Utility maximization example

WEBVTT
Kind: captions
Language: en

00:00:00.442 --> 00:00:02.291
- [Instructor] Here we have
an actual free response

00:00:02.291 --> 00:00:05.405
question from a previous
AP economics exam.

00:00:05.405 --> 00:00:08.306
And it tells us, utility
and price elasticity

00:00:08.306 --> 00:00:10.384
of demand are important concepts

00:00:10.384 --> 00:00:12.691
in explaining consumer behavior.

00:00:12.691 --> 00:00:14.187
I buy that.

00:00:14.187 --> 00:00:17.192
Now, part A, they say
define marginal utility.

00:00:17.192 --> 00:00:18.556
So before I do that, pause the video

00:00:18.556 --> 00:00:19.731
and at least think about it.

00:00:19.731 --> 00:00:21.283
How do you define it in your head?

00:00:21.283 --> 00:00:22.938
Or even better, try to
write it down in words

00:00:22.938 --> 00:00:24.197
because this is what you would actually

00:00:24.197 --> 00:00:26.795
have to do on an AP exam.

00:00:26.795 --> 00:00:29.966
So we have defined it in
previous videos in more depth,

00:00:29.966 --> 00:00:31.255
but if I were taking the exam,

00:00:31.255 --> 00:00:33.066
I would write something like,

00:00:33.066 --> 00:00:34.399
the incremental,

00:00:36.555 --> 00:00:38.222
incremental benefit,

00:00:40.148 --> 00:00:42.231
or gain, or satisfaction,

00:00:45.969 --> 00:00:46.969
satisfaction

00:00:48.403 --> 00:00:49.986
from one more unit,

00:00:51.099 --> 00:00:52.599
from one more unit

00:00:56.701 --> 00:00:57.784
of something.

00:00:58.974 --> 00:01:02.542
I think that is a good definition

00:01:02.542 --> 00:01:05.645
of a shorthand for marginal utility.

00:01:05.645 --> 00:01:08.106
And the marginal tells us that's what's

00:01:08.106 --> 00:01:09.963
for the next thing, not in total,

00:01:09.963 --> 00:01:11.935
but just for the next thing
that you're gonna get,

00:01:11.935 --> 00:01:15.298
how much gain or satisfaction
are you going to have.

00:01:15.298 --> 00:01:16.798
Now let's try part B.

00:01:16.798 --> 00:01:19.099
The table below shows
the quantities, prices,

00:01:19.099 --> 00:01:20.972
and marginal utilities of two goods,

00:01:20.972 --> 00:01:23.769
fudge and coffee, which Mandy purchases.

00:01:23.769 --> 00:01:25.721
So we see, I guess so far she's bought

00:01:25.721 --> 00:01:28.754
10 pounds of fudge and
seven pounds of coffee.

00:01:28.754 --> 00:01:29.870
Fudge is $2 a pound.

00:01:29.870 --> 00:01:31.746
Coffee is $4 a pound.

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And then the marginal utility

00:01:33.037 --> 00:01:37.453
of the last pound of fudge
is 12, is 20 for coffee.

00:01:37.453 --> 00:01:38.778
And this is interesting because you notice

00:01:38.778 --> 00:01:40.774
marginal utility doesn't have units.

00:01:40.774 --> 00:01:42.774
It tends to be unit-less

00:01:44.300 --> 00:01:46.897
or it kind of has your
marginal utility units.

00:01:46.897 --> 00:01:50.837
And what matters more
is how it might compare

00:01:50.837 --> 00:01:52.895
from the first pound to the second pound,

00:01:52.895 --> 00:01:55.127
as opposed to the absolute value here.

00:01:55.127 --> 00:01:57.620
But we're gonna, I'm suspecting
before even reading this,

00:01:57.620 --> 00:02:00.139
that we're gonna make
use of this in some way

00:02:00.139 --> 00:02:02.113
in this next part of this question.

00:02:02.113 --> 00:02:06.134
Mandy spends all her money
and buys only these two goods.

00:02:06.134 --> 00:02:08.583
In order to maximize her utility,

00:02:08.583 --> 00:02:10.235
maximize her utility,

00:02:10.235 --> 00:02:13.818
should Mandy purchase more
fudge and less coffee,

00:02:13.818 --> 00:02:16.163
purchase more coffee and less fudge,

00:02:16.163 --> 00:02:19.918
or maintain her current consumption?

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Explain.

00:02:21.756 --> 00:02:23.546
Okay, so some of you
might immediately say,

00:02:23.546 --> 00:02:24.379
all right, I'm just gonna look at

00:02:24.379 --> 00:02:28.057
how much gain, satisfaction she gets

00:02:28.057 --> 00:02:31.347
for the next pound of coffee
versus the next pound of fudge.

00:02:31.347 --> 00:02:34.526
It looks pretty clear
that 20 is higher than 12.

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And so you'd say, all right,

00:02:35.715 --> 00:02:38.843
she should just buy more coffee.

00:02:38.843 --> 00:02:40.613
But that would be a mistake.

00:02:40.613 --> 00:02:43.740
And to realize it, let me
give you another example.

00:02:43.740 --> 00:02:45.740
If your marginal utility

00:02:48.589 --> 00:02:51.004
of a Rolls-Royce is going to be higher

00:02:51.004 --> 00:02:53.598
than your marginal
utility of a strawberry,

00:02:53.598 --> 00:02:55.204
so if you use that logic that you just

00:02:55.204 --> 00:02:57.343
directly compare marginal utilities,

00:02:57.343 --> 00:02:58.424
you would always buy more

00:02:58.424 --> 00:03:00.630
Rolls-Royces and less strawberries.

00:03:00.630 --> 00:03:03.167
And the key idea here is, what matters is

00:03:03.167 --> 00:03:05.486
marginal utility per dollar,

00:03:05.486 --> 00:03:08.569
or marginal utility divided by price.

00:03:09.512 --> 00:03:10.583
Then, all of a sudden,

00:03:10.583 --> 00:03:12.492
a strawberry might be able to compete

00:03:12.492 --> 00:03:14.560
a little bit more with a Rolls-Royce.

00:03:14.560 --> 00:03:16.576
So we want to do the
exact same thing here.

00:03:16.576 --> 00:03:17.844
And on a test like the AP exam,

00:03:17.844 --> 00:03:19.752
you definitely wanna show your work.

00:03:19.752 --> 00:03:22.585
So let's say for fudge, for fudge,

00:03:24.703 --> 00:03:27.172
we could calculate the marginal utility

00:03:27.172 --> 00:03:29.932
per price is going to be equal to,

00:03:29.932 --> 00:03:31.771
here they give the
utility of the last pound.

00:03:31.771 --> 00:03:32.710
I guess this could be indicative

00:03:32.710 --> 00:03:35.279
of also the next pound.

00:03:35.279 --> 00:03:38.641
So this is going to be 12 divided by $2,

00:03:38.641 --> 00:03:40.196
which is equal to six.

00:03:40.196 --> 00:03:42.529
And then for coffee, coffee,

00:03:43.927 --> 00:03:47.602
it is the marginal
utility divided by price,

00:03:47.602 --> 00:03:49.956
is going to be equal to 20

00:03:49.956 --> 00:03:52.702
divided by four, which is equal to five.

00:03:52.702 --> 00:03:55.113
And so I would say that Mandy,

00:03:55.113 --> 00:03:56.488
so I would circle that,

00:03:56.488 --> 00:03:59.728
and I would say, so I would say,

00:03:59.728 --> 00:04:01.061
Mandy should buy

00:04:04.719 --> 00:04:06.972
or purchase more fudge,

00:04:06.972 --> 00:04:08.139
buy more fudge

00:04:11.278 --> 00:04:13.861
to maximize her utility because

00:04:15.130 --> 00:04:17.713
the marginal utility per dollar

00:04:20.400 --> 00:04:22.233
is higher than coffee.

00:04:28.861 --> 00:04:30.278
And we'd be done.


Comparing proportionality constants

WEBVTT
Kind: captions
Language: en

00:00:00.210 --> 00:00:02.295
- We're told that cars A, B, and C

00:00:02.295 --> 00:00:04.629
are traveling at constant speeds

00:00:04.629 --> 00:00:08.241
and they say select the car
that travels the fastest

00:00:08.241 --> 00:00:10.573
and we have these three scenarios here.

00:00:10.573 --> 00:00:12.443
So, I encourage you to pause this video

00:00:12.443 --> 00:00:14.984
and try to figure out
which of these three cars

00:00:14.984 --> 00:00:16.734
is traveling the fastest,

00:00:16.734 --> 00:00:18.651
car A, car B, or car C.

00:00:19.442 --> 00:00:21.333
Alright, let's work through
this together.

00:00:21.333 --> 00:00:22.333
So, car A,

00:00:22.333 --> 00:00:24.678
they clearly just give its speed,

00:00:24.678 --> 00:00:27.111
it's 50 kilometers per hour.

00:00:27.111 --> 00:00:27.944
Now, let's see,

00:00:27.944 --> 00:00:30.261
car B travels the distance of D kilometers

00:00:30.261 --> 00:00:31.094
in H hours

00:00:32.071 --> 00:00:34.688
based on the equation 55h

00:00:34.688 --> 00:00:37.402
is equal to D.

00:00:37.402 --> 00:00:40.078
Alright, now, let's see if
we can translate this somehow

00:00:40.078 --> 00:00:42.256
into kilometers per hour.

00:00:42.256 --> 00:00:44.060
So, 55h is equal to D

00:00:44.060 --> 00:00:45.663
or we could say D

00:00:45.663 --> 00:00:48.795
is equal to 55H

00:00:48.795 --> 00:00:50.542
and here I'm doing,

00:00:50.542 --> 00:00:51.968
this is this scenario right over here,

00:00:51.968 --> 00:00:53.445
not scenario A.

00:00:53.445 --> 00:00:54.929
And so, another way
to think about it is

00:00:54.929 --> 00:00:56.708
distance divided by time,

00:00:56.708 --> 00:00:58.664
so if we divide both sides by hours,

00:00:58.664 --> 00:01:01.527
we would have distance divided by time,

00:01:01.527 --> 00:01:03.777
and so if we have D over H,

00:01:04.812 --> 00:01:07.654
then we would just be left
with 55 on the right hand side.

00:01:07.654 --> 00:01:09.483
All I did is I divided both sides

00:01:09.483 --> 00:01:11.278
by H.

00:01:11.278 --> 00:01:13.986
Now, this is distance divided by time,

00:01:13.986 --> 00:01:15.861
so the units here are going to be,

00:01:15.861 --> 00:01:16.694
we're assuming,

00:01:16.694 --> 00:01:17.934
and it tells us D is in kilometers,

00:01:17.934 --> 00:01:19.185
H is in hours,

00:01:19.185 --> 00:01:20.861
so the units here are going to be

00:01:20.861 --> 00:01:22.528
kilometers per hour.

00:01:23.609 --> 00:01:26.634
So, car B is going 55 kilometers per hour

00:01:26.634 --> 00:01:30.057
while car A is only going
50 kilometers per hour.

00:01:30.057 --> 00:01:32.882
So, so far, car B is the fastest.

00:01:32.882 --> 00:01:36.965
Now, car C travels 135
kilometers in three hours.

00:01:38.671 --> 00:01:40.454
Well, let's just get the hourly rate

00:01:40.454 --> 00:01:42.376
or I guess you could say the unit rate.

00:01:42.376 --> 00:01:45.293
So, 135 kilometers

00:01:47.409 --> 00:01:50.409
in three hours,

00:01:50.409 --> 00:01:52.787
and so we can get the rate per hour,

00:01:52.787 --> 00:01:55.998
so 135 divided by three is what?

00:01:55.998 --> 00:01:57.481
That is going to be,

00:01:57.481 --> 00:01:58.734
let's do it in our head,

00:01:58.734 --> 00:01:59.644
I think it's 45

00:01:59.644 --> 00:02:00.806
but let me just verify that,

00:02:00.806 --> 00:02:03.022
three goes into 135,

00:02:03.022 --> 00:02:04.729
three goes into 13 four times,

00:02:04.729 --> 00:02:06.673
four times three is 12.

00:02:06.673 --> 00:02:08.205
You subtract,

00:02:08.205 --> 00:02:09.367
you get, yep,

00:02:09.367 --> 00:02:11.781
three goes into 15 five times,

00:02:11.781 --> 00:02:13.737
five times three is 15.

00:02:13.737 --> 00:02:14.981
Subtract zero.

00:02:14.981 --> 00:02:17.270
So, this is equal to 45

00:02:17.270 --> 00:02:18.937
kilometers per hour.

00:02:20.108 --> 00:02:21.894
So, car A is 50 kilometers per hour,

00:02:21.894 --> 00:02:24.186
car B is 55 kilometers per hour,

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car C is 45 kilometers per hour,

00:02:26.480 --> 00:02:29.190
so car B is the fastest.


Interpret proportionality constants

WEBVTT
Kind: captions
Language: en

00:00:00.775 --> 00:00:03.747
- [Instructor] We can
calculate the depth d of snow,

00:00:03.747 --> 00:00:06.929
in centimeters, that
accumulates in Harper's yard

00:00:06.929 --> 00:00:10.110
during the first h hours of a snowstorm

00:00:10.110 --> 00:00:14.359
using the equation d is
equal to five times h.

00:00:14.359 --> 00:00:16.193
So d is the depth of snow in centimeters,

00:00:16.193 --> 00:00:18.860
h is time that elapses in hours.

00:00:20.582 --> 00:00:23.926
How many hours does it take
for one centimeter of snow

00:00:23.926 --> 00:00:25.853
to accumulate in Harper's yard?

00:00:25.853 --> 00:00:28.337
Pause this video and see
if you can figure that out.

00:00:28.337 --> 00:00:30.543
All right so we wanna figure out what h

00:00:30.543 --> 00:00:33.399
gives us a d of one centimeter.

00:00:33.399 --> 00:00:35.582
Remember d is measured in centimeters.

00:00:35.582 --> 00:00:37.556
So we really just need
to solve the equation

00:00:37.556 --> 00:00:40.737
one centimeter, when d is equal to one,

00:00:40.737 --> 00:00:42.570
what is h going to be?

00:00:43.430 --> 00:00:45.381
And to solve for h we just need to divide

00:00:45.381 --> 00:00:47.424
both sides by five.

