GMAT: Math 22

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.820
We're on problem 115.

00:00:03.820 --> 00:00:07.450
Mary's income is 60% more
than Tim's income.

00:00:12.680 --> 00:00:16.650
So that's 1.6 times
Tim's income.

00:00:16.650 --> 00:00:19.920
You could write it as Tim's
income plus 60% of Tim's

00:00:19.920 --> 00:00:22.320
income, but hopefully it's
second nature that this is

00:00:22.320 --> 00:00:25.170
just 1.6 times Tim's income.

00:00:25.170 --> 00:00:30.270
So if Mary's income is equal to
1.6 times Tim's income, and

00:00:30.270 --> 00:00:34.430
Tim's income is 40% percent
less than Juan's income.

00:00:34.430 --> 00:00:39.070
So Tim's income, if it's 40%
less than Juan's income, that

00:00:39.070 --> 00:00:43.560
means that it's 60%
of Juan's income.

00:00:43.560 --> 00:00:47.110
You can do that as Juan's income
minus 40% percent of

00:00:47.110 --> 00:00:48.230
Juan's income.

00:00:48.230 --> 00:00:50.920
That's 0.6 times
Juan's income.

00:00:50.920 --> 00:00:53.850
What percentage of Juan's
income is Mary's income?

00:00:53.850 --> 00:00:55.540
So let's just do a little
substitution.

00:00:55.540 --> 00:00:58.230
Substitute this T over here.

00:00:58.230 --> 00:01:07.150
So you get Mary is equal to
1.6 times 0.6 times Juan.

00:01:07.150 --> 00:01:11.820
And then what's 6 times 16?

00:01:11.820 --> 00:01:21.770
60 plus 36.

00:01:21.770 --> 00:01:23.470
96.

00:01:23.470 --> 00:01:27.090
We have 2 numbers to the
right of the decimal.

00:01:27.090 --> 00:01:30.860
So it's 0.96 times
Juan's income.

00:01:30.860 --> 00:01:31.700
So what are they asking?

00:01:31.700 --> 00:01:33.910
What percentage of Juan's
income is Mary's income?

00:01:33.910 --> 00:01:37.280
Mary's income is 96%
of Juan's income.

00:01:37.280 --> 00:01:39.820
And that's choice C.

00:01:39.820 --> 00:01:43.110
0.96 is 96% of.

00:01:43.110 --> 00:01:46.350
Problem 116.

00:01:46.350 --> 00:01:49.470
Looks like something
I have to draw.

00:01:49.470 --> 00:01:53.150
Each dot in the mileage table
above represents an entry

00:01:53.150 --> 00:01:56.100
indicating the distance between
a pair of five cities.

00:01:56.100 --> 00:01:59.810
Let me try to draw
this out quickly.

00:01:59.810 --> 00:02:02.425
So we have five cities.

00:02:07.980 --> 00:02:18.870
It's 1, 2, 3, 4, and 5.

00:02:18.870 --> 00:02:20.920
And then we have to do
it the other way.

00:02:20.920 --> 00:02:28.595
1, 2, 3, and then 4.

00:02:31.550 --> 00:02:34.160
They have the same cities
on both axes.

00:02:34.160 --> 00:02:39.670
City A, B, C, D, and E.

00:02:39.670 --> 00:02:46.560
City A, B, C, D, and E.

00:02:46.560 --> 00:02:47.660
And they have some dots.

00:02:47.660 --> 00:02:55.030
That's a dot, a dot, a dot, a
dot, then dots down that way.

00:02:55.030 --> 00:02:57.730
Each dot in the mileage table
above represents an entry

00:02:57.730 --> 00:02:58.840
indicating the distance
between a

00:02:58.840 --> 00:03:00.440
pair of the five cities.

00:03:00.440 --> 00:03:03.360
If the table were extended to
represent the distances

00:03:03.360 --> 00:03:07.690
between all pairs of 30 cities,
and each distance were

00:03:07.690 --> 00:03:11.660
to be represented by only one
entry, how many entries would

00:03:11.660 --> 00:03:12.910
the table then have?

00:03:16.740 --> 00:03:19.440
So they're saying, OK, this
is essentially the

00:03:19.440 --> 00:03:21.480
case with five cities.

00:03:21.480 --> 00:03:24.110
You don't want to care about the
distance between A and A.

00:03:24.110 --> 00:03:30.450
They want the distance between
all pairs of 30 cities.

00:03:30.450 --> 00:03:33.990
So this is the case when you
have n is equal to 5.

00:03:33.990 --> 00:03:36.058
And the reason why you could say
the distance between A and

00:03:36.058 --> 00:03:38.010
B is this dot, you could have
also said the distance between

00:03:38.010 --> 00:03:40.800
B and A is this dot down here.

00:03:40.800 --> 00:03:43.480
But they didn't draw a dot
there, because they said and

00:03:43.480 --> 00:03:46.090
each distance were to be
represented by only one entry.

00:03:46.090 --> 00:03:49.120
So they don't want to double--
if this was 5, this

00:03:49.120 --> 00:03:49.790
would also be 5.

00:03:49.790 --> 00:03:51.730
They only want to write
it there once.

00:03:51.730 --> 00:03:54.670
So relative to 5, how would
we figure this out?

00:03:54.670 --> 00:04:00.900
So if you think about it, what
you're going to have is-- no

00:04:00.900 --> 00:04:02.790
matter how big this table
is, the diagonal

00:04:02.790 --> 00:04:04.910
is going to be empty.

00:04:04.910 --> 00:04:07.400
You don't have to fill out
the distance between

00:04:07.400 --> 00:04:09.840
the city and itself.

00:04:09.840 --> 00:04:13.530
And then if you just continue
the trend that they did, out

00:04:13.530 --> 00:04:15.840
of what's left over, you're just
going to fill out half of

00:04:15.840 --> 00:04:18.690
them, because the other half is
just redundant information.

00:04:18.690 --> 00:04:25.340
So for example, if you have 5
cities, you're going to have 5

00:04:25.340 --> 00:04:28.830
times 5 is equal
to 25 squares.

00:04:28.830 --> 00:04:30.380
How many squares are going
to be in the diagonal?

00:04:30.380 --> 00:04:32.880
We're going to have exactly
5 squares in the diagonal.

00:04:32.880 --> 00:04:34.700
1, 2, 3, 4, 5.

