.875
if you divide both of these by eight,

00:04:31.875 --> 00:04:33.542
one basketball costs

00:04:34.940 --> 00:04:36.690
six over eight shoes,

00:04:37.987 --> 00:04:40.345
all I did was eight
basketballs cost six shoes

00:04:40.345 --> 00:04:44.051
and one basketball's gonna
cost six divided by eight

00:04:44.051 --> 00:04:47.036
pairs of shoes and so
what is that gonna be?

00:04:47.036 --> 00:04:49.670
Well, six over eight is the
same thing as three fourths

00:04:49.670 --> 00:04:52.142
or three fourths of a pair of shoes,

00:04:52.142 --> 00:04:55.838
so one basketball costs three
fourths of a pair of shoes

00:04:55.838 --> 00:04:58.421
or we could say that as 0.75 S,

00:04:59.779 --> 00:05:02.716
where S is a pair of shoes

00:05:02.716 --> 00:05:04.571
for this is my simplified notation

00:05:04.571 --> 00:05:07.258
and what about in country B?

00:05:07.258 --> 00:05:08.843
Well, in country B,

00:05:08.843 --> 00:05:13.233
if I go from no basketballs
to four basketballs,

00:05:13.233 --> 00:05:16.123
then I would have given
up four pairs of shoes,

00:05:16.123 --> 00:05:18.600
I would have given up four pairs of shoes,

00:05:18.600 --> 00:05:21.183
so in country B, so in B, four,

00:05:23.065 --> 00:05:24.815
four basketballs cost

00:05:26.753 --> 00:05:28.236
four pairs of shoes

00:05:28.236 --> 00:05:31.276
or divide both by four, you
could have a basketball,

00:05:31.276 --> 00:05:34.526
one basketball costs one pair of shoes,

00:05:35.664 --> 00:05:39.831
so a basketball here in country
B costs one pair of shoes,

00:05:41.463 --> 00:05:45.807
so one pair of shoes, S once
again is a pair of shoes

00:05:45.807 --> 00:05:47.601
and you could have also gotten it

00:05:47.601 --> 00:05:49.465
from this information here,

00:05:49.465 --> 00:05:52.732
you could set up an
equation, you could say look,

00:05:52.732 --> 00:05:56.036
if I put all of, in country A, if I put,

00:05:56.036 --> 00:05:59.405
so let's look at this
part right over here,

00:05:59.405 --> 00:06:02.816
you could say in country A,
if I put all of my energy

00:06:02.816 --> 00:06:05.918
into basketballs, I could
produce eight basketballs,

00:06:05.918 --> 00:06:08.588
but if I put that same energy into shoes,

00:06:08.588 --> 00:06:11.871
I could produce six pairs of shoes,

00:06:11.871 --> 00:06:14.685
so with the same energy, I could
produce either one of these

00:06:14.685 --> 00:06:17.310
and then if I want the
opportunity costs for basketballs,

00:06:17.310 --> 00:06:19.021
I divide both by eight
and that's essentially

00:06:19.021 --> 00:06:22.688
what I did over here
and I get a basketball,

00:06:23.951 --> 00:06:26.829
it costs six eighths of a pair of shoes

00:06:26.829 --> 00:06:28.830
or three fourths of a pair of shoes,

00:06:28.830 --> 00:06:31.987
which is exactly what I have over here.

00:06:31.987 --> 00:06:33.582
Now let's do the opportunity cost

00:06:33.582 --> 00:06:36.001
for a pair of shoes in either country,

00:06:36.001 --> 00:06:38.039
well, there's a couple of
ways to think about it,

00:06:38.039 --> 00:06:39.960
you could just view it as the reciprocal

00:06:39.960 --> 00:06:43.834
or you could even go back to
this equation right over here,