00:00:47.424 --> 00:00:49.189
So you divide both sides by five,

00:00:49.189 --> 00:00:50.884
the coefficient on the h,

00:00:50.884 --> 00:00:54.831
and you are left with h is equal to 1/5.

00:00:54.831 --> 00:00:56.805
And the unit for h is hours,

00:00:56.805 --> 00:00:58.972
1/5 of an hour.

00:00:59.893 --> 00:01:02.238
So 1/5 of an hour.

00:01:02.238 --> 00:01:03.307
If they had minutes there,

00:01:03.307 --> 00:01:04.909
then you would say well 1/5 of an hour,

00:01:04.909 --> 00:01:07.022
there's 60 minutes,
well this is 12 minutes.

00:01:07.022 --> 00:01:09.460
But they just want it
as a number of hours,

00:01:09.460 --> 00:01:11.457
so 1/5 of an hour.

00:01:11.457 --> 00:01:13.291
How many centimeters of snow accumulates

00:01:13.291 --> 00:01:14.291
in per hour?

00:01:15.404 --> 00:01:17.331
Or this is a little bit of a typo.

00:01:17.331 --> 00:01:20.234
How many centimeters of snow accumulate in

00:01:20.234 --> 00:01:22.695
we could say one hour,

00:01:22.695 --> 00:01:24.297
in one hour, or they could have said

00:01:24.297 --> 00:01:27.386
how many centimeters of
snow accumulate per hour.

00:01:27.386 --> 00:01:29.058
That's another way of thinking about it.

00:01:29.058 --> 00:01:31.821
So we could get rid of per hour.

00:01:31.821 --> 00:01:35.350
So pause the video and see
if you can figure that out.

00:01:35.350 --> 00:01:38.067
Well there's a couple of
ways to think about it.

00:01:38.067 --> 00:01:39.530
Perhaps the easiest one is to say,

00:01:39.530 --> 00:01:42.757
well what is d when h is equal to one?

00:01:42.757 --> 00:01:44.429
And so we could just say

00:01:44.429 --> 00:01:46.400
d when h is equal to one,

00:01:46.400 --> 00:01:48.606
when only one hour has elapsed,

00:01:48.606 --> 00:01:50.417
well it's going to be five times one

00:01:50.417 --> 00:01:51.668
which is equal to five,

00:01:51.668 --> 00:01:54.872
and our units for d are in centimeters.

00:01:54.872 --> 00:01:56.915
So five centimeters.

00:01:56.915 --> 00:01:58.998
Let's do another example.

00:02:00.421 --> 00:02:03.510
Betty's Bakery calculates
the total price d

00:02:03.510 --> 00:02:05.600
in dollars for c cupcakes

00:02:05.600 --> 00:02:07.248
using the equation

00:02:07.248 --> 00:02:08.915
d is equal to two c.

00:02:10.545 --> 00:02:13.703
What does two mean in this situation?

00:02:13.703 --> 00:02:18.138
So pause this video and see if
you can answer this question.

00:02:18.138 --> 00:02:22.364
So remember d is in
dollars for c cupcakes.

00:02:22.364 --> 00:02:24.199
Now one way to think about it is,

00:02:24.199 --> 00:02:28.262
what happens if we take d
is equal to two times c,

00:02:28.262 --> 00:02:30.816
what happens if we divide both sides by c?

00:02:30.816 --> 00:02:32.233
You have d over c

00:02:33.069 --> 00:02:34.926
is equal to two.

00:02:34.926 --> 00:02:37.481
And so what would be the
units right over here?

00:02:37.481 --> 00:02:39.547
Well we have dollars, d dollars,

00:02:39.547 --> 00:02:41.358
over c cupcakes.

00:02:41.358 --> 00:02:43.025
So this would be $2,

00:02:44.191 --> 00:02:47.326
because that's the units
for d, per cupcake,

00:02:47.326 --> 00:02:49.741
dollars per cupcake.

00:02:49.741 --> 00:02:51.830
This is the unit rate per cupcake.

00:02:51.830 --> 00:02:54.129
How much do you have to pay per cupcake?

00:02:54.129 --> 00:02:56.265
So which of these
choices match up to that?

00:02:56.265 --> 00:02:58.402
The bakery charges $2 for each cupcake,

00:02:58.402 --> 00:03:01.072
yeah $2 per cupcake, that looks right.

00:03:01.072 --> 00:03:03.046
The bakery sells two
cupcakes for a dollar.

00:03:03.046 --> 00:03:05.275
No that would be two cupcakes per dollar,

00:03:05.275 --> 00:03:06.858
not $2 per cupcake.

00:03:08.154 --> 00:03:09.872
The bakery sells two types of cupcakes.

00:03:09.872 --> 00:03:11.405
No no we're definitely not
talking about two types

00:03:11.405 --> 00:03:13.797
of cupcakes, they're just
talking about cupcakes generally,

00:03:13.797 --> 00:03:15.979
or I guess one type of
cupcake, we don't know,

00:03:15.979 --> 00:03:19.710
but just cupcakes generally
is $2 per cupcake.


Percent word problem examples

WEBVTT
Kind: captions
Language: en

00:00:00.626 --> 00:00:01.908
- [Instructor] In a video game,

00:00:01.908 --> 00:00:06.097
Val scored 30% fewer points than Peeta.

00:00:06.097 --> 00:00:08.264
Peeta scored 1,060 points.

00:00:09.553 --> 00:00:12.003
How many points did Val score?

00:00:12.003 --> 00:00:14.148
Pause this video and see
if you can figure out

00:00:14.148 --> 00:00:16.398
how many points Val scored.

00:00:17.396 --> 00:00:19.381
Alright, now let's do this together.

00:00:19.381 --> 00:00:21.491
And there's a couple of ways
that you could think about it.

00:00:21.491 --> 00:00:25.874
One way to think about is
Peeta scored 1,060 points,

00:00:25.874 --> 00:00:28.236
and Val scored 30% fewer.

00:00:28.236 --> 00:00:30.601
When we're saying 30% fewer,

00:00:30.601 --> 00:00:33.751
we're saying essentially take 30% of 1060

00:00:33.751 --> 00:00:36.016
and subtract that from 1060.

00:00:36.016 --> 00:00:40.183
So 30%, so we could write
that as a decimal as 0.30.

00:00:41.788 --> 00:00:45.652
30/100 is the same thing as 30%,

00:00:45.652 --> 00:00:48.611
or we could even write this as .3.

00:00:48.611 --> 00:00:51.944
And then we would want 30% of 1060.

00:00:53.198 --> 00:00:55.076
So if you take Peeta's score

00:00:55.076 --> 00:00:57.702
and then subtract 30% of Peeta's score,

00:00:57.702 --> 00:00:59.469
then this would give you Val's score.

00:00:59.469 --> 00:01:01.486
So that's one way to calculate it.

00:01:01.486 --> 00:01:03.469
Another way to think about is

00:01:03.469 --> 00:01:06.688
whatever you're starting
with, let's call that 100%,

00:01:06.688 --> 00:01:08.985
and if you were to take out 30% of it,

00:01:08.985 --> 00:01:10.868
if you were to have 30% less,

00:01:10.868 --> 00:01:14.423
then you're going to have
70% of what you started with.

00:01:14.423 --> 00:01:15.765
So another way to think about it is

00:01:15.765 --> 00:01:19.932
we could take Peeta's score of
1060 and multiply it by 70%.

00:01:22.490 --> 00:01:25.278
And multiplying it by
70% is the same thing

00:01:25.278 --> 00:01:27.445
as multiplying it by 0.70,

00:01:28.870 --> 00:01:32.529
which is the same thing as
multiplying it by 70/100,

00:01:32.529 --> 00:01:35.065
is the same thing as 7/10.

00:01:35.065 --> 00:01:36.487
So let's just do this.

00:01:36.487 --> 00:01:39.904
So if I have 1060, and I multiply by 0.7,

00:01:42.097 --> 00:01:43.322
what do I get?

00:01:43.322 --> 00:01:44.975
Seven times zero is zero.

00:01:44.975 --> 00:01:46.808
Seven times six is 42.

00:01:48.163 --> 00:01:51.225
Seven times zero is
zero, plus four is four.

00:01:51.225 --> 00:01:53.187
Seven times one is seven.

00:01:53.187 --> 00:01:55.803
And I have one digit to
the right of the decimal.

00:01:55.803 --> 00:01:58.470
So there you have it, it is 742.

00:02:00.778 --> 00:02:01.611
742.

00:02:03.234 --> 00:02:05.860
That is how many points Val scored.

00:02:05.860 --> 00:02:07.943
Let's do another example.

00:02:10.083 --> 00:02:13.238
So we're told there are 20% more goblins

00:02:13.238 --> 00:02:15.951
than wizards in a magic club.

00:02:15.951 --> 00:02:20.118
There are 220 goblins and
wizards altogether in magic club.

00:02:21.263 --> 00:02:25.166
How many goblins are in the magic club?

00:02:25.166 --> 00:02:26.833
So pause the video and see

00:02:26.833 --> 00:02:30.158
if you can work through this on your own.

00:02:30.158 --> 00:02:31.949
So this one is an interesting one.

00:02:31.949 --> 00:02:34.853
It's gonna involve a
little bit of algebra here.

00:02:34.853 --> 00:02:36.968
So what we wanna do is
let's set a variable.

00:02:36.968 --> 00:02:40.410
Let's say w is the number of wizards.

00:02:40.410 --> 00:02:42.533
So that's the number of wizards.

00:02:42.533 --> 00:02:46.700
And then if we said g for
goblins, let's say g for goblins.

00:02:47.634 --> 00:02:49.967
So w plus g is equal to 220,

00:02:52.866 --> 00:02:54.737
is equal to 220.

00:02:54.737 --> 00:02:56.479
And you're like, well,
how does that help me?

00:02:56.479 --> 00:02:58.422
How does that help me actually figure out

00:02:58.422 --> 00:03:00.334
how many goblins are in the magic club?

00:03:00.334 --> 00:03:03.047
I have two variables
here with one equation.

00:03:03.047 --> 00:03:04.805
Well, one way to think about it is,

00:03:04.805 --> 00:03:06.566
remember, they give us
some more information.

00:03:06.566 --> 00:03:09.973
They tell us there are
20%, let me box that,

00:03:09.973 --> 00:03:13.770
there are 20% more goblins than wizards.

00:03:13.770 --> 00:03:15.900
So we also know one other thing.

00:03:15.900 --> 00:03:19.887
We know that the goblins,
we know that the goblins

00:03:19.887 --> 00:03:24.415
are equal to the number
of wizards plus 20%.

00:03:24.415 --> 00:03:27.165
So you could view this as wizards

00:03:28.014 --> 00:03:30.836
plus 20% of wizards.

00:03:30.836 --> 00:03:32.609
And I'm writing that as 20/100,

00:03:32.609 --> 00:03:34.751
or you could even write that as 2/10,

00:03:34.751 --> 00:03:37.793
plus 2/10 times the number of wizards.

00:03:37.793 --> 00:03:39.491
Or another way of thinking about it,

00:03:39.491 --> 00:03:42.728
goblins are equal to, if
I have one of something

00:03:42.728 --> 00:03:44.964
and then I have another
2/10 of that something,

00:03:44.964 --> 00:03:48.297
then I'm gonna have 1.2 of that something.

00:03:48.297 --> 00:03:52.047
So goblins is equal to
1.2 times the wizards.

00:03:53.006 --> 00:03:57.754
And so we could use that
to substitute back in here,

00:03:57.754 --> 00:04:00.557
and then we could say
the number of wizards

00:04:00.557 --> 00:04:04.724
plus the number of goblins,
which happens to be 20% more

00:04:05.742 --> 00:04:09.909
than the number of wizards,
is going to be equal to 220.

00:04:12.031 --> 00:04:13.618
Let me do that in that same color.

00:04:13.618 --> 00:04:15.275
Is equal to 220.

00:04:15.275 --> 00:04:18.067
Now this is pretty
straightforward to solve.

00:04:18.067 --> 00:04:19.734
What is w plus 1.2w?

00:04:20.811 --> 00:04:24.373
Well, that is going to be 2.2w, 2.2w.

00:04:24.373 --> 00:04:27.793
You could view this as one w plus 1.2w

00:04:27.793 --> 00:04:29.793
is 2.2w is equal to 220.

00:04:32.547 --> 00:04:34.639
And so just divide both sides.

00:04:34.639 --> 00:04:36.005
Let me scroll down a little bit.

00:04:36.005 --> 00:04:38.505
Divide both sides by 2.2, 2.2,

00:04:40.047 --> 00:04:41.664
and what do you get?

00:04:41.664 --> 00:04:45.414
You get w, the number
of wizards is equal to,

00:04:47.146 --> 00:04:50.169
let's see, this is going
to be equal to 100.

00:04:50.169 --> 00:04:52.384
The number of wizards is equal to 100.

00:04:52.384 --> 00:04:53.887
Now is that our answer?

00:04:53.887 --> 00:04:58.600
No, they're asking how many
goblins are in the magic club?

00:04:58.600 --> 00:05:02.370
Well, we know that goblins
are 1.2 times the wizards.

00:05:02.370 --> 00:05:06.537
So the number of goblins is
going to be 1.2 times 100,

00:05:08.246 --> 00:05:10.670
which is equal to 120.

00:05:10.670 --> 00:05:13.446
So there's 120 goblins,
does that make sense?

00:05:13.446 --> 00:05:16.633
120 is 20% more than 100.

00:05:16.633 --> 00:05:19.457
And if you add the 100
wizards to the 120 goblins,

00:05:19.457 --> 00:05:23.627
you get 220 goblins
and wizards altogether.

00:05:23.627 --> 00:05:26.087
Let's do another example.

00:05:26.087 --> 00:05:30.486
Here we're told Cody
was 165 centimeters tall

00:05:30.486 --> 00:05:32.414
on the first day of school this year,

00:05:32.414 --> 00:05:34.506
which was 10% taller than he was

00:05:34.506 --> 00:05:37.362
on the first day of school last year.

00:05:37.362 --> 00:05:42.096
How tall was Cody on the
first day of school last year?

00:05:42.096 --> 00:05:46.206
Pause this video, see if
you can figure that out.