00:04:34.700 --> 00:04:36.480
So you're going to subtract 5.

00:04:36.480 --> 00:04:39.700
So you're going to have
20 boxes left that

00:04:39.700 --> 00:04:40.740
aren't in the diagonal.

00:04:40.740 --> 00:04:43.620
And then you are going to have
to to fill out half of those.

00:04:43.620 --> 00:04:47.430
So you're going to fill out
exactly 10 boxes, so 1, 2, 3

00:04:47.430 --> 00:04:50.680
4, 5, 6, 7, 8, 9, 10.

00:04:50.680 --> 00:04:52.680
So let's do the same
exercise with 30.

00:04:52.680 --> 00:04:55.750
If you have 30 cities, you're
going to have 30 times 30

00:04:55.750 --> 00:04:59.380
boxes, which is equal to 900.

00:04:59.380 --> 00:05:02.400
You're going to take out the
30 boxes that are in the

00:05:02.400 --> 00:05:05.460
diagonal, so minus 30.

00:05:05.460 --> 00:05:09.050
So that's equal to 870.

00:05:09.050 --> 00:05:10.600
And then you are only going to
have to fill out half of

00:05:10.600 --> 00:05:12.390
those, because you don't want
redundant information.

00:05:12.390 --> 00:05:14.060
So you are only going to fill
out the top half they way they

00:05:14.060 --> 00:05:15.700
did it in this diagram.

00:05:15.700 --> 00:05:21.660
So if you divide this by 2,
you're going to get to what?

00:05:21.660 --> 00:05:25.160
870 divided by 2 is 435.

00:05:25.160 --> 00:05:27.250
And that's choice B.

00:05:30.510 --> 00:05:36.390
Next question, 117.

00:05:36.390 --> 00:05:39.690
Which of the following has
a value less than 1?

00:05:39.690 --> 00:05:44.630
So choice A is 2 times 7/13.

00:05:44.630 --> 00:05:46.780
2 times 7 is 14, so that's
greater than 13.

00:05:46.780 --> 00:05:49.040
So it's not A.

00:05:49.040 --> 00:05:57.100
B, square root of 10/2.

00:05:57.100 --> 00:05:59.620
The square root of 10, that's
going to be 3 point something,

00:05:59.620 --> 00:06:01.140
which is greater than 2.

00:06:01.140 --> 00:06:03.850
So that's not right.

00:06:03.850 --> 00:06:11.980
Choice C is 2 over the
square root of 2.

00:06:11.980 --> 00:06:14.770
The square root of
2 is less than 2.

00:06:14.770 --> 00:06:15.880
It's 1 point something.

00:06:15.880 --> 00:06:19.930
So this is going to also have
a value greater than 1.

00:06:19.930 --> 00:06:20.670
So it can't be this.

00:06:20.670 --> 00:06:22.600
We're looking for less than 1.

00:06:22.600 --> 00:06:25.520
Choice D says 1 divided
by 1/2.

00:06:25.520 --> 00:06:28.340
That's equal to 1 times 2,
which is equal to 2.

00:06:28.340 --> 00:06:31.040
So I can already tell you the
choice is going to be E.

00:06:31.040 --> 00:06:32.310
Let's look at it.

00:06:32.310 --> 00:06:35.430
So they have 9/10 squared.

00:06:35.430 --> 00:06:38.190
That's 81/100, which
sure enough is

00:06:38.190 --> 00:06:39.275
definitely less than 1.

00:06:39.275 --> 00:06:41.890
So the choice is E.

00:06:41.890 --> 00:06:51.580
Problem 118.

00:06:51.580 --> 00:06:54.540
The ratio of the length to the
width of a rectangular

00:06:54.540 --> 00:06:58.920
advertising display is
approximately 3.3:2.

00:06:58.920 --> 00:07:02.390
So the length is 3.3
and the width is 2.

00:07:02.390 --> 00:07:04.150
At least that's the ratios.

00:07:04.150 --> 00:07:14.620
If the width of the display
is 8 meters, what is the

00:07:14.620 --> 00:07:17.600
approximate length of the
display in meters?

00:07:17.600 --> 00:07:22.350
So they tell us the length to
the width is approximately

00:07:22.350 --> 00:07:26.190
3.3:2, which is approximately
equal to

00:07:26.190 --> 00:07:29.440
the length to 8 meters.

00:07:29.440 --> 00:07:32.010
They say the width of the
display is 8 meters.

00:07:32.010 --> 00:07:35.580
So we just have to solve this
as best as we can for L.

00:07:38.780 --> 00:07:42.130
The easiest way to think about
it is to go from 2 to 8, you

00:07:42.130 --> 00:07:43.530
multiply by 4.

00:07:43.530 --> 00:07:47.200
So to go from 3.3 to L we
have to multiply by 4.

00:07:47.200 --> 00:07:50.690
So L is equal to 3.3 times 4.

00:07:50.690 --> 00:07:51.250
What's that?

00:07:51.250 --> 00:07:53.810
33 times 4, or 3.3 times 4.

00:07:53.810 --> 00:07:57.980
4 times 3 is 12, plus 1 is 13.

00:07:57.980 --> 00:07:59.150
One number behind the
decimal point.

00:07:59.150 --> 00:08:01.160
13.2.

00:08:01.160 --> 00:08:03.340
So they say approximate,
and the closest one

00:08:03.340 --> 00:08:07.750
is choice C, 13.

00:08:07.750 --> 00:08:15.280
Problem 119.

00:08:15.280 --> 00:08:17.830
The average salary of 15
people in the shipping

00:08:17.830 --> 00:08:22.060
department of a certain
firm is $20,000.

00:08:22.060 --> 00:08:27.530
So the average of 15 people
is equal to $20,000.

00:08:27.530 --> 00:08:32.110
The salary of 5 of the employees
is $25,000 each, so

00:08:32.110 --> 00:08:34.470
we can even say the average of
those 5 is going to be $25,000

00:08:34.470 --> 00:08:37.390
because all of them
make $25,000.

00:08:37.390 --> 00:08:39.913
And the salary of 4 of the
employees is $16,000 each.

00:08:39.913 --> 00:08:44.440
The average of 4 of the
employees is $16,000 each.

00:08:44.440 --> 00:08:47.870
That's the salary, but if you
average 4 people making

00:08:47.870 --> 00:08:50.280
$16,000, the average is
going to be $16,000.