00:06:43.834 --> 00:06:48.523
if we are in country A,
we would say six shoes,

00:06:48.523 --> 00:06:49.710
if we put all our energy in shoes,

00:06:49.710 --> 00:06:51.794
we could produce six of them

00:06:51.794 --> 00:06:53.521
or six pairs of shoes, I should say

00:06:53.521 --> 00:06:55.841
and if we put all of our
energy into basketballs,

00:06:55.841 --> 00:06:58.308
we could produce eight basketballs,

00:06:58.308 --> 00:07:02.871
but if you divide by six,
you get per pair of shoes

00:07:02.871 --> 00:07:05.157
and so per each pair of shoes,

00:07:05.157 --> 00:07:08.750
the energy to produce one
pair of shoes is equivalent

00:07:08.750 --> 00:07:12.065
to the energy to produce
eight sixths of a basketball

00:07:12.065 --> 00:07:14.494
and eight sixths is the same thing

00:07:14.494 --> 00:07:16.986
as four thirds of a basketball

00:07:16.986 --> 00:07:19.200
and if we wanted to write it as a decimal

00:07:19.200 --> 00:07:21.818
just for simplicity or maybe
to make it easier to compare,

00:07:21.818 --> 00:07:24.669
we would say that this
is approximately 1.33,

00:07:24.669 --> 00:07:27.524
obviously the 3s just keep
going on, it repeats forever,

00:07:27.524 --> 00:07:30.998
but approximately 1.33 basketballs

00:07:30.998 --> 00:07:32.532
is the cost of producing a shoe

00:07:32.532 --> 00:07:34.800
and the opportunity
cost of producing a shoe

00:07:34.800 --> 00:07:37.300
in country A, 1.33 basketballs

00:07:38.945 --> 00:07:41.077
and what about in country B?

00:07:41.077 --> 00:07:42.353
Well, in country B

00:07:42.353 --> 00:07:44.883
we could set up a
similar type of equation,

00:07:44.883 --> 00:07:47.366
where the same energy for four shoes,

00:07:47.366 --> 00:07:49.281
I could produce four basketballs

00:07:49.281 --> 00:07:50.645
and that's essentially what we set up

00:07:50.645 --> 00:07:53.672
right over here on the left,
you divide both sides by four,

00:07:53.672 --> 00:07:56.956
the energy of a shoe is equal
to the energy of a basketball,

00:07:56.956 --> 00:07:59.298
or I should say the
energy of a pair of shoes

00:07:59.298 --> 00:08:01.802
is equal to the energy
of making a basketball,

00:08:01.802 --> 00:08:05.037
so the opportunity cost
of making a pair of shoes

00:08:05.037 --> 00:08:07.287
is equal to one basketball.

00:08:08.455 --> 00:08:11.582
So now we're ready to draw the connection,

00:08:11.582 --> 00:08:14.872
given the opportunity
costs that we calculated,

00:08:14.872 --> 00:08:19.039
what country has the comparative
advantage in basketballs?

00:08:20.053 --> 00:08:21.703
Pause this video and try to figure it out.

00:08:21.703 --> 00:08:23.677
So now let's look at the opportunity cost

00:08:23.677 --> 00:08:26.543
of producing a basketball
in either country.

00:08:26.543 --> 00:08:30.951
In country A, each
basketball costs a worker

00:08:30.951 --> 00:08:32.642
three fourths of a pair of shoes,

00:08:32.642 --> 00:08:36.897
while in country B, it costs
them a whole pair of shoes,

00:08:36.897 --> 00:08:40.893
so country A actually has
a lower opportunity cost

00:08:40.893 --> 00:08:42.875
of producing basketballs

00:08:42.875 --> 00:08:45.829
and so it has the
comparative advantage here,

00:08:45.829 --> 00:08:48.662
comparative, comparative advantage

00:08:51.192 --> 00:08:55.558
and then if we look at shoes,
it goes the other way around,

00:08:55.558 --> 00:08:57.694
country A has an opportunity cost

00:08:57.694 --> 00:09:01.590
of one and one third basketballs
for every pair of shoes,