00:05:46.206 --> 00:05:47.930
So let's just define a variable here.

00:05:47.930 --> 00:05:50.877
Let's just say that his
height on the first day

00:05:50.877 --> 00:05:54.660
of school last year,
let's say that that is x.

00:05:54.660 --> 00:05:58.069
So his height on the first
day of school last year is x.

00:05:58.069 --> 00:06:00.627
This year he is 10% taller,

00:06:00.627 --> 00:06:04.368
so we would add 10%, which
we could say is 10/100,

00:06:04.368 --> 00:06:07.174
or we could even say that as 1/10.

00:06:07.174 --> 00:06:09.262
He's 1/10 taller.

00:06:09.262 --> 00:06:11.272
So whatever his height was last year,

00:06:11.272 --> 00:06:14.068
we're going to add 1/10
of that same height again

00:06:14.068 --> 00:06:15.710
to get to his height this year,

00:06:15.710 --> 00:06:19.578
which they tell us is 165 centimeters.

00:06:19.578 --> 00:06:23.893
And so here we could say
well, one x plus 1/10 of an x

00:06:23.893 --> 00:06:28.060
is going to be one and
1/10x is equal to 165.

00:06:29.121 --> 00:06:31.265
And now to solve for x, which
remember, was his height

00:06:31.265 --> 00:06:33.490
on the first day of school last year,

00:06:33.490 --> 00:06:35.823
we divide both sides by 1.1.

00:06:36.693 --> 00:06:38.529
And so x is equal to,

00:06:38.529 --> 00:06:40.402
well, let's see what this is going to be.

00:06:40.402 --> 00:06:43.485
If I were to take 165 divided by 1.1,

00:06:46.153 --> 00:06:48.580
the first thing I would wanna do

00:06:48.580 --> 00:06:51.171
is multiply them both by 10.

00:06:51.171 --> 00:06:53.216
So that has the effect of moving

00:06:53.216 --> 00:06:55.786
this decimal place one to the right.

00:06:55.786 --> 00:06:57.632
So really, I am now trying to figure out

00:06:57.632 --> 00:06:59.882
what 11 goes into 1,650 is.

00:07:02.401 --> 00:07:05.230
And so let's see, let me
just do that step by step.

00:07:05.230 --> 00:07:08.080
11 goes into 16 one time.

00:07:08.080 --> 00:07:09.885
One times 11 is 11.

00:07:09.885 --> 00:07:13.860
Subtract, we get a five,
and we bring down a five.

00:07:13.860 --> 00:07:16.480
11 goes into 55 exactly five times.

00:07:16.480 --> 00:07:18.643
Five times 11 is 55.

00:07:18.643 --> 00:07:20.041
Subtract.

00:07:20.041 --> 00:07:22.691
We have no remainder, but
then we bring down this zero

00:07:22.691 --> 00:07:23.721
and we do it one more time.

00:07:23.721 --> 00:07:25.924
11 goes into zero zero times.

00:07:25.924 --> 00:07:28.459
Remember, the decimal
place is right over here.

00:07:28.459 --> 00:07:31.455
Zero times 11 is zero, and
then we have no remainder.

00:07:31.455 --> 00:07:35.010
So last year he was 150 centimeters.

00:07:35.010 --> 00:07:37.027
And it's always good
to do a reality check.

00:07:37.027 --> 00:07:38.909
Make sure, if for example,
if I divided wrong

00:07:38.909 --> 00:07:41.504
and I somehow got 15, or I got 1500,

00:07:41.504 --> 00:07:43.689
just to make sure that that
wouldn't make any sense.

00:07:43.689 --> 00:07:46.539
150 centimeters, you add 10% of that.

00:07:46.539 --> 00:07:49.467
10% of 150 is 15 centimeters.

00:07:49.467 --> 00:07:52.068
So you add 10% of that, you indeed do get

00:07:52.068 --> 00:07:56.235
to 165 centimeters for the
first day of school this year.


Proportional relationships example

WEBVTT
Kind: captions
Language: en

00:00:00.459 --> 00:00:03.578
- [Instructor] We're told
that Mael mixes 15 milliliters

00:00:03.578 --> 00:00:06.495
of bleach with 3.75 liters of water

00:00:07.936 --> 00:00:10.893
to make a sanitizing
solution for a daycare.

00:00:10.893 --> 00:00:12.832
The amounts of bleach and water

00:00:12.832 --> 00:00:15.046
always have to be proportional

00:00:15.046 --> 00:00:17.810
when he makes the sanitizing solution.

00:00:17.810 --> 00:00:20.666
Which of the following could
be combinations of volumes

00:00:20.666 --> 00:00:24.607
of bleach and water for
Mael's sanitizing solution?

00:00:24.607 --> 00:00:26.598
And they gave us,
actually they gave us five

00:00:26.598 --> 00:00:29.463
potential combinations,
they say pick three.

00:00:29.463 --> 00:00:30.967
So I encourage you to pause this video

00:00:30.967 --> 00:00:32.227
and try to figure it out.

00:00:32.227 --> 00:00:34.929
Remember, he mixes 15
milliliters of bleach

00:00:34.929 --> 00:00:37.512
for every 3.75 liters of water.

00:00:40.565 --> 00:00:43.352
Alright, now let's try
to work this together.

00:00:43.352 --> 00:00:44.781
So I'm gonna make a table here.

00:00:44.781 --> 00:00:49.761
So let's say this is bleach,
bleach in milliliters.

00:00:49.761 --> 00:00:52.844
And lets say this is water in liters.

00:00:54.657 --> 00:00:58.490
And they tell us that
he mixes 15 milliliters,

00:01:00.642 --> 00:01:01.883
the unit here is milliliters,

00:01:01.883 --> 00:01:04.474
for every 15 milliliters of bleach

00:01:04.474 --> 00:01:07.057
for every 3.75 liters of water.

00:01:09.955 --> 00:01:12.983
So what is the
proportionality constant here?

00:01:12.983 --> 00:01:16.529
If you said the water is
equal to some constant

00:01:16.529 --> 00:01:19.222
times the bleach, well what's going on?

00:01:19.222 --> 00:01:22.482
Well let's see, what would
he have to multiply by?

00:01:22.482 --> 00:01:25.982
He would have to multiply by 3.75 over 15.

00:01:29.381 --> 00:01:31.964
Now what is 3.75 divided by 15?

00:01:33.009 --> 00:01:34.998
Let me actually do it right over here,

00:01:34.998 --> 00:01:36.498
15 goes into 3.75.

00:01:39.044 --> 00:01:42.053
Let's see, 15 goes into 37 two times,

00:01:42.053 --> 00:01:43.261
we have our little
decimal right over here,

00:01:43.261 --> 00:01:46.178
two times 15 is 30, subtract seven,

00:01:48.272 --> 00:01:52.505
bring down the five and
then 15 times five is 75,

00:01:52.505 --> 00:01:55.858
five times 15 is 75, it all works out.

00:01:55.858 --> 00:01:58.945
So we see to go from bleach to water

00:01:58.945 --> 00:02:03.112
we're multiplying by a
proportionality constant of 0.25.

00:02:05.855 --> 00:02:07.328
So we have to see which of these

00:02:07.328 --> 00:02:09.807
have the same exact
proportionality constant

00:02:09.807 --> 00:02:11.961
going from bleach to water.

00:02:11.961 --> 00:02:15.628
So let's see, this next
one is 12 and three.

00:02:18.064 --> 00:02:20.481
So if we multiply 12 by 0.25,

00:02:23.254 --> 00:02:24.514
do we get three?

00:02:24.514 --> 00:02:27.776
Yeah, three is one fourth
of 12, 0.25, 25 hundredths

00:02:27.776 --> 00:02:31.655
is the same thing as one
fourth so this one checks out.

00:02:31.655 --> 00:02:34.405
What about going from six to 1.5?

00:02:36.845 --> 00:02:39.095
Are we multiplying by 0.25?

00:02:40.211 --> 00:02:42.711
Yeah, 1.5 is one fourth of six

00:02:44.077 --> 00:02:46.103
or another way to think about it is

00:02:46.103 --> 00:02:47.853
what is six times 25?

00:02:49.951 --> 00:02:52.909
It is a 150 so six times 25 hundredths

00:02:52.909 --> 00:02:57.318
would be a 150 hundredths
which is the same thing as 1.5.

00:02:57.318 --> 00:02:58.903
So this one works.

00:02:58.903 --> 00:03:01.693
What about three and 0.75?

00:03:01.693 --> 00:03:03.193
So three and 0.75.

00:03:05.796 --> 00:03:07.879
Am I multiplying by 0.25?

00:03:08.757 --> 00:03:11.427
Yeah, if I multiply three
times 25 hundredths,

00:03:11.427 --> 00:03:14.384
I get 75 hundredths so that works.

00:03:14.384 --> 00:03:16.965
So actually the first three
choices are our three answers

00:03:16.965 --> 00:03:18.680
but let's just verify that the next two

00:03:18.680 --> 00:03:21.069
are not good answers.

00:03:21.069 --> 00:03:24.152
So let's see, if I go from 20 to 5.5,

00:03:27.671 --> 00:03:30.338
and so am I multiplying by 0.25?

00:03:31.663 --> 00:03:34.975
No, 0.25 which is the
same thing as one fourth,

00:03:34.975 --> 00:03:38.629
one fourth times 20 is five, not 5.5.

00:03:38.629 --> 00:03:39.892
So that doesn't work.

00:03:39.892 --> 00:03:42.475
And then going from 11 to 3.75,

00:03:44.483 --> 00:03:46.200
well we definitely know
that's not gonna work

00:03:46.200 --> 00:03:48.640
because notice we have
the same amount of water

00:03:48.640 --> 00:03:50.640
but we have less bleach.

00:03:51.572 --> 00:03:53.973
Or you could say what's one fourth of 11?

00:03:53.973 --> 00:03:57.327
Well that's going to be less than 3.75

00:03:57.327 --> 00:04:00.744
so we can rule both of these choices out.


Tax, discount and tip examples

WEBVTT
Kind: captions
Language: en

00:00:00.491 --> 00:00:02.877
- We're told that Casey buys a bracelet.

00:00:02.877 --> 00:00:07.708
She pays for the bracelet and
pays 72 cents in sales tax.

00:00:07.708 --> 00:00:10.208
The sales tax rate is 6%.

00:00:11.283 --> 00:00:15.987
What is the original price
of the bracelet, before tax?

00:00:15.987 --> 00:00:19.839
So pause this video and see
if you can figure this out.

00:00:19.839 --> 00:00:23.294
Well, let's think about how
your sales tax is calculated.

00:00:23.294 --> 00:00:27.461
Your sales tax would be
equal to the sales tax rate,

00:00:29.608 --> 00:00:32.180
I'll just call that the
rate, so that's going to be

00:00:32.180 --> 00:00:35.847
equal to the rate times
your original price.

00:00:39.609 --> 00:00:41.435
And what do we know here?

00:00:41.435 --> 00:00:44.369
Well they tell us what our sales tax is.

00:00:44.369 --> 00:00:48.536
Our sales tax is 72 cents,
so I'll write that here.

00:00:49.424 --> 00:00:53.563
So 72 cents is going to be
equal to my sales tax rate,

00:00:53.563 --> 00:00:56.127
they also tell us that, that is 6%,

00:00:56.127 --> 00:01:00.446
is going to be equal to
6% times my original price

00:01:00.446 --> 00:01:03.062
and that's what they're asking
for, the original price.

00:01:03.062 --> 00:01:06.482
So times, I'm just going
to call it P for short,

00:01:06.482 --> 00:01:07.754
original price.

00:01:07.754 --> 00:01:08.587
And so you have it.

00:01:08.587 --> 00:01:10.965
We have a fairly
straightforward equation now.

00:01:10.965 --> 00:01:13.235
And now I'm just going to
rewrite everything as decimal,

00:01:13.235 --> 00:01:16.735
so we could write 0.72 is equal to 6%

00:01:19.117 --> 00:01:22.229
is the same thing as six per 100,

00:01:22.229 --> 00:01:25.094
which is the same thing as six hundredths,

00:01:25.094 --> 00:01:27.511
0.06, six hundredths times p.

00:01:30.085 --> 00:01:32.819
And now we can just divide
both sides by six hundredths

00:01:32.819 --> 00:01:37.375
to solve for the original
price, so let's do that.

00:01:37.375 --> 00:01:38.208
Alright.

00:01:39.317 --> 00:01:41.206
So on the right-hand side I have a P.

00:01:41.206 --> 00:01:44.495
On the left-hand side
if I have 72 hundredths

00:01:44.495 --> 00:01:47.676
divided by six hundredths,
well that's going to be

00:01:47.676 --> 00:01:50.863
equal to 12, 12 times six is 72.

00:01:50.863 --> 00:01:54.247
Six hundredths times 12
would be 72 hundredths.

00:01:54.247 --> 00:01:57.497
And so this is going to be equal to 12.

00:01:58.612 --> 00:02:02.860
So the original price of the
bracelet, before tax is $12.

00:02:02.860 --> 00:02:04.092
And you can verify that.

00:02:04.092 --> 00:02:08.259
6% of $12, 0.06 times 12
is indeed 72 hundredths.

00:02:12.513 --> 00:02:14.596
Let's do another example.

00:02:15.582 --> 00:02:18.999
A store has a 25% off sale on coats.

00:02:20.149 --> 00:02:24.316
With this discount, the
price of one coat is $34.50.

00:02:25.446 --> 00:02:28.877
What is the original price of the coat?

00:02:28.877 --> 00:02:30.335
So once again, pause this video

00:02:30.335 --> 00:02:32.510
and see if you can figure it out.

00:02:32.510 --> 00:02:34.464
These examples, they're
actually quite useful

00:02:34.464 --> 00:02:36.713
because you will encounter
this all of the time

00:02:36.713 --> 00:02:39.261
when you are shopping or
you're trying to calculate tax

00:02:39.261 --> 00:02:41.548
or you're buying something,
so think about this.

00:02:41.548 --> 00:02:44.806
Can you figure out the
original price of the coat?

00:02:44.806 --> 00:02:46.284
Alright, well let's just write it out

00:02:46.284 --> 00:02:48.253
the same way we did last time.