00:08:50.280 --> 00:08:54.860
What is the average salary of
the remaining employees?

00:08:54.860 --> 00:08:57.260
Let's think about this.

00:08:57.260 --> 00:08:59.860
Let's try to find the average
of all of the employees.

00:08:59.860 --> 00:09:04.420
So if we find the average of all
15 employees, we're going

00:09:04.420 --> 00:09:07.920
to sum up all their salaries,
divide by 15.

00:09:07.920 --> 00:09:14.950
And we know that that is going
to be equal to $20,000.

00:09:14.950 --> 00:09:18.860
Now we have these 5 guys making
$25,000, so the sum of

00:09:18.860 --> 00:09:23.940
their salaries is going
to be 5 times $25,000.

00:09:23.940 --> 00:09:26.890
And then you have these 4 people
making $16,000, so it's

00:09:26.890 --> 00:09:28.760
plus 4 times 16.

00:09:28.760 --> 00:09:30.710
I'm just trying to sum up
everyone's salaries up here in

00:09:30.710 --> 00:09:32.830
the numerator, and I'm going
to divide by 15.

00:09:32.830 --> 00:09:35.880
And then I'm going
to have what?

00:09:35.880 --> 00:09:37.440
The average of the
remaining people.

00:09:37.440 --> 00:09:38.740
How many people are remaining?

00:09:38.740 --> 00:09:43.430
I've already done 9 people, so
there's 6 people remaining

00:09:43.430 --> 00:09:45.460
because there's a total
of 15 people.

00:09:45.460 --> 00:09:51.310
So the average of the 6
remaining people times 6.

00:09:51.310 --> 00:09:54.670
Whatever their average is, times
6, that'll give you the

00:09:54.670 --> 00:09:56.760
sum of their salaries.

00:09:56.760 --> 00:10:00.730
And now we just solve for
the average of those 6.

00:10:00.730 --> 00:10:02.280
Let's see what we can do.

00:10:02.280 --> 00:10:05.970
So 5 times 25, that's 125.

00:10:05.970 --> 00:10:08.420
Plus 4 times 16 is 64.

00:10:08.420 --> 00:10:12.310
Plus 6 times-- let's
call it A-- 6A is

00:10:12.310 --> 00:10:15.170
equal to 20 times 15.

00:10:15.170 --> 00:10:16.430
So what's 20 times 15?

00:10:16.430 --> 00:10:18.370
20 times 10 is 200.

00:10:18.370 --> 00:10:20.120
20 times 5 is 100.

00:10:20.120 --> 00:10:29.380
So it's equal to 300.

00:10:29.380 --> 00:10:38.670
125 plus 64 is 189 plus
6A, is equal to 300.

00:10:38.670 --> 00:10:41.600
Subtract 189 from both sides.

00:10:41.600 --> 00:10:47.040
You get 6A is equal
to 300 minus 189.

00:10:47.040 --> 00:10:54.790
That's 111.

00:10:54.790 --> 00:10:57.330
89, add 11 to it,
you get to 200.

00:10:57.330 --> 00:10:58.540
And then you have to add
another 100 to get to

00:10:58.540 --> 00:11:00.290
300, so it's 111.

00:11:00.290 --> 00:11:02.150
300 minus 189.

00:11:02.150 --> 00:11:04.540
And then how many times
does 6 go into 111?

00:11:04.540 --> 00:11:05.890
Divide both sides by 6.

00:11:05.890 --> 00:11:13.800
6 goes into 111-- 6 goes
into 51 eight times.

00:11:13.800 --> 00:11:17.610
8 times 48.

00:11:17.610 --> 00:11:19.960
18.5 times.

00:11:19.960 --> 00:11:23.950
$18,500 is the average
of their salary.

00:11:23.950 --> 00:11:26.350
And luckily enough, that's
one of the choices.

00:11:26.350 --> 00:11:28.780
That's choice B.

00:11:28.780 --> 00:11:30.430
$18,500.

00:11:30.430 --> 00:11:32.030
We did everything in
[? thousandths ?]

00:11:32.030 --> 00:11:35.530
so we got 18.5, but
that's $18,500.

00:11:35.530 --> 00:11:37.590
See you in the next video.


GMAT: Math 23

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:04.920
We're on problem 120.

00:00:04.920 --> 00:00:17.540
David has d books, which is 3
times as many as Jeff, and 1/2

00:00:17.540 --> 00:00:18.890
as many as Paula.

00:00:18.890 --> 00:00:23.740
So d is equal to 1/2 the number
of books Paula has.

00:00:23.740 --> 00:00:26.020
How many books do the
three of them have

00:00:26.020 --> 00:00:28.770
together in terms of d?

00:00:28.770 --> 00:00:33.620
So they want to know how
much d plus j plus p

00:00:33.620 --> 00:00:36.130
is, in terms of d.

00:00:36.130 --> 00:00:38.120
So what's j in terms of d?

00:00:38.120 --> 00:00:42.950
j is equal to-- divide both
sides by 3-- is equal to d/3.

00:00:42.950 --> 00:00:47.040
And what's p in terms of d?

00:00:47.040 --> 00:00:51.700
If we multiply both sides by 2,
you get p is equal to 2d.

00:00:51.700 --> 00:00:57.970
So this turns into d plus
j, which is d/3,

00:00:57.970 --> 00:01:01.180
plus p, which is 2d.

00:01:01.180 --> 00:01:02.410
Let's find the common
denominator.

00:01:02.410 --> 00:01:04.890
Well, 3.

00:01:04.890 --> 00:01:09.366
1d is the same thing
as 3d over 3.

00:01:09.366 --> 00:01:11.690
d over 3, well, that's still
just going to be d/3.

00:01:14.520 --> 00:01:19.880
2d, that's the same thing
is 6d over 3.

00:01:19.880 --> 00:01:29.220
That's equal to 3, 4, plus 6 is
10d over 3, or 10 thirds d.

00:01:29.220 --> 00:01:31.390
And that's choice C.

00:01:31.390 --> 00:01:34.800
10 thirds d.

00:01:34.800 --> 00:01:40.620
Problem 121.

00:01:40.620 --> 00:01:44.090
There are 8 teams in a certain
league, and each team plays

00:01:44.090 --> 00:01:46.260
each of the other teams
exactly once.

00:01:51.120 --> 00:01:56.490
If each game is played by 2
teams-- it's a traditional

00:01:56.490 --> 00:01:59.953
game-- what is the total
number of games played?