00:09:01.590 --> 00:09:03.624
while country B has an opportunity cost

00:09:03.624 --> 00:09:05.928
of only one basketball per pair of shoes,

00:09:05.928 --> 00:09:08.767
so it has a lower opportunity cost

00:09:08.767 --> 00:09:10.514
and this one actually might be

00:09:10.514 --> 00:09:12.467
a little bit counterintuitive,

00:09:12.467 --> 00:09:15.875
because if you look on the
shoe axis right over here,

00:09:15.875 --> 00:09:20.348
country A has the absolute
advantage in producing shoes,

00:09:20.348 --> 00:09:22.416
a worker per day in country A

00:09:22.416 --> 00:09:24.470
can produce six pairs of shoes,

00:09:24.470 --> 00:09:25.845
while a worker in country B

00:09:25.845 --> 00:09:28.494
can only produce four pairs of shoes,

00:09:28.494 --> 00:09:31.576
but even though country A
has the absolute advantage,

00:09:31.576 --> 00:09:33.924
it would actually make sense for country A

00:09:33.924 --> 00:09:38.592
to focus on basketballs, while
country B focuses on shoes

00:09:38.592 --> 00:09:41.570
and in the next video,
we'll see how they can

00:09:41.570 --> 00:09:43.880
trade with each other
to get to a scenario,

00:09:43.880 --> 00:09:47.300
that is beyond their
production possibility curves

00:09:47.300 --> 00:09:49.754
and why focusing on your
comparative advantage,

00:09:49.754 --> 00:09:51.304
at least in this theoretical,

00:09:51.304 --> 00:09:54.221
very simplified world, makes sense.


PPCs for increasing, decreasing and constant opportunity cost

WEBVTT
Kind: captions
Language: en

00:00:00.466 --> 00:00:01.901
- [Instructor] So we have three different

00:00:01.901 --> 00:00:04.817
possible production possibility curves

00:00:04.817 --> 00:00:07.320
for rabbits and berries
here, which we've already

00:00:07.320 --> 00:00:09.615
talked about in other
videos, but the reason why

00:00:09.615 --> 00:00:11.336
I'm showing you three different curves

00:00:11.336 --> 00:00:12.847
is because these three different curves

00:00:12.847 --> 00:00:15.292
clearly have different shapes,
and we wanna think about

00:00:15.292 --> 00:00:18.062
why you would have and
under what scenarios

00:00:18.062 --> 00:00:19.850
would you have these different shapes?

00:00:19.850 --> 00:00:23.660
Here, our production
possibility curve, or our PPC,

00:00:23.660 --> 00:00:25.597
it looks like a straight line.

00:00:25.597 --> 00:00:28.102
Here, it looks like it's
bowed out from the origin,

00:00:28.102 --> 00:00:30.486
it looks like it's popping
out in that direction.

00:00:30.486 --> 00:00:33.484
And here, it looks like
it's bowed in to the origin,

00:00:33.484 --> 00:00:35.861
it's popping in in this direction.

00:00:35.861 --> 00:00:38.954
So the first thing I'm going
to do is ask you a question.

00:00:38.954 --> 00:00:41.544
Which one describes the scenario where

00:00:41.544 --> 00:00:45.425
for every extra rabbit I catch,
every incremental rabbit,

00:00:45.425 --> 00:00:48.472
I'm giving up more and
more in terms of berries?

00:00:48.472 --> 00:00:50.072
Or another way of thinking about it is,

00:00:50.072 --> 00:00:53.294
as I catch more and more
rabbits, the opportunity cost

00:00:53.294 --> 00:00:56.377
in terms of berries is increasing.

00:00:56.377 --> 00:00:59.817
Which one of these curves describes that?

00:00:59.817 --> 00:01:01.145
Well some of you might have already seen

00:01:01.145 --> 00:01:04.775
the video on KhanAcademy, on
increasing opportunity cost,

00:01:04.775 --> 00:01:06.771
and you might recognize
that this curve here.