00:02:48.253 --> 00:02:52.420
Let's say our original
price and then there's a 25%

00:02:55.566 --> 00:02:59.649
discount, so minus 25%
of the original price

00:03:05.678 --> 00:03:08.178
is going to be equal to $34.50

00:03:12.310 --> 00:03:14.873
because that the price with the discount.

00:03:14.873 --> 00:03:17.448
So can we solve for the original price?

00:03:17.448 --> 00:03:20.022
So one way to think about it, this is 100%

00:03:20.022 --> 00:03:22.766
of the original price or
one times the original price

00:03:22.766 --> 00:03:26.536
minus 25% of the original
price is going to equal this.

00:03:26.536 --> 00:03:30.703
Well, 100% minus 25% of
something is going to be 75%

00:03:32.682 --> 00:03:34.515
of our original price,

00:03:37.323 --> 00:03:39.103
this is why people use
letters for variables,

00:03:39.103 --> 00:03:41.478
so they don't have to keep
writing this over and over again,

00:03:41.478 --> 00:03:44.061
is going to be equal to $34.50.

00:03:45.518 --> 00:03:48.074
Now to solve for our
original price we just divide

00:03:48.074 --> 00:03:50.350
both sides by 75%.

00:03:50.350 --> 00:03:52.155
And 75%, that's the same thing as

00:03:52.155 --> 00:03:54.488
75 per 100 or 75 hundredths.

00:03:56.123 --> 00:03:59.956
So I could divide both
sides by 75 hundredths.

00:04:02.612 --> 00:04:04.273
On the left-hand side
these two are equivalent,

00:04:04.273 --> 00:04:06.039
so I'll be left with my original price.

00:04:06.039 --> 00:04:08.684
And now I just have to figure
out what is this going to be.

00:04:08.684 --> 00:04:11.401
So if I take 75 hundredths
and let me actually

00:04:11.401 --> 00:04:13.387
multiply both of these times 100.

00:04:13.387 --> 00:04:15.207
So this is going to be the same thing as

00:04:15.207 --> 00:04:16.874
3,450 divided by 75.

00:04:18.888 --> 00:04:20.639
I just did that to get
rid of the decimals.

00:04:20.639 --> 00:04:22.825
I moved, essentially, both of the decimals

00:04:22.825 --> 00:04:24.733
to the right two places.

00:04:24.733 --> 00:04:27.703
Multiplying the numerator
and the denominator by 100.

00:04:27.703 --> 00:04:31.620
So let's see how many
times 75 goes into 3,450.

00:04:32.850 --> 00:04:35.089
Let's see, 75 doesn't go into three.

00:04:35.089 --> 00:04:36.426
Doesn't go into 34.

00:04:36.426 --> 00:04:38.093
It does go into 345.

00:04:39.242 --> 00:04:41.598
What is that, four times?

00:04:41.598 --> 00:04:43.431
Four times five is 20.

00:04:45.710 --> 00:04:48.793
Four times seven is 28 plus two, 300.

00:04:49.723 --> 00:04:50.653
And then you subtract.

00:04:50.653 --> 00:04:51.813
Get 45.

00:04:51.813 --> 00:04:52.845
Bring down the zero.

00:04:52.845 --> 00:04:54.124
450.

00:04:54.124 --> 00:04:56.374
75 goes into 450 six times.

00:04:57.841 --> 00:05:00.162
Six times five is 30.

00:05:00.162 --> 00:05:03.582
Six times seven is 42
plus three is indeed 45

00:05:03.582 --> 00:05:05.380
and we luckily have no remainder.

00:05:05.380 --> 00:05:09.649
So this right over here is
going to be equal to $46.

00:05:09.649 --> 00:05:11.560
Our original price is $46.

00:05:11.560 --> 00:05:15.790
You take off one-fourth or
25%, you're left with $34.50.

00:05:15.790 --> 00:05:18.022
And that makes intuitive sense as well.

00:05:18.022 --> 00:05:20.563
Always good to just take
whatever answer you have

00:05:20.563 --> 00:05:23.356
and put it back in and
see if it makes sense.

00:05:23.356 --> 00:05:26.759
In terms of if you somehow
got $460 and you said taking

00:05:26.759 --> 00:05:30.466
25% off you get to $34.50, you'd
say "Well that sounds off."

00:05:30.466 --> 00:05:33.710
Or if you got $4 and you
said you take 25% off

00:05:33.710 --> 00:05:36.602
and you got $34.50 that
would have also felt off,

00:05:36.602 --> 00:05:38.467
but $46 feels right.

00:05:38.467 --> 00:05:39.475
And you can even calculate that.

00:05:39.475 --> 00:05:44.092
What's 25% of $46, subtract that from $46

00:05:44.092 --> 00:05:46.259
and you should get $34.50.


Exploring scale copies

WEBVTT
Kind: captions
Language: en

00:00:00.633 --> 00:00:03.475
- [Instructor] We are
told, "Drag the sliders."

00:00:03.475 --> 00:00:05.383
And then they say, "Which
slider creates a scale

00:00:05.383 --> 00:00:06.968
"copy of the shape?"

00:00:06.968 --> 00:00:09.893
Or, "Which slider creates
scale copies of the shape?"

00:00:09.893 --> 00:00:11.392
So let's just see and
explore this a little bit.

00:00:11.392 --> 00:00:14.565
Okay, that's pretty neat,
these sliders seem to change

00:00:14.565 --> 00:00:18.382
the shape in some way,
and in different ways.

00:00:18.382 --> 00:00:21.835
So shape B right over
here, so it starts off,

00:00:21.835 --> 00:00:24.552
it looks like the width
is a little bit bigger

00:00:24.552 --> 00:00:26.530
than the height, I'm just
trying to eyeball it,

00:00:26.530 --> 00:00:28.280
we don't know the exact numbers.

00:00:28.280 --> 00:00:30.150
And so in order to create a scaled copy,

00:00:30.150 --> 00:00:33.357
you'd want to scale the
width, you'd want to scale

00:00:33.357 --> 00:00:36.810
this bottom side, and the top
side, and all of the sides.

00:00:36.810 --> 00:00:39.086
You'd want to scale by the same factor.

00:00:39.086 --> 00:00:41.396
But as we move this slider,
it seems like it's only

00:00:41.396 --> 00:00:44.567
scaling the width, it's
not scaling the height.

00:00:44.567 --> 00:00:46.917
So this slider, shape B right over here,

00:00:46.917 --> 00:00:49.032
the slider for shape B is not creating

00:00:49.032 --> 00:00:52.771
scale copies of itself, it's
only increasing the width,

00:00:52.771 --> 00:00:54.115
not the height.

00:00:54.115 --> 00:00:56.835
While shape A, it looks
like it is increasing

00:00:56.835 --> 00:00:59.589
both the width and the height,

00:00:59.589 --> 00:01:02.096
so that would be a scale copy.

00:01:02.096 --> 00:01:05.992
So for example, that looks
like a scale copy of this,

00:01:05.992 --> 00:01:07.372
which looks like a scale copy of this,

00:01:07.372 --> 00:01:09.492
which looks like a scale copy of that,

00:01:09.492 --> 00:01:11.046
which was our original shape.

00:01:11.046 --> 00:01:13.879
That is not a scaled copy of this.

00:01:14.721 --> 00:01:16.804
Let's do another example.

00:01:17.814 --> 00:01:20.128
So once again they say,
"Drag the sliders."

00:01:20.128 --> 00:01:22.115
And they say, "Which
slider creates a scale copy

00:01:22.115 --> 00:01:23.455
"of the shape?"

00:01:23.455 --> 00:01:25.166
All right, let's get shape A.

00:01:25.166 --> 00:01:27.468
So this does look like we're scaling down,

00:01:27.468 --> 00:01:29.178
but we're scaling both
the width and the height

00:01:29.178 --> 00:01:32.134
by the same factor, so this shape A slider

00:01:32.134 --> 00:01:36.090
does look like it's creating
scale copies of the shape.

00:01:36.090 --> 00:01:39.088
B right over here, well
now we're only scaling,

00:01:39.088 --> 00:01:40.918
it looks like we're
only scaling the height,

00:01:40.918 --> 00:01:43.967
but not the width, so this
is not creating scale copies

00:01:43.967 --> 00:01:45.622
of our original shape.

00:01:45.622 --> 00:01:48.754
It's elongating it, it's
increasing its height,

00:01:48.754 --> 00:01:50.254
but not the width.


Proportionality constant from table

WEBVTT
Kind: captions
Language: en

00:00:00.543 --> 00:00:01.681
- [Instructor] We're told the quantities

00:00:01.681 --> 00:00:04.281
x and y are proportional,

00:00:04.281 --> 00:00:06.302
and then they give us a
table where they give us

00:00:06.302 --> 00:00:09.181
a bunch of x's and they give
us the corresponding y's.

00:00:09.181 --> 00:00:11.108
When x is four, y is 10.

00:00:11.108 --> 00:00:15.288
When x is five, y is 12.5,
and so on and so forth.

00:00:15.288 --> 00:00:19.165
Find the constant of proportionality, r,

00:00:19.165 --> 00:00:22.498
in the equation y is equal to r times x.

00:00:23.461 --> 00:00:27.455
So pause this video and see
if you can figure that out.

00:00:27.455 --> 00:00:29.336
All right so we wanna find the constant,

00:00:29.336 --> 00:00:31.426
the constant of proportionality,

00:00:31.426 --> 00:00:32.842
I sometimes have trouble saying that,

00:00:32.842 --> 00:00:35.512
r, where if I for any x,

00:00:35.512 --> 00:00:38.012
if I multiply it by r I get y.

00:00:38.949 --> 00:00:40.876
Well we just have to
look at each of these x's

00:00:40.876 --> 00:00:43.059
and figure out well what
are we multiplying by

00:00:43.059 --> 00:00:44.847
to get to y.

00:00:44.847 --> 00:00:47.726
So what do I have to multiply
four by to get to 10?

00:00:47.726 --> 00:00:50.861
Well if you multiply it by 10/4,

00:00:50.861 --> 00:00:54.297
if you multiply it by 10/4,
then you're gonna get to 10.

00:00:54.297 --> 00:00:58.430
And 10/4 is the same thing
as multiplying it by,

00:00:58.430 --> 00:01:00.613
let's see that is 2 1/2.

00:01:00.613 --> 00:01:03.121
So this is times 2 1/2.

00:01:03.121 --> 00:01:04.607
So let's see does that hold?

00:01:04.607 --> 00:01:06.975
In every case if I multiply by 2 1/2,

00:01:06.975 --> 00:01:10.017
if I multiply x by 2 1/2, do I get y?

00:01:10.017 --> 00:01:12.014
So five times 2 1/2,

00:01:12.014 --> 00:01:15.358
five times two is 10, and
then another 1/2 is 2.5.

00:01:15.358 --> 00:01:17.935
It indeed does equals 12.5.

00:01:17.935 --> 00:01:19.816
Five times 2.5.

00:01:19.816 --> 00:01:21.697
And then here 10 times 2.5

00:01:21.697 --> 00:01:22.864
is clearly 25.

00:01:24.367 --> 00:01:25.551
And you can work it out by hand.

00:01:25.551 --> 00:01:27.920
I did the little bit of the
multiplication in my head,

00:01:27.920 --> 00:01:29.475
but you can see in every case,

00:01:29.475 --> 00:01:33.678
I take my x, I multiply
it by 2.5, I get my y.

00:01:33.678 --> 00:01:35.977
So in every case I take
my x and multiply it

00:01:35.977 --> 00:01:39.878
by the constant of
proportionality 2.5, I get my y.

00:01:39.878 --> 00:01:41.527
So what's the constant of proportionality?

00:01:41.527 --> 00:01:42.360
It is 2.5.


Constant of proportionality from equation

WEBVTT
Kind: captions
Language: en

00:00:00.707 --> 00:00:02.705
- [Instructor] We are
asked, "What is the constant

00:00:02.705 --> 00:00:06.872
"of proportionality in the
equation 4y is equal to 8x?"

00:00:07.723 --> 00:00:11.025
Pause this video and have
a go at this question.

00:00:11.025 --> 00:00:12.419
All right, so we might be used

00:00:12.419 --> 00:00:14.469
to seeing constants of proportionality

00:00:14.469 --> 00:00:17.356
when we have equations in
a slightly different form.

00:00:17.356 --> 00:00:19.449
A constant of proportionality is

00:00:19.449 --> 00:00:22.502
what do you multiply x by to get to y?

00:00:22.502 --> 00:00:23.979
So y would be equal

00:00:23.979 --> 00:00:27.562
to our constant of
proportionality times x.

00:00:31.359 --> 00:00:33.711
But this isn't written in that form,

00:00:33.711 --> 00:00:36.194
so what we do is
manipulate it a little bit

00:00:36.194 --> 00:00:38.166
so that we can see it in that form.

00:00:38.166 --> 00:00:40.857
And the obvious thing is we
just need to solve for y.

00:00:40.857 --> 00:00:44.567
So right now it says 4y is equal to 8x.

00:00:44.567 --> 00:00:45.716
Well, if we wanna solve for y,

00:00:45.716 --> 00:00:48.572
we can just divide both sides by four,

00:00:48.572 --> 00:00:52.016
and we are left with y is
equal to eight divided by four,

00:00:52.016 --> 00:00:54.151
which is two times x.

00:00:54.151 --> 00:00:57.919
Well, now the constant of
proportionality jumps out at us.

00:00:57.919 --> 00:01:01.169
To get y, we have multiply x by two.

00:01:02.048 --> 00:01:04.811
That is our constant of proportionality.

00:01:04.811 --> 00:01:06.894
Let's do another example.

00:01:08.214 --> 00:01:11.924
Here we're asked, "Which
equation has a constant

00:01:11.924 --> 00:01:15.287
"of proportionality equal to 1/2?"

00:01:15.287 --> 00:01:17.238
Again, pause the video.

00:01:17.238 --> 00:01:19.782
Try to answer it yourself.

00:01:19.782 --> 00:01:22.498
Okay, so I'm just gonna
go equation by equation

00:01:22.498 --> 00:01:26.177
and calculate their
constants of proportionality

00:01:26.177 --> 00:01:28.458
and see which one has a constant

00:01:28.458 --> 00:01:30.734
of proportionality equal to 1/2.