00:02:04.040 --> 00:02:06.840
Let's think about
it a little bit.

00:02:06.840 --> 00:02:10.550
They each play each other
exactly once.

00:02:10.550 --> 00:02:15.900
So this turns into that grid
problem that we did last time.

00:02:15.900 --> 00:02:24.160
If I have 8 teams, 1,
2, 3, 4, 5, 6, 7, 8.

00:02:24.160 --> 00:02:25.520
And I'm going to write
the same thing.

00:02:25.520 --> 00:02:34.170
1, 2, 3, 4, 5, 6, 7, 8.

00:02:34.170 --> 00:02:37.190
They can't play each other.

00:02:37.190 --> 00:02:38.290
1 is never going to play 1.

00:02:38.290 --> 00:02:39.360
2 is never going to play 2.

00:02:39.360 --> 00:02:40.580
3 is never going to play 3.

00:02:40.580 --> 00:02:42.460
4 is never going to play
4, et cetera, all the

00:02:42.460 --> 00:02:43.290
way down the diagonal.

00:02:43.290 --> 00:02:46.050
You can't play yourself.

00:02:46.050 --> 00:02:49.860
And 1 is going to play
2 exactly 1 game.

00:02:49.860 --> 00:02:50.890
That's 1 game.

00:02:50.890 --> 00:02:53.170
And we're not going to write
2 played 1, because 1

00:02:53.170 --> 00:02:54.090
already played 2.

00:02:54.090 --> 00:02:55.580
They only play each
other once.

00:02:55.580 --> 00:02:59.210
So you don't want to count
this one down here.

00:02:59.210 --> 00:03:01.370
You don't want to
count this one.

00:03:01.370 --> 00:03:03.120
This is when 1 plays 3.

00:03:03.120 --> 00:03:04.650
This is when 2 plays 3.

00:03:04.650 --> 00:03:08.080
So, essentially, you want to
figure out all of these spots

00:03:08.080 --> 00:03:10.330
that are above the diagonal.

00:03:10.330 --> 00:03:14.370
Let's think about how many
spots there are.

00:03:14.370 --> 00:03:21.260
If we have an 8 by 8 square,
there are 8 times 8 squares on

00:03:21.260 --> 00:03:22.880
this grid, if I actually
drew them as squares.

00:03:22.880 --> 00:03:25.170
I did that a couple
of problems ago.

00:03:25.170 --> 00:03:30.240
If I actually drew it like that,
then you might see the

00:03:30.240 --> 00:03:31.480
grid a little bit better.

00:03:31.480 --> 00:03:32.360
But I think you get the idea.

00:03:32.360 --> 00:03:36.880
You have 8 by 8 spots on this
grid, because it's 8 spots

00:03:36.880 --> 00:03:38.260
high, 8 spots wide.

00:03:38.260 --> 00:03:41.370
So you have 64 spots.

00:03:41.370 --> 00:03:43.850
Now, you're not going to have
one of the games along the

00:03:43.850 --> 00:03:46.400
diagonal, because that would
be a team playing itself.

00:03:46.400 --> 00:03:47.950
So how many spots are
along the diagonal?

00:03:47.950 --> 00:03:48.820
There's 8.

00:03:48.820 --> 00:03:52.010
There's 1 against 1,
2 and 2, 3 and 3.

00:03:52.010 --> 00:03:53.300
So that's each team
playing itself.

00:03:53.300 --> 00:03:56.180
So there's 8 spots along
the diagonal.

00:03:56.180 --> 00:04:02.240
So 64 minus 8 is 56.

00:04:02.240 --> 00:04:04.460
And so 56 would give us the
number of squares that aren't

00:04:04.460 --> 00:04:05.530
in the diagonal.

00:04:05.530 --> 00:04:09.310
But if we counted all of them,
we would be double counting.

00:04:09.310 --> 00:04:14.000
We'd be counting 1 playing 2
up here, and 2 playing 1.

00:04:14.000 --> 00:04:16.750
So we only want to count half
of the remaining squares.

00:04:16.750 --> 00:04:20.519
So if you divide this
by 2, you get 28.

00:04:20.519 --> 00:04:23.320
So there'd be a total
of 28 games.

00:04:23.320 --> 00:04:27.750
And that's choice C.

00:04:27.750 --> 00:04:32.750
Question 122.

00:04:32.750 --> 00:04:40.230
An operation theta is defined
by the equation a theta b is

00:04:40.230 --> 00:04:47.840
equal to a minus b over a plus b
for all numbers a and b such

00:04:47.840 --> 00:04:51.460
that a does not equal minus b.

00:04:51.460 --> 00:04:53.560
That ensures that the
denominator does not equal 0.

00:04:53.560 --> 00:04:54.930
So they're defining
the domain.

00:04:54.930 --> 00:04:57.060
If a equaled minus b, then it
would be undefined, because

00:04:57.060 --> 00:04:59.360
you'd have something
divided by 0.

00:04:59.360 --> 00:05:12.050
If a does not equal negative c,
and a theta c is equal to

00:05:12.050 --> 00:05:17.840
0, then c is equal to-- So
they're telling us right now

00:05:17.840 --> 00:05:22.420
that a does not equal minus c.

00:05:22.420 --> 00:05:30.670
So let's just write it down. a
theta c is equal to a minus c

00:05:30.670 --> 00:05:33.670
over a plus c.

00:05:33.670 --> 00:05:36.020
And then, that we want
to set equal to 0.

00:05:36.020 --> 00:05:39.780
Because they're telling us
a theta c is equal to 0.

00:05:39.780 --> 00:05:43.780
They're telling us that a
does not equal minus c.

00:05:46.900 --> 00:05:49.220
So that's essentially saying
if a was minus c, then the

00:05:49.220 --> 00:05:50.590
denominator would be 0,
and we would have

00:05:50.590 --> 00:05:51.470
something really strange.

00:05:51.470 --> 00:05:54.720
So they are telling us the
denominator does not equal 0.

00:05:54.720 --> 00:05:57.850
So if the denominator does not
equal 0, and this fraction

00:05:57.850 --> 00:06:01.320
equals 0, then the numerator
of this fraction has to be

00:06:01.320 --> 00:06:02.390
equal to 0.