00:01:06.771 --> 00:01:09.105
But let's just review it,
so there's a world where

00:01:09.105 --> 00:01:11.979
I'm eating all berries,
and I can get, I can pick

00:01:11.979 --> 00:01:14.661
300 berries a day, but
maybe I decide to go after

00:01:14.661 --> 00:01:16.723
that first rabbit that
just likes to hang out

00:01:16.723 --> 00:01:19.829
and play with my knives,
and so when I catch that,

00:01:19.829 --> 00:01:22.138
it's very easy to catch,
so I don't give up a lot

00:01:22.138 --> 00:01:24.854
in terms of berries, especially
because I'm probably not,

00:01:24.854 --> 00:01:26.498
the berries I'm giving up are probably

00:01:26.498 --> 00:01:28.584
the ones that are hardest to pick.

00:01:28.584 --> 00:01:32.183
And so let's say that first
rabbit, the opportunity cost,

00:01:32.183 --> 00:01:34.917
I pick 20 less berries,
so notice, when I increase

00:01:34.917 --> 00:01:37.923
the rabbits by one, my
berries go down by 20,

00:01:37.923 --> 00:01:42.223
so my opportunity cost is 20
berries for that first rabbit.

00:01:42.223 --> 00:01:43.397
But let's say that second rabbit is

00:01:43.397 --> 00:01:46.463
a little bit harder to
catch, and I'm not giving up

00:01:46.463 --> 00:01:48.658
the quite so hard to pick berries,

00:01:48.658 --> 00:01:51.053
and so when I pick that next,
or when I hunt that next

00:01:51.053 --> 00:01:55.031
rabbit, I should say, then
I've given up 40 berries.

00:01:55.031 --> 00:01:57.649
So notice, my opportunity
cost has increased.

00:01:57.649 --> 00:02:01.570
For that first rabbit, my
opportunity cost was 20 berries.

00:02:01.570 --> 00:02:05.221
For that second rabbit, my
opportunity cost is 40 berries.

00:02:05.221 --> 00:02:07.860
And it keeps going, then third rabbit,

00:02:07.860 --> 00:02:10.772
I'm going to give up 60 berries.

00:02:10.772 --> 00:02:14.855
That fourth rabbit, I'm
gonna give up 80 berries,

00:02:16.075 --> 00:02:18.996
80 berries, and then last but not least,

00:02:18.996 --> 00:02:21.361
that fifth rabbit, which
is the most that I can hunt

00:02:21.361 --> 00:02:24.261
in a day, I'm gonna give up 100 berries

00:02:24.261 --> 00:02:26.942
'cuz here, I'm going after
the really nimble rabbit,

00:02:26.942 --> 00:02:29.453
the really sly rabbit, and
I'm giving up literally

00:02:29.453 --> 00:02:31.492
the low-hanging fruit in terms of berries,

00:02:31.492 --> 00:02:34.344
the one, they might be on the ground,

00:02:34.344 --> 00:02:36.724
just ready for me to pick up, and so,

00:02:36.724 --> 00:02:38.989
the important realization from this video

00:02:38.989 --> 00:02:41.242
is this bowed out shape right over here,

00:02:41.242 --> 00:02:44.552
this is describing an
increasing opportunity cost.

00:02:44.552 --> 00:02:47.223
Let me write that down, increasing,

00:02:47.223 --> 00:02:50.390
increasing, O.C. for opportunity cost.

00:02:51.263 --> 00:02:53.674
So with that out of
the way, which of these

00:02:53.674 --> 00:02:57.092
would describe a decreasing
opportunity cost?

00:02:57.092 --> 00:02:58.799
Maybe you could imagine a scenario where

00:02:58.799 --> 00:03:00.782
every incremental rabbit I catch,

00:03:00.782 --> 00:03:03.916
I get better and better
at catching rabbits.

00:03:03.916 --> 00:03:05.491
Well you might guess that, well look,

00:03:05.491 --> 00:03:07.795
if this one is increasing
and I'm bowed out,

00:03:07.795 --> 00:03:11.599
then being bowed in would be
a decreasing opportunity cost.