00:01:30.734 --> 00:01:33.189
So this one right over here,
choice A clearly has a constant

00:01:33.189 --> 00:01:37.085
of proportionality of 1/8,
so we can just rule that out.

00:01:37.085 --> 00:01:39.420
Equation B right over here
clearly has a constant

00:01:39.420 --> 00:01:42.036
of proportionality of four, not 1/2,

00:01:42.036 --> 00:01:43.830
so we can rule that one out.

00:01:43.830 --> 00:01:47.770
Let's see, the constant of
proportionality for equation C,

00:01:47.770 --> 00:01:51.617
if we wanna solve for y, we
could divide both sides by six.

00:01:51.617 --> 00:01:55.784
And so we're gonna get y
is equal to 3/6 times x.

00:01:56.748 --> 00:01:59.800
Well, 3/6 is the same
thing as 1/2 times x,

00:01:59.800 --> 00:02:00.857
and so there you have it.

00:02:00.857 --> 00:02:03.574
We have a constant of
proportionality of 1/2.

00:02:03.574 --> 00:02:05.155
That's the choice I like.

00:02:05.155 --> 00:02:07.329
And we can verify that
this one doesn't work.

00:02:07.329 --> 00:02:12.070
If you wanna solve for y, you
divide both sides by three,

00:02:12.070 --> 00:02:16.896
and you get y is equal to
nine divided by three is 3x,

00:02:16.896 --> 00:02:19.462
so here our constant of
proportionality is three,

00:02:19.462 --> 00:02:21.893
so we can feel good about choice C.


Identifying constant of proportionality graphically

WEBVTT
Kind: captions
Language: en

00:00:00.775 --> 00:00:03.414
- We're asked what is the
constant of proportionality

00:00:03.414 --> 00:00:05.957
between Y and X in the graph?

00:00:05.957 --> 00:00:07.915
And just as a reminder,
when we're talking about

00:00:07.915 --> 00:00:09.794
constant of proportionality,

00:00:09.794 --> 00:00:11.099
it sounds like a very fancy thing,

00:00:11.099 --> 00:00:12.784
but it's not too bad.

00:00:12.784 --> 00:00:16.631
If we're thinking about
any X-Y pair on this,

00:00:16.631 --> 00:00:20.250
on this line, let's say, right over here.

00:00:20.250 --> 00:00:22.167
We have some X comma Y.

00:00:23.235 --> 00:00:27.413
If Y is proportionate,
or is proportional to X,

00:00:27.413 --> 00:00:31.853
then that means we can say that
Y is equal to some constant.

00:00:31.853 --> 00:00:34.853
Y is equal to some constant times X.

00:00:35.970 --> 00:00:40.930
And that constant, that is our
constant of proportionality,

00:00:40.930 --> 00:00:42.701
right over there.

00:00:42.701 --> 00:00:44.285
Sometimes you'll see this expressed,

00:00:44.285 --> 00:00:46.240
if you divide both sides by X,

00:00:46.240 --> 00:00:49.980
sometimes you'll see this as Y over X,

00:00:49.980 --> 00:00:53.647
is equal to the constant
of proportionality.

00:00:54.949 --> 00:00:58.912
It shows for any X-Y pair, if
you take your Y divided by X,

00:00:58.912 --> 00:01:00.393
what do you get?

00:01:00.393 --> 00:01:02.529
That's the same, same thing.

00:01:02.529 --> 00:01:03.990
So with that out of the way,

00:01:03.990 --> 00:01:05.890
see if you can answer their question.

00:01:05.890 --> 00:01:07.531
What is the constant of proportionality

00:01:07.531 --> 00:01:09.528
between Y and X in the graph?

00:01:09.528 --> 00:01:12.341
Well they very clearly give
us a point right over here,

00:01:12.341 --> 00:01:16.258
this point is the point
three, three comma two.

00:01:19.806 --> 00:01:21.520
And so we could set it up a few ways.

00:01:21.520 --> 00:01:23.736
We could say, look when Y is equal to two,

00:01:23.736 --> 00:01:28.461
X is equal to three, and so
two would need to be equal to

00:01:28.461 --> 00:01:32.926
some constant of
proportionality times three.

00:01:32.926 --> 00:01:34.272
And if you wanted to solve for this,

00:01:34.272 --> 00:01:37.247
you just divide both sides by the three.

00:01:37.247 --> 00:01:39.598
So divide both sides by three,

00:01:39.598 --> 00:01:41.673
and you would get your
constant of proportionality

00:01:41.673 --> 00:01:43.147
is two-thirds.

00:01:43.147 --> 00:01:45.040
Another way to do it, right over here.

00:01:45.040 --> 00:01:46.890
Well here, we've kind
of already solved for

00:01:46.890 --> 00:01:48.360
our constant of proportionality.

00:01:48.360 --> 00:01:51.892
When Y, or we could say, when X is three,

00:01:51.892 --> 00:01:54.809
when X is three, Y is equal to two.

00:01:56.244 --> 00:01:57.949
In either case, our
constant of proportionality

00:01:57.949 --> 00:01:59.714
is two-thirds.

00:01:59.714 --> 00:02:01.797
Let's do another example.

00:02:04.546 --> 00:02:08.850
So here we have which line has
a constant of proportionality

00:02:08.850 --> 00:02:11.683
between Y and X of five over four.

00:02:12.561 --> 00:02:16.281
So pause the video and see
if you can figure that out.

00:02:16.281 --> 00:02:18.586
So the key realization
is we should test points

00:02:18.586 --> 00:02:22.040
on these lines, we should test X-Y pairs,

00:02:22.040 --> 00:02:24.534
and say, look if we
take our Y divided by X,

00:02:24.534 --> 00:02:26.594
do we get five-fourths?

00:02:26.594 --> 00:02:29.384
Because that would be our
constant of proportionality.

00:02:29.384 --> 00:02:33.551
So let's first try, let's
try line A right over here.

00:02:34.668 --> 00:02:37.059
So line A, let me find
a point that sits on it.

00:02:37.059 --> 00:02:39.466
so that looks like a
point that sits on it.

00:02:39.466 --> 00:02:43.633
And so if I take, this is
the point two comma five.

00:02:45.237 --> 00:02:47.904
And so if I took Y divided by X,

00:02:50.245 --> 00:02:54.412
I would get a constant of
proportionality as five halves.

00:02:56.464 --> 00:02:59.055
So A is not going to be our answer.

00:02:59.055 --> 00:03:00.448
We wanna get to a constant
of proportionality

00:03:00.448 --> 00:03:01.781
of five-fourths.

00:03:02.881 --> 00:03:07.540
Alright, let's try B. Okay
B, let me find a point on B.

00:03:07.540 --> 00:03:09.297
Looks like this is a point on B.

00:03:09.297 --> 00:03:13.464
That is the point four
comma five, four comma five.

00:03:14.440 --> 00:03:18.154
And so in this situation, K
would be our Y, which is five,

00:03:18.154 --> 00:03:20.280
divided by our X, which is four.

00:03:20.280 --> 00:03:23.098
So it looks like B is our choice.

00:03:23.098 --> 00:03:24.538
For kicks, you could also look at the

00:03:24.538 --> 00:03:26.847
constant of proportionality
right over here.

00:03:26.847 --> 00:03:29.076
Now there is one interesting example

00:03:29.076 --> 00:03:31.922
that I just wanna touch on
before we finish these examples.

00:03:31.922 --> 00:03:35.672
What about a situation
where Y is equal to X?

00:03:37.067 --> 00:03:39.958
What is the constant of
proportionality then?

00:03:39.958 --> 00:03:43.051
And what would it look like as a line?

00:03:43.051 --> 00:03:45.157
Pause this video and think about it.

00:03:45.157 --> 00:03:47.455
Well, there's really nothing new here.

00:03:47.455 --> 00:03:49.131
It's just, you might not really see

00:03:49.131 --> 00:03:50.647
the constant of proportionality

00:03:50.647 --> 00:03:52.318
when you see it expressed this way.

00:03:52.318 --> 00:03:54.502
But Y is equal to X, is the same thing as

00:03:54.502 --> 00:03:57.064
Y is equal to one times X.

00:03:57.064 --> 00:03:58.701
And so then it might jump out at you that

00:03:58.701 --> 00:04:00.933
the constant of proportionality is one

00:04:00.933 --> 00:04:02.894
in this scenario right over here.

00:04:02.894 --> 00:04:06.452
Or if you took Y divided by X, Y over...

00:04:06.452 --> 00:04:08.647
Let me do it in that black color.

00:04:08.647 --> 00:04:13.215
Or if you took Y over X,
you divided both sides by X,

00:04:13.215 --> 00:04:15.657
you would be left with the
constant of proportionality,

00:04:15.657 --> 00:04:17.200
which would be equal to one.

00:04:17.200 --> 00:04:18.520
And if you wanted to graph it,

00:04:18.520 --> 00:04:20.674
well it would just look like this.

00:04:20.674 --> 00:04:23.591
Y would be equal to X, for all X's.

00:04:25.973 --> 00:04:29.516
So that's what when your
constant of proportionality

00:04:29.516 --> 00:04:31.420
is one, you would...

00:04:31.420 --> 00:04:34.489
Those would represent
points on this orange line

00:04:34.489 --> 00:04:36.105
that I just constructed.


Identifying values in scale drawings

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:01.913
- [Instructor] We're told that Figure A

00:00:01.913 --> 00:00:03.921
is a scale image of Figure B,

00:00:03.921 --> 00:00:06.559
so that's Figure A, this is Figure B here.

00:00:06.559 --> 00:00:08.568
The scale that maps Figure A

00:00:08.568 --> 00:00:13.142
to Figure B is one to two and 1/2.

00:00:13.142 --> 00:00:15.059
What is the value of X?

00:00:15.917 --> 00:00:17.287
All right, pause this video

00:00:17.287 --> 00:00:20.169
and see if you can figure it out.

00:00:20.169 --> 00:00:21.678
All right, so X is the length

00:00:21.678 --> 00:00:25.471
of this side right over here on Figure A,

00:00:25.471 --> 00:00:29.638
and the corresponding side
on Figure B has length five.

00:00:30.591 --> 00:00:34.411
And so, one way to think about
it is the ratio between X

00:00:34.411 --> 00:00:38.115
and five should be one to two and a half

00:00:38.115 --> 00:00:38.959
because that's the scale

00:00:38.959 --> 00:00:40.779
that goes from Figure A to Figure B.

00:00:40.779 --> 00:00:42.540
Figure A to Figure B.

00:00:42.540 --> 00:00:44.758
So, the ratio, let me write this out.

00:00:44.758 --> 00:00:47.175
The ratio between X and five,

00:00:49.062 --> 00:00:51.229
so the ratio of X to five,

00:00:53.451 --> 00:00:57.711
this should be an equivalent ratio as one,

00:00:57.711 --> 00:00:59.544
one to two and a half.

00:01:01.145 --> 00:01:04.450
One to two and a half, and
that one is hard to read,

00:01:04.450 --> 00:01:06.880
let me make it a deeper
blue, there you go.

00:01:06.880 --> 00:01:08.760
All right, so let's just
think about how to do this.

00:01:08.760 --> 00:01:12.927
To go from from two and a
half to five, to go that way,

00:01:13.926 --> 00:01:15.760
you would multiply be two,

00:01:15.760 --> 00:01:20.637
so to go from one to X, you
would also multiple by two.

00:01:20.637 --> 00:01:23.086
So, the value of X is equal to two.

00:01:23.086 --> 00:01:26.996
If these are scaled up, you
multiple this by two and a half,

00:01:26.996 --> 00:01:28.329
you get to five,

00:01:29.292 --> 00:01:32.275
so the scale factor is
one to two and a half.

00:01:32.275 --> 00:01:34.358
Let's do another example.

00:01:35.732 --> 00:01:39.737
We're told Figure A is a
scale image of Figure B,

00:01:39.737 --> 00:01:41.699
and we see them both right over here,

00:01:41.699 --> 00:01:43.438
and once again, we gotta figure out

00:01:43.438 --> 00:01:44.816
what X is going to be.

00:01:44.816 --> 00:01:46.860
And they don't give us the scale factor,

00:01:46.860 --> 00:01:49.066
but we can figure out the scale factor.

00:01:49.066 --> 00:01:50.154
How do we do that?

00:01:50.154 --> 00:01:52.940
Well, we can see, when you go from,

00:01:52.940 --> 00:01:56.118
this side right over here
corresponds to this side.

00:01:56.118 --> 00:01:59.613
It's the shorter side
that forms a right angle

00:01:59.613 --> 00:02:00.655
with the base,

00:02:00.655 --> 00:02:02.660
this is the longer side
that forms a right angle

00:02:02.660 --> 00:02:03.702
with the base,

00:02:03.702 --> 00:02:06.033
and so, you could set up some ratios.

00:02:06.033 --> 00:02:09.036
You can say, look, the
ratio of three to four,

00:02:09.036 --> 00:02:11.536
you can say the ratio of three

00:02:12.574 --> 00:02:13.407
to four,

00:02:15.250 --> 00:02:16.687
we need that blue color,

00:02:16.687 --> 00:02:18.774
the ratio of three to four is going

00:02:18.774 --> 00:02:22.958
to be the same thing as
the ratio of X to 7.2.

00:02:22.958 --> 00:02:24.473
The ratio of X

00:02:24.473 --> 00:02:25.306
to 7.2,

00:02:27.384 --> 00:02:28.960
X

00:02:28.960 --> 00:02:30.443
to

00:02:30.443 --> 00:02:31.276
7.2,

00:02:32.716 --> 00:02:35.490
and so how do we figure
out what X is going to be?

00:02:35.490 --> 00:02:38.358
Well, how do you go from four to 7.2?

00:02:38.358 --> 00:02:40.224
What do you have to multiply by?

00:02:40.224 --> 00:02:41.385
You might wanna get out a calculator

00:02:41.385 --> 00:02:43.248
or you might be able to
do this in your head.

00:02:43.248 --> 00:02:47.738
Four 72 is two times 36,
which is two times 18,

00:02:47.738 --> 00:02:50.536
so four times 18 would be 72

00:02:50.536 --> 00:02:53.119
or four times 1.8 would be 7.2,

00:02:54.237 --> 00:02:56.477
and if you don't feel good
about that mental arithmetic,

00:02:56.477 --> 00:02:58.808
you could just do the division.