00:06:02.390 --> 00:06:06.460
Or another way you could say it,
a minus c over a plus c is

00:06:06.460 --> 00:06:07.030
equal to 0.

00:06:07.030 --> 00:06:08.940
We know a plus c is
not equal to 0.

00:06:08.940 --> 00:06:11.390
So let's multiply both sides
of the equation times it.

00:06:11.390 --> 00:06:14.450
And you get a minus c is equal
to 0 times a plus c.

00:06:14.450 --> 00:06:15.295
That's just 0.

00:06:15.295 --> 00:06:17.625
In other words, if a fraction
is equal 0, the

00:06:17.625 --> 00:06:18.880
denominator is not 0.

00:06:18.880 --> 00:06:20.015
That would make it undefined.

00:06:20.015 --> 00:06:22.520
Then the numerator has
to be equal to 0.

00:06:22.520 --> 00:06:26.100
So then, just solving, you
get a is equal to c.

00:06:26.100 --> 00:06:27.100
Add c to both sides.

00:06:27.100 --> 00:06:28.170
So they want to know
what c is equal to.

00:06:28.170 --> 00:06:30.200
So c is equal to a.

00:06:30.200 --> 00:06:31.710
And that is choice E.

00:06:35.140 --> 00:06:36.310
Next question.

00:06:36.310 --> 00:06:38.140
Let me switch colors.

00:06:38.140 --> 00:06:41.960
123.

00:06:41.960 --> 00:06:50.380
The price of a lunch for 15
people was $207, including a

00:06:50.380 --> 00:06:52.840
15% gratuity for service.

00:06:52.840 --> 00:06:56.130
So that was plus 15%.

00:06:56.130 --> 00:06:58.920
What was the average
price per person,

00:06:58.920 --> 00:07:04.220
excluding the gratuity?

00:07:04.220 --> 00:07:13.210
So essentially, we want to
figure out the price of 15

00:07:13.210 --> 00:07:17.650
before gratuity times-- so
this is after gratuity.

00:07:17.650 --> 00:07:20.050
So I'll call this, price
before gratuity.

00:07:20.050 --> 00:07:22.400
I don't want to get too
complicated with my notation--

00:07:22.400 --> 00:07:28.250
Price before gratuity times
1.15, because we're going to

00:07:28.250 --> 00:07:35.840
increase by 15, is
equal to $207.

00:07:35.840 --> 00:07:40.430
So the price before gratuity of
the 15 people is equal to

00:07:40.430 --> 00:07:44.820
207 divided by 1.15.

00:07:44.820 --> 00:07:47.630
And we want to figure out the
average price per person.

00:07:47.630 --> 00:07:57.630
So we want to figure out
this divided by 15.

00:07:57.630 --> 00:08:01.350
This looks, at its heart,
kind mathy.

00:08:01.350 --> 00:08:06.310
But let's just chug through it
and see where we can go.

00:08:06.310 --> 00:08:19.310
So this is the same thing
is 207/1.15 times 1/15.

00:08:19.310 --> 00:08:28.120
And that's equal to 207 divided
by 15 times 1.15, the

00:08:28.120 --> 00:08:29.550
denominator.

00:08:29.550 --> 00:08:33.020
Let's figure out the
denominator first.

00:08:33.020 --> 00:08:39.580
Well, maybe the numerator--
207 divided by 1.15.

00:08:39.580 --> 00:08:44.610
1.15 goes into 207.

00:08:44.610 --> 00:08:46.780
Add some decimals.

00:08:46.780 --> 00:08:48.620
Push this two to the right,
two to the right.

00:08:48.620 --> 00:08:50.320
The decimal goes there.

00:08:50.320 --> 00:08:52.930
115 goes into 207 one time.

00:08:52.930 --> 00:08:56.830
1 times 115 is 115.

00:08:56.830 --> 00:09:02.010
Then, it's 85 plus 7.

00:09:02.010 --> 00:09:06.610
85 plus 7 is 92.

00:09:06.610 --> 00:09:09.180
920.

00:09:09.180 --> 00:09:13.460
And then then this should
go roughly 8 times.

00:09:13.460 --> 00:09:15.870
8 times 115.

00:09:15.870 --> 00:09:18.260
8 times 5 is 40.

00:09:18.260 --> 00:09:21.490
8 times 1 is 8 plus 4 is 12.

00:09:21.490 --> 00:09:22.360
Carry the 1.

00:09:22.360 --> 00:09:24.830
8 times 1 is 8, plus 1 is 9.

00:09:24.830 --> 00:09:26.710
It actually equals
exactly 920.

00:09:26.710 --> 00:09:27.960
Great.

00:09:30.140 --> 00:09:34.420
So before gratuity,
it was $180.

00:09:34.420 --> 00:09:36.750
Before gratuity, our
price was $180.

00:09:36.750 --> 00:09:38.260
And now, we have to divide
that by 15 to

00:09:38.260 --> 00:09:40.220
get the average price.

00:09:40.220 --> 00:09:43.820
So 15 goes into 180
how many times?

00:09:43.820 --> 00:09:46.200
1 times 15, 30.

00:09:46.200 --> 00:09:48.190
15 goes into 30 two times.

00:09:48.190 --> 00:09:54.250
So the average price before
gratuity was exactly $12, B.

00:09:54.250 --> 00:09:56.120
And I'm out of time.

00:09:56.120 --> 00:09:57.870
See you in the next video.


GMAT: Math 21

WEBVTT
Kind: captions
Language: en

00:00:01.100 --> 00:00:04.630
Problem 110.

00:00:04.630 --> 00:00:07.820
If candy bars that regularly
sell for $0.40 each are on

00:00:07.820 --> 00:00:11.940
sale at 2 for $0.75-- so they
used to be $0.40, and now

00:00:11.940 --> 00:00:17.400
they're going 2 for $0.75-- what
is the percent reduction

00:00:17.400 --> 00:00:19.880
in the price of 2
such candy bars

00:00:19.880 --> 00:00:22.110
purchased at the sale price?

00:00:22.110 --> 00:00:24.130
What is the price reduction?

00:00:24.130 --> 00:00:26.950
So before, if I was going to
buy 2 candy bars, I'd be

00:00:26.950 --> 00:00:33.140
paying $0.80, and now for 2
candy bars I'm paying $0.75.

00:00:33.140 --> 00:00:36.380
So I'm saving $0.05
or I'm getting

00:00:36.380 --> 00:00:38.240
a $0.05 price reduction.