00:03:11.599 --> 00:03:15.444
Decreasing opportunity
cost, and let's make sure

00:03:15.444 --> 00:03:17.674
that it makes sense, so we
could go back to the scenario

00:03:17.674 --> 00:03:19.998
where we're doing nothing
but picking berries,

00:03:19.998 --> 00:03:22.382
and let's say that first
rabbit, so we're gonna

00:03:22.382 --> 00:03:24.839
talk about a different scenario
now, that first rabbit,

00:03:24.839 --> 00:03:27.992
I had to train myself to
be able to get rabbits,

00:03:27.992 --> 00:03:30.144
I have to buy the tools,
I have to stretch,

00:03:30.144 --> 00:03:34.810
it takes me a lot of effort
to get that first rabbit.

00:03:34.810 --> 00:03:37.966
And so, there, I give
up 100 berries, so my

00:03:37.966 --> 00:03:41.184
opportunity cost for that
first rabbit was 100 berries.

00:03:41.184 --> 00:03:42.952
But then for that second rabbit,

00:03:42.952 --> 00:03:46.175
my opportunity cost is 80 berries.

00:03:46.175 --> 00:03:48.095
Maybe now, I've kind of
gotten the hang of it.

00:03:48.095 --> 00:03:50.917
I've already bought my
rabbit catching shoes.

00:03:50.917 --> 00:03:52.682
I've already invested in that.

00:03:52.682 --> 00:03:57.055
I'm all stretched and
limber, maybe those rabbits

00:03:57.055 --> 00:04:00.017
like to hang out together,
and so that keeps on going.

00:04:00.017 --> 00:04:03.758
So that third rabbit, my
opportunity cost is 60 berries.

00:04:03.758 --> 00:04:06.398
I'm getting really good
at catching rabbits,

00:04:06.398 --> 00:04:09.569
so clearly, you see here, that
for each incremental rabbit

00:04:09.569 --> 00:04:13.704
I get, my opportunity cost is decreasing,

00:04:13.704 --> 00:04:15.126
all the way to that fifth rabbit,

00:04:15.126 --> 00:04:18.459
maybe my opportunity cost is 20 berries.

00:04:19.674 --> 00:04:21.074
To catch that next extra rabbit,

00:04:21.074 --> 00:04:23.231
I'm giving up those 20 berries.

00:04:23.231 --> 00:04:26.527
So very clearly, you see a
decreasing opportunity cost.

00:04:26.527 --> 00:04:29.275
And so, by deductive reasoning,
you might be able to say,

00:04:29.275 --> 00:04:31.443
"Well, okay, this straight
line must represent

00:04:31.443 --> 00:04:33.943
"a constant opportunity cost."

00:04:35.088 --> 00:04:37.259
And that is, indeed, what it shows.

00:04:37.259 --> 00:04:40.328
For every rabbit, every rabbit you catch,

00:04:40.328 --> 00:04:44.526
you're giving up exactly,
you're giving up exactly

00:04:44.526 --> 00:04:47.448
60 berries, every time I catch a rabbit,

00:04:47.448 --> 00:04:50.410
I give up 60 berries,
so my opportunity cost

00:04:50.410 --> 00:04:54.271
for rabbits, in terms of
berries, is just a constant 60.

00:04:54.271 --> 00:04:55.999
And so this is a scenario,
if you were imagining

00:04:55.999 --> 00:04:58.365
in this fictional world we created, where

00:04:58.365 --> 00:05:01.713
every rabbit is about as easy
to catch as any other one,

00:05:01.713 --> 00:05:04.055
and every berry is about
as easy to pick or find

00:05:04.055 --> 00:05:07.030
as any other one, and so, the trade off,

00:05:07.030 --> 00:05:09.532
the amount of time I spent
for each incremental rabbit,

00:05:09.532 --> 00:05:11.759
I'm giving up a fixed amount of berries.

00:05:11.759 --> 00:05:14.548
No matter how many rabbits I go for,

00:05:14.548 --> 00:05:18.631
and no matter how many
berries I am currently at,

00:05:20.002 --> 00:05:22.968
so that's a constant opportunity cost,

00:05:22.968 --> 00:05:24.843
when you have a straight line.