00:02:58.808 --> 00:03:00.364
Four goes into seven one time,

00:03:00.364 --> 00:03:01.463
one times four is four,

00:03:01.463 --> 00:03:03.402
subtract, you get a three,

00:03:03.402 --> 00:03:04.711
you're gonna have your
decimal right over there,

00:03:04.711 --> 00:03:06.345
bring down the two, 32,

00:03:06.345 --> 00:03:08.142
four goes into 32 eight times,

00:03:08.142 --> 00:03:10.008
eight times four is 32,

00:03:10.008 --> 00:03:11.375
and we're done.

00:03:11.375 --> 00:03:13.395
So, to go from four to 7.2,

00:03:13.395 --> 00:03:15.670
you have to multiply by 1.8

00:03:15.670 --> 00:03:18.294
and so, to go from three to X,

00:03:18.294 --> 00:03:21.611
you also have to multiply by 1.8.

00:03:21.611 --> 00:03:23.965
And so, X is three times 1.8,

00:03:23.965 --> 00:03:25.175
what is that going to be?

00:03:25.175 --> 00:03:27.819
Well, three times 18 is, what?

00:03:27.819 --> 00:03:31.152
30, it's 54, so this is going to be 5.4.

00:03:33.313 --> 00:03:35.121
Let me verify that or show you that.

00:03:35.121 --> 00:03:37.640
18 or 1.8 times three,

00:03:37.640 --> 00:03:40.179
three times eight is 24,

00:03:40.179 --> 00:03:44.510
three times one is
three, plus two is five,

00:03:44.510 --> 00:03:48.677
one number behind the decimal
point, 5.4 and we're done.


Scale factors and area

WEBVTT
Kind: captions
Language: en

00:00:00.439 --> 00:00:02.726
- [Instructor] We're told that Polygon Q

00:00:02.726 --> 00:00:05.640
is a scaled copy of Polygon P

00:00:05.640 --> 00:00:09.195
using a scale factor of one half.

00:00:09.195 --> 00:00:13.362
Polygon Q's area is what
fraction of Polygon P's area?

00:00:15.616 --> 00:00:19.913
Pause this video and see
if you can figure that out.

00:00:19.913 --> 00:00:22.983
Alright, my brain wants to
make this a little bit tangible

00:00:22.983 --> 00:00:24.571
and once we get some
practice, you might be able

00:00:24.571 --> 00:00:26.520
to do it without drawing pictures

00:00:26.520 --> 00:00:29.641
but they're saying some
arbitrary Polygon Q and P

00:00:29.641 --> 00:00:31.017
so let's just make a simple one.

00:00:31.017 --> 00:00:32.824
Since we're talking about
area, I like to deal

00:00:32.824 --> 00:00:35.279
with rectangles since it's easy to think

00:00:35.279 --> 00:00:37.195
about areas of rectangles.

00:00:37.195 --> 00:00:40.279
And so, let's see, Polygon Q
is a scaled copy of Polygon P.

00:00:40.279 --> 00:00:42.555
So let's start with Polygon P.

00:00:42.555 --> 00:00:44.695
And I will do this in red.

00:00:44.695 --> 00:00:47.790
So Polygon P, let's just say,

00:00:47.790 --> 00:00:50.677
I'm just gonna create
an arbitrary polygon.

00:00:50.677 --> 00:00:52.242
So let's say that this is,

00:00:52.242 --> 00:00:53.574
and I'm gonna scale it by one half

00:00:53.574 --> 00:00:56.549
so I'm gonna make its sides
have nice even numbers.

00:00:56.549 --> 00:00:59.282
So let's say this side
right over here is four

00:00:59.282 --> 00:01:02.962
and this side right over
here is equal to eight,

00:01:02.962 --> 00:01:06.161
this is Polygon P right over
here, it's a quadrilateral,

00:01:06.161 --> 00:01:07.453
it's in fact a rectangle

00:01:07.453 --> 00:01:11.954
and its area is just going
to be four times eight

00:01:11.954 --> 00:01:13.743
which is 32.

00:01:13.743 --> 00:01:16.962
Now let's create Polygon Q, and remember,

00:01:16.962 --> 00:01:19.725
Polygon Q is a scaled copy of P

00:01:19.725 --> 00:01:21.969
using a scale factor of one half.

00:01:21.969 --> 00:01:24.612
So we're gonna scale it by one half.

00:01:24.612 --> 00:01:29.272
So instead of this side being
four, it's going to be two

00:01:29.272 --> 00:01:32.191
and instead of this side
over here being eight,

00:01:32.191 --> 00:01:34.496
the corresponding side
in the scaled version

00:01:34.496 --> 00:01:36.689
is going to be four.

00:01:36.689 --> 00:01:38.997
So there you go, we've
scaled it by one half,

00:01:38.997 --> 00:01:41.413
and now what is our area going to be?

00:01:41.413 --> 00:01:44.268
Well our area, and this Polygon Q,

00:01:44.268 --> 00:01:47.917
and so our area is going
to be two times four

00:01:47.917 --> 00:01:50.181
which is equal to eight.

00:01:50.181 --> 00:01:53.931
So notice that Polygon
Q's area is one fourth

00:01:54.967 --> 00:01:58.604
of Polygon P's area and that makes sense

00:01:58.604 --> 00:02:01.609
because when you scale the dimensions

00:02:01.609 --> 00:02:05.836
of the Polygon by one half,
the area is going to change

00:02:05.836 --> 00:02:07.427
by the square of that.

00:02:07.427 --> 00:02:10.042
One half squared is one fourth

00:02:10.042 --> 00:02:13.110
and so the area has been changed
by a factor of one fourth

00:02:13.110 --> 00:02:14.686
or another way to answer this question,

00:02:14.686 --> 00:02:18.973
Polygon Q's area is what
fraction of Polygon P's area?

00:02:18.973 --> 00:02:23.140
Well it's going to be one
fourth of Polygon P's area.

00:02:24.273 --> 00:02:27.852
And the big takeaway here
is if you scale something,

00:02:27.852 --> 00:02:32.107
if you scale the sides of
a figure by one half each,

00:02:32.107 --> 00:02:36.393
then the area is going
to be the square of that

00:02:36.393 --> 00:02:39.810
and so one half squared is one over four.

00:02:41.067 --> 00:02:42.608
If it was scaled by one third,

00:02:42.608 --> 00:02:45.932
then the area would be scaled, or the area

00:02:45.932 --> 00:02:47.264
would be one ninth.

00:02:47.264 --> 00:02:49.702
If it was scaled by a factor of two,

00:02:49.702 --> 00:02:54.123
then our area would have
grown by a factor of four.

00:02:54.123 --> 00:02:57.039
Let's do another example.

00:02:57.039 --> 00:03:00.763
Here we're told, Rectangle N has an area

00:03:00.763 --> 00:03:02.877
of five square units.

00:03:02.877 --> 00:03:04.633
Let me do this in a different color.

00:03:04.633 --> 00:03:08.626
So Rectangle N has an
area of five square units.

00:03:08.626 --> 00:03:11.625
James drew a scaled version of Rectangle N

00:03:11.625 --> 00:03:13.442
and labeled it Rectangle P.

00:03:13.442 --> 00:03:14.934
So they have that right over here.

00:03:14.934 --> 00:03:17.882
This is a scaled version of Rectangle N.

00:03:17.882 --> 00:03:20.004
What scale factor did James use

00:03:20.004 --> 00:03:23.863
to go from Rectangle N to Rectangle P?

00:03:23.863 --> 00:03:24.779
So let's think about it.

00:03:24.779 --> 00:03:28.581
They gave us Rectangle P right over here,

00:03:28.581 --> 00:03:30.360
and let's think about its dimensions.

00:03:30.360 --> 00:03:34.250
This height is one,
two, three, four, five,

00:03:34.250 --> 00:03:37.318
it's five high, and it
is, one, two, three,

00:03:37.318 --> 00:03:41.503
four, five, six, seven, eight, nine wide,

00:03:41.503 --> 00:03:44.086
and so its area is equal to 45.

00:03:47.639 --> 00:03:51.806
Now, Rectangle N had an
area of five square units.

00:03:54.404 --> 00:03:57.167
So our area, let me write this down,

00:03:57.167 --> 00:03:58.834
so N area to P area,

00:04:03.820 --> 00:04:07.987
N area to P area, we are
multiplying by a factor of nine.

00:04:09.094 --> 00:04:10.975
If we're going from an
area of five square units

00:04:10.975 --> 00:04:14.808
to 45 square units, you
notice N area is five,

00:04:16.773 --> 00:04:20.897
N's area is five square units, P's area,

00:04:20.897 --> 00:04:23.263
we just figured out is 45 square units,

00:04:23.263 --> 00:04:27.717
and so we have it growing
by a factor of nine.

00:04:27.717 --> 00:04:30.277
Now what would be the
scale factor if our area

00:04:30.277 --> 00:04:32.138
grew by a factor of nine?

00:04:32.138 --> 00:04:36.305
Well we just talked about
the idea that area will grow,

00:04:37.217 --> 00:04:39.481
the factor with which area grows

00:04:39.481 --> 00:04:41.838
is the square of the scale factor.

00:04:41.838 --> 00:04:45.272
So one way to think
about it is scale factor,

00:04:45.272 --> 00:04:46.939
scale factor squared

00:04:49.303 --> 00:04:51.323
is going to be equal to nine,

00:04:51.323 --> 00:04:52.286
or another way to think about it,

00:04:52.286 --> 00:04:56.119
our scale factor is going
to be equal to three

00:04:58.962 --> 00:05:01.134
to go from N to P.

00:05:01.134 --> 00:05:03.857
Now let's verify that we
answered their question

00:05:03.857 --> 00:05:05.728
but I just want us to feel good about it.

00:05:05.728 --> 00:05:10.114
Let's draw a rectangle
that is scaled down from P

00:05:10.114 --> 00:05:12.005
by a factor of three.

00:05:12.005 --> 00:05:13.753
So a rectangle if we were to scale it up

00:05:13.753 --> 00:05:16.738
by a factor of three, we get rectangle P.

00:05:16.738 --> 00:05:20.905
So its bottom would have
length three instead of nine.

00:05:21.912 --> 00:05:23.749
So it'd be like this.

00:05:23.749 --> 00:05:26.260
So that would be three, and its height

00:05:26.260 --> 00:05:29.194
instead of being five,
it would be five thirds.

00:05:29.194 --> 00:05:31.115
Five thirds is one and two thirds,

00:05:31.115 --> 00:05:33.153
so it'd go about that high,

00:05:33.153 --> 00:05:34.639
it would look something like that.

00:05:34.639 --> 00:05:36.639
It would be five thirds.

00:05:37.504 --> 00:05:39.337
And so our Rectangle N

00:05:41.153 --> 00:05:43.692
would look like this,
and what is its area?

00:05:43.692 --> 00:05:46.841
Well five thirds times three
is indeed five square units.

00:05:46.841 --> 00:05:50.792
So notice, when we have the area growing

00:05:50.792 --> 00:05:54.545
by a factor of nine, the
scale factor of the size

00:05:54.545 --> 00:05:58.289
to go from five thirds to
five, you multiply by three.

00:05:58.289 --> 00:06:02.289
To go from three to nine,
you multiply by three.


Identifying scale factors

WEBVTT
Kind: captions
Language: en

00:00:00.592 --> 00:00:01.770
- [Instructor] So right over here,

00:00:01.770 --> 00:00:05.684
Figure B is a scaled copy of Figure A

00:00:05.684 --> 00:00:07.440
and what we wanna do is figure out

00:00:07.440 --> 00:00:11.400
what is the scale factor to
go from Figure A to Figure B?

00:00:11.400 --> 00:00:15.238
Pause the video and see if
you can figure that out.

00:00:15.238 --> 00:00:17.926
Well, all we have to do is
look at corresponding sides

00:00:17.926 --> 00:00:20.491
and think about how much
they have been scaled by.

00:00:20.491 --> 00:00:24.202
So for example, this side right over here

00:00:24.202 --> 00:00:28.846
would correspond to this side
right over here on Figure B

00:00:28.846 --> 00:00:30.929
and over here, it had length two

00:00:30.929 --> 00:00:32.476
and over here, it has length

00:00:32.476 --> 00:00:35.857
one, two, three, four, five, six.

00:00:35.857 --> 00:00:39.203
So it looks like that
side has been scaled up

00:00:39.203 --> 00:00:40.870
by a factor of three

00:00:41.896 --> 00:00:45.232
and so if Figure B truly is a scaled copy

00:00:45.232 --> 00:00:48.660
then every side should be
scaled up by a factor of three

00:00:48.660 --> 00:00:49.677
and we could verify that.

00:00:49.677 --> 00:00:50.936
We don't have to do it with every side.

00:00:50.936 --> 00:00:53.380
We're being told that
these are scaled copies

00:00:53.380 --> 00:00:55.092
but we can see that this is the case.

00:00:55.092 --> 00:00:56.885
For example, this side right over here

00:00:56.885 --> 00:01:00.392
corresponds to this base right over here.

00:01:00.392 --> 00:01:02.312
This has length three.

00:01:02.312 --> 00:01:05.656
So if we're scaling up
by a factor of three,

00:01:05.656 --> 00:01:07.406
we should multiply that by three

00:01:07.406 --> 00:01:08.956
and this should be of length nine.

00:01:08.956 --> 00:01:10.350
Let's see if that's the case.

00:01:10.350 --> 00:01:14.517
One, two, three, four, five,
six, seven, eight and nine.

00:01:17.212 --> 00:01:19.748
And so you can see we can feel pretty good

00:01:19.748 --> 00:01:22.084
that Figure B is a scaled copy of Figure A

00:01:22.084 --> 00:01:25.460
and that scaling factor is three.

00:01:25.460 --> 00:01:28.279
Let's do another example.

00:01:28.279 --> 00:01:32.078
So here, we are told
Ismael made a scaled copy

00:01:32.078 --> 00:01:34.253
of the following quadrilateral.

00:01:34.253 --> 00:01:38.086
He used a scale factor
less than one, alright,

00:01:39.375 --> 00:01:41.256
and then they say what could be the length

00:01:41.256 --> 00:01:44.028
of the side that corresponds to AD?

00:01:44.028 --> 00:01:46.109
So AD is right over here.