00:00:38.240 --> 00:00:40.130
So they want to know the percent
price reduction.

00:00:40.130 --> 00:00:44.720
So it's essentially what percent
is $0.05 of $0.80?

00:00:44.720 --> 00:00:50.940
5 over 80 is equal to
what percentage?

00:00:50.940 --> 00:00:52.880
So let's just figure it out.

00:00:52.880 --> 00:00:57.980
80 goes into 5-- and actually,
we could divide the top and

00:00:57.980 --> 00:01:02.140
the bottom, make it a
little bit easier.

00:01:02.140 --> 00:01:06.450
5.00, add some 0's there.

00:01:06.450 --> 00:01:09.170
So 80 does not go into 50.

00:01:09.170 --> 00:01:13.460
80 goes into 500, so how many
times does 8-- It goes into it

00:01:13.460 --> 00:01:16.420
six times, because
8 times 6 is 48.

00:01:16.420 --> 00:01:20.465
6 times 80 is 480.

00:01:20.465 --> 00:01:22.940
500 minus 480 is 200.

00:01:22.940 --> 00:01:26.780
80 goes into 200 two times.

00:01:26.780 --> 00:01:31.780
2 times 80 is 160, and then
remainder of 400.

00:01:31.780 --> 00:01:34.170
800 goes into 400 five times.

00:01:34.170 --> 00:01:40.200
So that's 0.0625, which is
the same thing as 6.25%.

00:01:40.200 --> 00:01:44.280
And that is choice B, 6 1/4%.

00:01:44.280 --> 00:01:45.570
Same thing.

00:01:45.570 --> 00:01:47.950
Choice B.

00:01:47.950 --> 00:01:52.690
Problem 111.

00:01:52.690 --> 00:02:01.780
If s is greater than 0, and the
square root of r over s is

00:02:01.780 --> 00:02:08.410
equal to s, what is
r in terms of s?

00:02:08.410 --> 00:02:11.900
OK, so let's just square both
sides of this equation, so you

00:02:11.900 --> 00:02:13.540
get r over s.

00:02:13.540 --> 00:02:16.450
The square root of r over s
squared is just r over s.

00:02:16.450 --> 00:02:19.650
And that's going to be
equal to s squared.

00:02:19.650 --> 00:02:21.990
We've got to square both
sides of the equation.

00:02:21.990 --> 00:02:25.790
Now let's multiply both sides
of the equation times s, and

00:02:25.790 --> 00:02:30.160
we get r is equal to
s to the third.

00:02:37.040 --> 00:02:39.690
r is equal to s to the
third, either way.

00:02:39.690 --> 00:02:44.000
And that is choice D.

00:02:44.000 --> 00:02:46.560
Next question, 112.

00:02:46.560 --> 00:02:48.540
They've drawn us a neat
little diagram here.

00:02:52.930 --> 00:02:56.210
Let me see if I can draw
what they have drawn.

00:02:56.210 --> 00:03:01.720
They drew it in a dark color, so
I'll draw it-- That's kind

00:03:01.720 --> 00:03:05.590
of overly bright, but it
gets the job done.

00:03:05.590 --> 00:03:09.400
And then they have two
boxes inside of it.

00:03:09.400 --> 00:03:12.070
It looks like that.

00:03:12.070 --> 00:03:16.110
One box, something like that.

00:03:16.110 --> 00:03:21.570
One box, and then there's
another box in there.

00:03:21.570 --> 00:03:22.880
It looks something like that,
although they look like the

00:03:22.880 --> 00:03:24.560
same size in the picture.

00:03:24.560 --> 00:03:32.440
And they tell us that this
distance is 6 feet.

00:03:32.440 --> 00:03:36.230
They tell us that this
distance is 8 feet.

00:03:36.230 --> 00:03:40.700
The front of a 6 by 8 foot
rectangular door has brass

00:03:40.700 --> 00:03:42.930
rectangular trim, as indicated
by the shading

00:03:42.930 --> 00:03:43.560
in the figure above.

00:03:43.560 --> 00:03:45.560
OK, this green stuff is brass.

00:03:45.560 --> 00:03:50.340
If the trim is uniformly 1 foot
wide-- so that's 1 foot--

00:03:50.340 --> 00:03:52.880
what fraction of the door's
front surface

00:03:52.880 --> 00:03:55.590
is covered by trim?

00:03:55.590 --> 00:03:57.390
So we need to figure
out the area that

00:03:57.390 --> 00:03:59.400
is essentially green.

00:03:59.400 --> 00:04:02.040
Well, the easier thing might
be to figure out the area

00:04:02.040 --> 00:04:03.620
that's black, and
then subtract it

00:04:03.620 --> 00:04:04.470
from the total area.

00:04:04.470 --> 00:04:07.195
So what's the area
that's black?

00:04:07.195 --> 00:04:10.955
So both of these-- well,
they don't tell us.

00:04:10.955 --> 00:04:13.325
Well, it doesn't matter,
we could figure it out.

00:04:13.325 --> 00:04:14.860
Well, there's a couple of
ways to figure it out.

00:04:14.860 --> 00:04:18.200
Let's just figure out the
area that's green.

00:04:18.200 --> 00:04:20.680
I'm going to do it in
different color.

00:04:20.680 --> 00:04:25.350
So let's figure out this
area right here.

00:04:25.350 --> 00:04:27.060
I hope you can see that.

00:04:27.060 --> 00:04:29.180
So that area is what?

00:04:29.180 --> 00:04:31.400
Let me actually draw all of the
different boxes and then

00:04:31.400 --> 00:04:32.650
we can figure out the areas.

00:04:35.200 --> 00:04:38.660
Then we have that one.

00:04:38.660 --> 00:04:41.560
And then we have this one.

00:04:41.560 --> 00:04:43.440
And then we have this
one right here.

00:04:48.450 --> 00:04:49.690
What's the area of this?

00:04:49.690 --> 00:04:54.450
Its height is 8, this is
8, and its width is 1.

00:04:54.450 --> 00:04:57.980
So the area here is 8, and so
the area here is going to be 8

00:04:57.980 --> 00:05:01.190
of this box right here,
that whole box.

00:05:01.190 --> 00:05:02.970
What's the width of this?

00:05:02.970 --> 00:05:04.800
Well, it's not the whole
6, because you have

00:05:04.800 --> 00:05:06.860
1 here and 1 here.