00:01:46.109 --> 00:01:50.276
AD has length 16 units in
our original quadrilateral.

00:01:51.244 --> 00:01:52.228
What could be the length

00:01:52.228 --> 00:01:53.902
of the side that corresponds with AD

00:01:53.902 --> 00:01:57.121
on the scaled copy of the quadrilateral?

00:01:57.121 --> 00:01:59.124
So it's a scale factor less than one.

00:01:59.124 --> 00:02:03.012
So we're gonna get something
that is less than 16

00:02:03.012 --> 00:02:04.199
for that side

00:02:04.199 --> 00:02:08.316
and the rest of it will all
be scaled by the same factor.

00:02:08.316 --> 00:02:11.988
So the resulting quadrilateral
might look something,

00:02:11.988 --> 00:02:13.335
might look something like this.

00:02:13.335 --> 00:02:15.487
This is just my hand-drawn version.

00:02:15.487 --> 00:02:16.999
So the key realization is

00:02:16.999 --> 00:02:19.369
is if our scale factor is less than one,

00:02:19.369 --> 00:02:20.711
this thing right over here

00:02:20.711 --> 00:02:24.112
is going to be less than 16 units.

00:02:24.112 --> 00:02:25.379
So let's look at the choices

00:02:25.379 --> 00:02:28.442
and it says choose three answers.

00:02:28.442 --> 00:02:30.936
So pause the video and
which of these would match

00:02:30.936 --> 00:02:34.402
if we're scaling by a
factor of less than one?

00:02:34.402 --> 00:02:35.426
Well, we just have to see which of these

00:02:35.426 --> 00:02:37.109
are less than 16 units.

00:02:37.109 --> 00:02:38.700
This is less than 16.

00:02:38.700 --> 00:02:40.143
This is less than 16.

00:02:40.143 --> 00:02:41.456
This is less than 16

00:02:41.456 --> 00:02:44.142
and those are the only
three that are less than 16.

00:02:44.142 --> 00:02:47.710
32 units, this would be
a scale factor of two.

00:02:47.710 --> 00:02:50.161
64 units, this would be
a scale factor of four.

00:02:50.161 --> 00:02:54.244
Clearly a scale factor
that is not less than one.


Identifying scaled copies

WEBVTT
Kind: captions
Language: en

00:00:00.494 --> 00:00:01.870
- [Instructor] What we're
going to do in this video

00:00:01.870 --> 00:00:03.630
is look at pairs of figures,

00:00:03.630 --> 00:00:05.950
and see if they are scaled
copies of each other.

00:00:05.950 --> 00:00:08.054
For example, in this diagram,

00:00:08.054 --> 00:00:11.471
is Figure B a scaled version of Figure A?

00:00:12.693 --> 00:00:15.965
Pause the video and see if
you can figure that out.

00:00:15.965 --> 00:00:18.261
There's multiple ways that
you could approach this.

00:00:18.261 --> 00:00:19.125
One way is to say,

00:00:19.125 --> 00:00:21.774
"Well, let's see what the
scaling factor would be."

00:00:21.774 --> 00:00:23.926
We could look at the side lengths.

00:00:23.926 --> 00:00:27.479
This side right over here
has length three on Figure A.

00:00:27.479 --> 00:00:29.455
This side length right over here

00:00:29.455 --> 00:00:33.407
has length one, two, three, four, five.

00:00:33.407 --> 00:00:37.358
This side length has length five as well.

00:00:37.358 --> 00:00:39.286
This has length five.

00:00:39.286 --> 00:00:40.966
This length, we could figure it out

00:00:40.966 --> 00:00:42.494
with the Pythagorean Theorem,

00:00:42.494 --> 00:00:44.734
but I won't even look
at that one just yet.

00:00:44.734 --> 00:00:46.552
But let's look at corresponding sides.

00:00:46.552 --> 00:00:48.353
To go from this side.

00:00:48.353 --> 00:00:51.466
If we scale up, the
corresponding side to that

00:00:51.466 --> 00:00:55.033
would be this side right over here.

00:00:55.033 --> 00:00:57.161
What is its length?

00:00:57.161 --> 00:01:01.328
Well, its length, when you
scale it up, looks like five.

00:01:02.177 --> 00:01:04.427
So to go from three to five

00:01:05.986 --> 00:01:10.273
you would have to multiply by 5/3.

00:01:10.273 --> 00:01:11.984
5/3.

00:01:11.984 --> 00:01:15.444
But let's look at this side now.

00:01:15.444 --> 00:01:17.136
It's five in Figure A.

00:01:17.136 --> 00:01:19.488
What length is it in Figure B?

00:01:19.488 --> 00:01:21.696
Well, it is one, two, three, four, five.

00:01:21.696 --> 00:01:24.856
It's still five, so to
go from five to five,

00:01:24.856 --> 00:01:27.848
you have to multiply it by one,

00:01:27.848 --> 00:01:29.656
and so you have a different scaling factor

00:01:29.656 --> 00:01:30.760
for corresponding,

00:01:30.760 --> 00:01:33.224
or what could have been
corresponding sides.

00:01:33.224 --> 00:01:36.072
This side right over here,
you're scaling up by 5/3,

00:01:36.072 --> 00:01:38.344
while this bottom side,
this base right here,

00:01:38.344 --> 00:01:39.728
you're not scaling at all.

00:01:39.728 --> 00:01:43.608
So these actually are not
scaled versions of each other.

00:01:43.608 --> 00:01:45.691
Let's do another example.

00:01:46.568 --> 00:01:50.735
In this example, is Figure B
a scaled version of Figure A?

00:01:52.232 --> 00:01:54.912
Pause the video and see
if you can figure it out.

00:01:54.912 --> 00:01:56.384
All right, well we're
gonna do the same exercise,

00:01:56.384 --> 00:01:57.408
and here they've given us

00:01:57.408 --> 00:02:00.167
the measures of the different sides.

00:02:00.167 --> 00:02:02.144
This side has length two.

00:02:02.144 --> 00:02:03.600
This side has length

00:02:03.600 --> 00:02:04.664
the corresponding side,

00:02:04.664 --> 00:02:07.392
or what could be the
corresponding side has length six.

00:02:07.392 --> 00:02:12.016
To go from two to six, you
have to multiply by three.

00:02:12.016 --> 00:02:15.944
If we look at these two
potentially corresponding sides,

00:02:15.944 --> 00:02:18.928
that side and that side, once again,

00:02:18.928 --> 00:02:22.744
to go from four to twelve,
you would multiply by three,

00:02:22.744 --> 00:02:25.936
so that is looking good as well.

00:02:25.936 --> 00:02:30.680
Now to go from this side down
here, this has length six.

00:02:30.680 --> 00:02:32.632
The potentially corresponding
side right over here

00:02:32.632 --> 00:02:34.432
has the length 14.

00:02:34.432 --> 00:02:36.704
Well here we're not multiplying by three.

00:02:36.704 --> 00:02:37.768
If these were scaled

00:02:37.768 --> 00:02:40.064
If Figure B was a scaled-up
version of Figure A,

00:02:40.064 --> 00:02:42.328
we would multiply by three,

00:02:42.328 --> 00:02:44.888
but six times three is not 14, it's 18,

00:02:44.888 --> 00:02:46.144
so these actually are not

00:02:46.144 --> 00:02:50.120
Figure B is not a scaled
version of Figure A.

00:02:50.120 --> 00:02:53.080
Let's do one more example.

00:02:53.080 --> 00:02:54.640
Once again, pause this video

00:02:54.640 --> 00:02:59.248
and see if Figure B is a
scaled version of Figure A.

00:02:59.248 --> 00:03:00.648
We're gonna do the same exercise.

00:03:00.648 --> 00:03:02.904
Let's look at potentially
corresponding sides.

00:03:02.904 --> 00:03:05.582
That side to that side,

00:03:05.582 --> 00:03:08.134
to go from four to 12,

00:03:08.134 --> 00:03:11.198
we would multiply by three,

00:03:11.198 --> 00:03:15.782
and then we could look at
this side and this side,

00:03:15.782 --> 00:03:20.190
to go from four to 12, once
again you multiply by three,

00:03:20.190 --> 00:03:22.158
so that's looking good so far.

00:03:22.158 --> 00:03:24.966
We could look at this side and this side,

00:03:24.966 --> 00:03:26.998
potentially corresponding sides.

00:03:26.998 --> 00:03:29.273
Once again, we're going from four to 12,

00:03:29.273 --> 00:03:31.481
multiplying by three.

00:03:31.481 --> 00:03:33.177
Looks good so far.

00:03:33.177 --> 00:03:37.969
And then we could look at
this side and this side.

00:03:37.969 --> 00:03:42.425
2.2 to 6.6, once again
multiplying by three.

00:03:42.425 --> 00:03:43.721
Looking really good.

00:03:43.721 --> 00:03:45.737
And then we only have
one last one to check.

00:03:45.737 --> 00:03:49.404
2.2 to 6.6, once again
multiplying by three,

00:03:52.089 --> 00:03:55.577
so all of the side lengths
have been scaled up by three,

00:03:55.577 --> 00:03:57.281
so we can feel pretty good that

00:03:57.281 --> 00:04:01.448
Figure B is indeed a scaled-up
representation of Figure A.


Corresponding points and sides of scaled shapes

WEBVTT
Kind: captions
Language: en

00:00:00.348 --> 00:00:02.148
- [Instructor] We are told figure two

00:00:02.148 --> 00:00:04.917
is a scaled copy of figure one.

00:00:04.917 --> 00:00:06.712
So this is figure two here.

00:00:06.712 --> 00:00:07.895
This is figure one.

00:00:07.895 --> 00:00:10.771
Looks like figure two not
only has it been scaled down,

00:00:10.771 --> 00:00:11.992
it's a smaller version,

00:00:11.992 --> 00:00:15.875
but it also looks like it
has been rotated 180 degrees

00:00:15.875 --> 00:00:18.909
or you can say it's been
flipped upside down.

00:00:18.909 --> 00:00:21.892
We're asked to identify
the side in figure two

00:00:21.892 --> 00:00:26.141
that corresponds to side EA in figure one.

00:00:26.141 --> 00:00:29.038
So this is EA in figure one.

00:00:29.038 --> 00:00:31.863
Pause this video and
see if you can identify

00:00:31.863 --> 00:00:35.236
the side of figure two that corresponds

00:00:35.236 --> 00:00:37.319
to side EA in figure one.

00:00:38.931 --> 00:00:40.887
Alright, now let's work
through this together.

00:00:40.887 --> 00:00:42.803
Some of you might immediately intuit

00:00:42.803 --> 00:00:45.985
or be able to spot which
side corresponds to EA

00:00:45.985 --> 00:00:48.597
but I'll do it slightly more
methodically in this video

00:00:48.597 --> 00:00:49.860
just so that we can see maybe all

00:00:49.860 --> 00:00:52.462
of the corresponding sides and points.

00:00:52.462 --> 00:00:54.071
So one way to think about it

00:00:54.071 --> 00:00:57.734
is well, it looks like the
longest side in figure one

00:00:57.734 --> 00:00:58.651
is side BC.

00:01:00.408 --> 00:01:03.027
So this is the longest side in figure one

00:01:03.027 --> 00:01:05.586
and then the longest
side in figure two is HL

00:01:05.586 --> 00:01:08.669
so it looks like BC corresponds to HL

00:01:11.714 --> 00:01:15.473
and then touching on that longest side,

00:01:15.473 --> 00:01:18.484
that longest side has two right angles.

00:01:18.484 --> 00:01:21.499
It forms two right angles
with the sides next to it.

00:01:21.499 --> 00:01:24.393
So you have these two right
angles right over here

00:01:24.393 --> 00:01:26.389
and then of those two right angles,

00:01:26.389 --> 00:01:30.301
you have the shorter side
connected to the longest side

00:01:30.301 --> 00:01:32.953
and so that is BA over here

00:01:32.953 --> 00:01:36.989
and so that would correspond
to LK right over here.

00:01:36.989 --> 00:01:40.164
That is the shorter side
connected to the long side

00:01:40.164 --> 00:01:42.041
that forms a right angle

00:01:42.041 --> 00:01:45.739
and then if we just keep
following that side of the shape,

00:01:45.739 --> 00:01:50.152
we then go to side AE which
would correspond now to

00:01:50.152 --> 00:01:53.985
and this is answering
our question to side KJ.

00:01:55.010 --> 00:01:59.010
So the side in figure two
that corresponds to EA

00:01:59.850 --> 00:02:02.662
in figure one, that is going to be KJ

00:02:02.662 --> 00:02:04.579
or we could say side JK

00:02:06.873 --> 00:02:08.156
and if someone wanted to say

00:02:08.156 --> 00:02:10.594
what points correspond to which point,

00:02:10.594 --> 00:02:14.177
well, you can see that
BA corresponds to LK

00:02:15.402 --> 00:02:18.010
and BC corresponds to LH

00:02:18.010 --> 00:02:19.561
so we would know for example

00:02:19.561 --> 00:02:23.005
that this point B right over here, point B

00:02:23.005 --> 00:02:26.391
would correspond to point L
in figure two right over here

00:02:26.391 --> 00:02:29.077
and we could use a similar
logic for the other points.

00:02:29.077 --> 00:02:31.172
So we've answered the
questions, the question,

00:02:31.172 --> 00:02:32.257
but the real thing is to see

00:02:32.257 --> 00:02:34.750
well, which sides or points

00:02:34.750 --> 00:02:37.601
have similar features on a relative basis?

00:02:37.601 --> 00:02:39.040
So I said what is the longest side?

00:02:39.040 --> 00:02:39.975
What is the longest side?

00:02:39.975 --> 00:02:41.616
Where do we see right angles?

00:02:41.616 --> 00:02:43.239
And sometimes your brain
might just immediately

00:02:43.239 --> 00:02:44.743
be able to spot out the shape.

00:02:44.743 --> 00:02:47.852
If you were to rotate this
one all the way around,

00:02:47.852 --> 00:02:49.567
it would jump out at
you a little bit faster

00:02:49.567 --> 00:02:52.817
that JK and AE are corresponding sides.


Fraction decimal and percent from visual model

WEBVTT
Kind: captions
Language: en

00:00:00.292 --> 00:00:02.781
- [Instructor] So let's
assume that this entire square

00:00:02.781 --> 00:00:04.904
represents a whole.

00:00:04.904 --> 00:00:08.682
And we can see that part
of it is shaded in in blue.

00:00:08.682 --> 00:00:10.499
What we're going to do in this video

00:00:10.499 --> 00:00:14.160
is try to represent the
part that is shaded in blue

00:00:14.160 --> 00:00:15.327
as a fraction,

00:00:17.308 --> 00:00:18.391
as a decimal,

00:00:24.621 --> 00:00:26.038
and as a percent.

00:00:26.966 --> 00:00:30.512
So pause the video and
see if you can do that.

00:00:30.512 --> 00:00:34.598
Well, let's first think
about it as a fraction.

00:00:34.598 --> 00:00:38.075
So, the whole is split
into one, two, three, four,

00:00:38.075 --> 00:00:42.463
five, six, seven, eight,
nine, 10 equal sections,

00:00:42.463 --> 00:00:45.564
and six of them are filled in.

00:00:45.564 --> 00:00:49.433
So the blue represents 6/10 of a whole,

00:00:49.433 --> 00:00:52.168
or it represents, you
could just say, 6/10.

00:00:52.168 --> 00:00:53.574
And you could also rewrite that.

00:00:53.574 --> 00:00:56.554
If you divide the numerator
and the denominator by two,

00:00:56.554 --> 00:00:59.971
that's the same thing as three over five.

00:01:01.204 --> 00:01:02.383
Fair enough.

00:01:02.383 --> 00:01:04.723
Now let's represent it as a decimal.

00:01:04.723 --> 00:01:05.930
What decimal would it be?

00:01:05.930 --> 00:01:08.797
Pause the video again and
see if you can do that.

00:01:08.797 --> 00:01:12.964
Well, 6/10, we could literally
just go to our place value.

00:01:13.831 --> 00:01:16.132
So that's the ones
place, we have a decimal.

00:01:16.132 --> 00:01:18.366
And then you have your tenths place.

00:01:18.366 --> 00:01:20.642
And so we have 6/10,

00:01:20.642 --> 00:01:22.677
so you could just put it right over there.

00:01:22.677 --> 00:01:26.844
We are putting a six in the
tenths place to represent 6/10.

00:01:28.763 --> 00:01:31.275
Now, what about a percentage?

00:01:31.275 --> 00:01:34.339
Well, percent means per 100,

00:01:34.339 --> 00:01:37.922
so one way to think
about it is six over 10

00:01:39.039 --> 00:01:42.545
is the same thing as what per 100?

00:01:42.545 --> 00:01:44.950
That is equal to, if we
multiply the numerator

00:01:44.950 --> 00:01:46.727
and the denominator by 10,

00:01:46.727 --> 00:01:49.727
that's the same thing as 60 per 100.

00:01:51.796 --> 00:01:53.085
Or another way of thinking about it,

00:01:53.085 --> 00:01:56.536
60 per, instead of 100 you could say cent.

00:01:56.536 --> 00:01:59.786
And so this would be equal to 60%.

00:02:01.455 --> 00:02:03.538
Let's do another example.

00:02:04.910 --> 00:02:09.293
So here, once again, our entire
square represents a whole.

00:02:09.293 --> 00:02:11.235
So see if you can represent this

00:02:11.235 --> 00:02:13.777
as the part that's shaded
in blue as a fraction.

00:02:13.777 --> 00:02:16.185
Pause the video and do that.

00:02:16.185 --> 00:02:19.188
Well, you can see that
this is a 10 by 10 grid,

00:02:19.188 --> 00:02:22.111
so there's 100 equal sections here.

00:02:22.111 --> 00:02:24.325
100 equal sections.

00:02:24.325 --> 00:02:26.996
Each of these squares represents 1/100.

00:02:26.996 --> 00:02:28.890
And how many of them are there?

00:02:28.890 --> 00:02:32.557
Well, let's see, this
row is 10, 20, 30, 40,

00:02:34.163 --> 00:02:36.655
and then one, two, three, four.

00:02:36.655 --> 00:02:40.155
So this is 44 over 100, 44/100.

00:02:41.966 --> 00:02:44.954
And we could actually
represent this in other ways.

00:02:44.954 --> 00:02:49.161
We could divide the numerator
and the denominator by four,

00:02:49.161 --> 00:02:52.859
in which case you would get 11 over 25.

00:02:52.859 --> 00:02:56.581
That's another way to
represent this same fraction.

00:02:56.581 --> 00:02:58.391
Now, what about as a decimal?

00:02:58.391 --> 00:03:01.433
Well, 44/100, you could say,

00:03:01.433 --> 00:03:03.441
well, you have your ones place,

00:03:03.441 --> 00:03:05.539
and then this is the same thing.

00:03:05.539 --> 00:03:08.737
You could literally
just say this is 44/100.

00:03:08.737 --> 00:03:10.913
This is another way of
representing 44/100.

00:03:10.913 --> 00:03:15.080
It's 4/10 and 4/100 is 44/100.

00:03:16.061 --> 00:03:17.796
And then if you were to do a percent,

00:03:17.796 --> 00:03:21.382
well, this is 44 per 100, or 44/100,

00:03:21.382 --> 00:03:25.549
but even here I like look
at it as 44 per 100 or 44%.

00:03:26.597 --> 00:03:29.264
So this is going to be 44%.

00:03:30.981 --> 00:03:32.231
And we're done.


Rate problems

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:03.435
- [Instructor] So we're told
that Lynnette can wash 95 cars

00:00:03.435 --> 00:00:05.176
in five days.

00:00:05.176 --> 00:00:08.785
How many cars can
Lynnette wash in 11 days?

00:00:08.785 --> 00:00:10.217
So like always, pause this video

00:00:10.217 --> 00:00:12.755
and see if you can figure this out.

00:00:12.755 --> 00:00:14.785
So the way that I would like to tackle it

00:00:14.785 --> 00:00:18.952
is given the information they
gave us, 95 cars in five days,

00:00:20.309 --> 00:00:23.760
can we figure out how many
cars she can wash per day?

00:00:23.760 --> 00:00:25.867
How many cars in one day?

00:00:25.867 --> 00:00:27.755
And then we could just multiply that by 11

00:00:27.755 --> 00:00:30.516
to figure out how many
she could wash in 11 days.

00:00:30.516 --> 00:00:33.099
So there's 95 cars in five days

00:00:37.686 --> 00:00:39.115
and so instead of five,

00:00:39.115 --> 00:00:42.032
if we were to say one day, one day,

00:00:43.359 --> 00:00:46.583
well, to go from five days to
one day, we divide by five.

00:00:46.583 --> 00:00:48.869
So the number of cars
she can do in one day,

00:00:48.869 --> 00:00:50.745
that would be that divided by five.

00:00:50.745 --> 00:00:53.844
In either case, we would divide by five

00:00:53.844 --> 00:00:55.995
and what's 95 divided by five?

00:00:55.995 --> 00:00:57.596
You might recognize that well, look,

00:00:57.596 --> 00:01:00.045
five times 20 is equal to 100.

00:01:00.045 --> 00:01:02.462
So five times 19 is 95.

00:01:02.462 --> 00:01:06.460
So you might recognize
that's 19 cars in one day

00:01:06.460 --> 00:01:08.868
or if that math I just did
seems a little bit too fast,

00:01:08.868 --> 00:01:12.421
you could just take five into 95.

00:01:12.421 --> 00:01:15.366
Five goes into nine one
time, one times five is five,

00:01:15.366 --> 00:01:19.888
subtract, nine minus five is
four, bring down that five,

00:01:19.888 --> 00:01:23.969
five goes into 45 nine
times, nine times five is 45

00:01:23.969 --> 00:01:25.216
and it goes in perfectly.

00:01:25.216 --> 00:01:26.296
There's no remainder.

00:01:26.296 --> 00:01:29.432
So all I did is I divided
both of these numbers by five

00:01:29.432 --> 00:01:32.712
to figure out how many
she can wash per day.

00:01:32.712 --> 00:01:37.286
So she can wash 19 cars in
one day, 19 cars per day

00:01:37.286 --> 00:01:38.496
and so if I wanna figure out

00:01:38.496 --> 00:01:42.496
how many she can wash in
11 days, so in 11 days,

00:01:44.578 --> 00:01:47.637
well, now, I am just multiplying by 11.

00:01:47.637 --> 00:01:51.804
So if I multiply 19 times
11, what am I going to get?

00:01:53.898 --> 00:01:55.856
Well, if you're good at
the mental mathematics,

00:01:55.856 --> 00:01:59.490
you might recognize 19
times 10 would be 190

00:01:59.490 --> 00:02:01.737
and then you would have one more 19

00:02:01.737 --> 00:02:03.820
so that would be 209 cars

00:02:05.041 --> 00:02:09.061
or you could just multiply 19 times 11.

00:02:09.061 --> 00:02:12.978
One times 19 is 19 and
then one times 19 is 19.

00:02:14.293 --> 00:02:16.793
Add 'em together, you get 209.

00:02:19.029 --> 00:02:22.029
So she can wash 209 cars in 11 days.

00:02:23.554 --> 00:02:25.637
Let's do another example.

00:02:26.718 --> 00:02:30.885
Here, we are told at the market,
eight batteries cost $10.

00:02:32.592 --> 00:02:34.558
How much do six batteries cost?

00:02:34.558 --> 00:02:37.419
Once again, pause this video
and try to figure it out.

00:02:37.419 --> 00:02:39.388
Well, I'm gonna do the same technique.

00:02:39.388 --> 00:02:41.399
If eight batteries cost $10,

00:02:41.399 --> 00:02:44.182
let's figure out how
much one battery costs.

00:02:44.182 --> 00:02:46.349
So one battery costs what?

00:02:50.472 --> 00:02:51.493
Figure that out.

00:02:51.493 --> 00:02:53.931
Well, it's going to be 1/8th.

00:02:53.931 --> 00:02:56.374
To go from eight to
one, I divide by eight.

00:02:56.374 --> 00:02:59.112
So what happens when I divide 10 by eight?

00:02:59.112 --> 00:03:02.749
That's gonna be my per battery cost.

00:03:02.749 --> 00:03:06.916
So 10 divided by eight
is the same thing as 5/4

00:03:09.404 --> 00:03:12.545
which is the same thing as one and 1/4

00:03:12.545 --> 00:03:14.875
which is the same thing if
we're talking in terms of money,

00:03:14.875 --> 00:03:17.176
it might be more useful to write it like

00:03:17.176 --> 00:03:19.404
one and 25 hundredths

00:03:19.404 --> 00:03:21.654
or one battery costs $1.25.

00:03:24.632 --> 00:03:26.352
And so when they're saying six batteries,

00:03:26.352 --> 00:03:28.598
how much do six batteries cost,

00:03:28.598 --> 00:03:32.181
well, if we're going
to six batteries cost,

00:03:34.971 --> 00:03:37.997
we're multiplying the number
of batteries times six now.

00:03:37.997 --> 00:03:40.726
So what's $1.25 times six?

00:03:40.726 --> 00:03:44.109
So it's gonna be $6 plus another $1.50

00:03:44.109 --> 00:03:46.723
'cause six times 25 is 150.

00:03:46.723 --> 00:03:49.333
So it's gonna be $7.50

00:03:49.333 --> 00:03:51.297
and just to make sure the
math I just did was right,

00:03:51.297 --> 00:03:53.880
let's multiply $1.25 times six.

00:03:55.548 --> 00:03:57.424
Six times five is 30.

00:03:57.424 --> 00:04:00.811
Six times two is 12 plus three is 15.

00:04:00.811 --> 00:04:03.180
Six times one is six plus one is seven

00:04:03.180 --> 00:04:06.036
and we have a total of two
digits behind the decimal

00:04:06.036 --> 00:04:08.860
so there you have it, you have $7.50.

00:04:08.860 --> 00:04:11.839
Now, there's other ways
that you could tackle this.

00:04:11.839 --> 00:04:14.004
You could say that hey, this is,

00:04:14.004 --> 00:04:16.530
six batteries is gonna cost 6/8ths

00:04:16.530 --> 00:04:18.123
as much as eight batteries

00:04:18.123 --> 00:04:20.369
but that's a little bit
confusing, at least for my brain.

00:04:20.369 --> 00:04:23.063
I like to go just down to
the unit cost you could say,

00:04:23.063 --> 00:04:26.298
the cost per battery and
then multiply it by six.

00:04:26.298 --> 00:04:29.742
It gives me more comfort in
understanding what I'm doing.


Relative position

WEBVTT
Kind: captions
Language: en

00:00:00.236 --> 00:00:02.677
- [Instructor] In which picture is the cow

00:00:02.677 --> 00:00:07.215
beside the little chicken
here, the little chick?

00:00:07.215 --> 00:00:09.092
Choose one answer.

00:00:09.092 --> 00:00:11.592
So which one do you think it is?

00:00:11.592 --> 00:00:14.207
Well, over here, they
aren't beside each other.

00:00:14.207 --> 00:00:15.630
Over here, they are.

00:00:15.630 --> 00:00:18.246
So I would pick choice B.

00:00:18.246 --> 00:00:20.329
Let's do another example.

00:00:21.437 --> 00:00:24.437
Which elephant is behind the bricks?

00:00:25.400 --> 00:00:26.306
Think about it.

00:00:26.306 --> 00:00:29.556
Which one is behind the bricks, A or B?

00:00:30.431 --> 00:00:34.104
Well, in choice A, the elephant
is in front of the bricks.

00:00:34.104 --> 00:00:35.820
It's in choice B that the elephant

00:00:35.820 --> 00:00:37.775
it looks like it's
hiding behind the bricks

00:00:37.775 --> 00:00:39.323
just sneaking around.

00:00:39.323 --> 00:00:41.895
Alright, I'm gonna go with choice B.

00:00:41.895 --> 00:00:44.062
Let's do one more example.

00:00:45.787 --> 00:00:49.954
Which shape is below this
green rectangle over here?

00:00:51.800 --> 00:00:54.535
So if I look over here,
this is our green rectangle

00:00:54.535 --> 00:00:56.660
and which one is below it?

00:00:56.660 --> 00:00:59.429
Well, we see this red triangle.

00:00:59.429 --> 00:01:03.596
So this is below the green
triangle and we are done.