00:05:06.860 --> 00:05:12.090
So the width is going to be 4 by
1, so this is going to have

00:05:12.090 --> 00:05:13.450
an area of 4.

00:05:13.450 --> 00:05:18.460
This is the same thing, width of
4 by 1, and that's 4 and 4.

00:05:18.460 --> 00:05:22.500
So all of the green stuff has
an area of-- 4 plus 4, plus

00:05:22.500 --> 00:05:32.510
4-- that's 12-- plus 8, plus
8, plus 16, is equal to 28.

00:05:32.510 --> 00:05:34.390
They want to know what fraction
of the door's front

00:05:34.390 --> 00:05:35.500
surface is covered
by the trim.

00:05:35.500 --> 00:05:39.190
So 28 is the amount of square
feet covered by the trim.

00:05:39.190 --> 00:05:41.870
And what's the total surface
area of the front?

00:05:41.870 --> 00:05:45.550
The total area's going to be 6
times 8, which is equal to 48.

00:05:45.550 --> 00:05:47.580
So if you want to know the
fraction that's covered by the

00:05:47.580 --> 00:05:51.890
trim, it'd be 28 over 48, which
is equal to-- let's see,

00:05:51.890 --> 00:05:58.210
you divide the top and bottom
by 4-- 7 over 12, 7/12.

00:05:58.210 --> 00:06:01.420
And that's choice D.

00:06:01.420 --> 00:06:07.410
Next question, 113.

00:06:07.410 --> 00:06:15.070
If a is equal to minus
0.3, which of the

00:06:15.070 --> 00:06:18.510
following is true?

00:06:18.510 --> 00:06:22.260
OK, so they have a bunch of
statements that relate a-- I'm

00:06:22.260 --> 00:06:30.040
just trying to see the pattern
in the-- So in all of these

00:06:30.040 --> 00:06:35.530
choices, they're relating a to
a squared, to a to the third,

00:06:35.530 --> 00:06:37.620
and then they're putting
it in order.

00:06:37.620 --> 00:06:39.750
They're saying which of these
is the least, then next, and

00:06:39.750 --> 00:06:42.910
then the greatest. So let's
just figure it out.

00:06:42.910 --> 00:06:46.420
a is minus 0.3.

00:06:46.420 --> 00:06:49.270
What's a squared?

00:06:49.270 --> 00:06:50.330
So it's going to be positive.

00:06:50.330 --> 00:06:52.350
Negative times a negative
is a positive.

00:06:52.350 --> 00:06:54.590
And then 3 times 3, and we're
going to have two digits

00:06:54.590 --> 00:06:55.460
behind the decimal, right?

00:06:55.460 --> 00:06:57.370
Because 0.3 times 0.3.

00:06:57.370 --> 00:07:01.410
So it's going to be
positive 0.09.

00:07:01.410 --> 00:07:02.940
And then what's a
to the third?

00:07:02.940 --> 00:07:05.350
Well, 0.09 times 0.3.

00:07:05.350 --> 00:07:07.085
9 times 3 is 27.

00:07:07.085 --> 00:07:09.780
And we're going to have 1, 2,
3 numbers behind the decimal

00:07:09.780 --> 00:07:13.210
spot, so it'd be 0.027.

00:07:13.210 --> 00:07:14.830
And we have a negative
times a positive, so

00:07:14.830 --> 00:07:16.230
it's going to be negative.

00:07:16.230 --> 00:07:17.880
So let's put these in order.

00:07:17.880 --> 00:07:21.690
Which of these is the
smallest number?

00:07:21.690 --> 00:07:22.800
So think about it.

00:07:22.800 --> 00:07:24.410
The smallest number is going to
be the one that's actually

00:07:24.410 --> 00:07:26.020
the most negative.

00:07:26.020 --> 00:07:29.560
So this right here is
the most negative.

00:07:29.560 --> 00:07:33.310
Minus 0.3 is the
most negative.

00:07:33.310 --> 00:07:35.860
We could even draw a
number line here.

00:07:35.860 --> 00:07:42.240
You could have minus
0.3, which is a.

00:07:42.240 --> 00:07:47.120
Then you have minus 0.027,
which is a to the third.

00:07:47.120 --> 00:07:48.700
And then you have
0 some place.

00:07:48.700 --> 00:07:53.450
And then you have 0.09,
which is a squared.

00:07:53.450 --> 00:07:57.130
So the correct answer should
be a is less than a to the

00:07:57.130 --> 00:08:00.420
third, which is less
than a squared.

00:08:00.420 --> 00:08:05.130
And that is choice B.

00:08:10.250 --> 00:08:12.040
Problem 114.

00:08:12.040 --> 00:08:14.930
Let me switch colors.

00:08:14.930 --> 00:08:18.240
Which of the following is the
product of two integers

00:08:18.240 --> 00:08:19.500
whose sum is 11.

00:08:19.500 --> 00:08:24.040
So they're telling us that
x plus y is equal to 11.

00:08:24.040 --> 00:08:25.440
And which one is the product?

00:08:25.440 --> 00:08:29.580
So xy is equal to what?

00:08:29.580 --> 00:08:31.000
So let's think about it.

00:08:31.000 --> 00:08:33.250
Which of the following is a
product of two integers

00:08:33.250 --> 00:08:37.619
whose sum is 11?

00:08:37.619 --> 00:08:39.440
So this is interesting.

00:08:39.440 --> 00:08:45.200
Maybe we just have to do some
experimentation right now.

00:08:45.200 --> 00:08:49.230
We could we could be dealing
with negative numbers here.

00:08:49.230 --> 00:08:54.190
So let's think about how
we can rearrange this.

00:08:54.190 --> 00:08:55.240
I'm just experimenting.

00:08:55.240 --> 00:08:58.340
Let's see, if we have y is equal
to 11 minus x, does that

00:08:58.340 --> 00:08:59.140
give us an intuition?

00:08:59.140 --> 00:09:03.990
Then xy is equal to 11 minus
x, because y is 11 minus x.

00:09:03.990 --> 00:09:10.810
That equals 11x minus
x squared, where x

00:09:10.810 --> 00:09:14.040
is really any integer.

00:09:14.040 --> 00:09:24.830
So if you put a 1 there, you
could you could get a 10.

00:09:24.830 --> 00:09:28.480
They don't tell us that they're
greater than 0.

00:09:28.480 --> 00:09:30.780
So let me think about this.

00:09:30.780 --> 00:09:36.360
So 32 is 8 and 4.

00:09:36.360 --> 00:09:43.770
26 is 13.

00:09:43.770 --> 00:09:44.840
Let me look at all
those choices.

00:09:44.840 --> 00:09:47.190
This one has me temporarily
stumped, hopefully

00:09:47.190 --> 00:09:48.530
temporarily.

00:09:48.530 --> 00:09:51.270
So minus 42, what are
the different

00:09:51.270 --> 00:09:52.240
ways I could get there?

00:09:52.240 --> 00:09:54.280
I can get 21 and 2.

00:09:54.280 --> 00:09:57.270
I could get 7 and 6, but one
would have to be positive and

00:09:57.270 --> 00:10:00.050
one would have to be negative.

00:10:00.050 --> 00:10:08.405
I can get 14 and
3, 14 times 3.

00:10:08.405 --> 00:10:13.610
And actually if I have 14
times minus 3, what

00:10:13.610 --> 00:10:14.500
is that equal to?

00:10:14.500 --> 00:10:19.016
It's minus 42, and then
14 plus minus 3,

00:10:19.016 --> 00:10:20.990
and that equals 11.

00:10:20.990 --> 00:10:23.110
So it's answer A.

00:10:23.110 --> 00:10:25.620
I wish I could give you an
analytical way of doing it

00:10:25.620 --> 00:10:28.000
instead of just experimenting
with numbers.

00:10:28.000 --> 00:10:32.080
I was just trying to figure
out what equals 42.

00:10:32.080 --> 00:10:34.380
Which of the following is a
product of two integers

00:10:34.380 --> 00:10:37.740
whose sum is 11?

00:10:37.740 --> 00:10:40.960
I guess a more systematic way of
doing it might have been to

00:10:40.960 --> 00:10:46.210
just say what are the
factors of 42?

00:10:46.210 --> 00:10:57.600
It'd be 1 and 42, 2 and 21, 3
and 14, and then 6 and 7.

00:10:57.600 --> 00:11:00.370
And you say OK, if I-- and
since this is a negative

00:11:00.370 --> 00:11:02.350
number, one of these has to be
positive and one has to be

00:11:02.350 --> 00:11:05.780
negative-- so if I were to take
a positive 1 of one of

00:11:05.780 --> 00:11:08.230
these, and a negative 1 of one
of these, and add them up, can

00:11:08.230 --> 00:11:09.210
I get to 11?

00:11:09.210 --> 00:11:12.410
And I guess that's where your
brain should say, oh, 14 minus

00:11:12.410 --> 00:11:14.620
3, that is equal to 11.

00:11:14.620 --> 00:11:16.280
And luckily it was choice
A, and you didn't

00:11:16.280 --> 00:11:16.990
have to go any further.

00:11:16.990 --> 00:11:19.810
But you would have to do this
with every one from there.

00:11:19.810 --> 00:11:20.620
You might want to look
at the solutions.

00:11:20.620 --> 00:11:22.650
Maybe they have more elegant
solution to this.

00:11:22.650 --> 00:11:25.010
Anyway, I'll see you
in the next video.


GMAT: Math 20

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:05.180
We're on problem 106.

00:00:05.180 --> 00:00:08.610
The present ratio of students
to teachers at a certain

00:00:08.610 --> 00:00:10.110
school is 30:1.

00:00:10.110 --> 00:00:15.890
So the ratio of students to
teachers is equal to 30:1.

00:00:15.890 --> 00:00:19.770
If the student enrollment were
to increase by 50 students,

00:00:19.770 --> 00:00:23.040
and the number of teachers were
increased by 5, the ratio

00:00:23.040 --> 00:00:26.220
of students to teachers
would then be 25:1.

00:00:26.220 --> 00:00:33.100
So if we increase the students
by 50 students, and increase

00:00:33.100 --> 00:00:43.700
the teachers by 5, the new ratio
is going to be 25:1.

00:00:43.700 --> 00:00:49.200
What is the present number
of teachers?

00:00:49.200 --> 00:00:50.660
Let's just simplify these
a little bit.

00:00:50.660 --> 00:00:52.010
We could cross-multiply.

00:00:52.010 --> 00:00:54.730
We could just write s over T.

00:00:54.730 --> 00:01:01.050
30 over 1 is equal to 30,
or s is equal to 30T.

00:01:01.050 --> 00:01:02.790
And let's substitute that into
this equation, because we're

00:01:02.790 --> 00:01:05.050
going to have to solve for the
present number of teachers,

00:01:05.050 --> 00:01:07.990
which is just T.

00:01:07.990 --> 00:01:15.830
So substitute this here, so we
get 30T plus 50, over T plus

00:01:15.830 --> 00:01:21.920
5, is equal to-- well, 25
over 1, that's just 25.

00:01:21.920 --> 00:01:26.705
And then let's multiply both
sides of this equation by T

00:01:26.705 --> 00:01:36.900
plus 5, and you get 30T plus
50 is equal to 25T, plus--

00:01:36.900 --> 00:01:44.000
what's 25 times 5?-- 125.

00:01:44.000 --> 00:01:46.240
Let's subtract 25T
from both sides.

00:01:46.240 --> 00:01:51.980
You get 5T plus 50
is equal to 125.

00:01:51.980 --> 00:01:58.900
Subtract 50 from both sides,
you get 5T is equal to 75.

00:01:58.900 --> 00:02:07.950
And then T is equal to 15.

00:02:07.950 --> 00:02:11.140
15 times 5 is 50, plus
25, right, 75.

00:02:11.140 --> 00:02:12.190
T is 15.

00:02:12.190 --> 00:02:14.390
So the answer is E.

00:02:14.390 --> 00:02:15.126
Next question.

00:02:15.126 --> 00:02:16.940
I don't want to run out of
space, so I'll just do it

00:02:16.940 --> 00:02:18.360
right here.

00:02:18.360 --> 00:02:22.130
Problem 107.

00:02:22.130 --> 00:02:29.950
What is the smallest integer,
n, for which 25 to the n is

00:02:29.950 --> 00:02:33.030
greater than 5 to the 12th?

00:02:33.030 --> 00:02:36.650
Whenever you see some type of
an equation or inequality

00:02:36.650 --> 00:02:38.940
where the variables are under
the exponent, and you're

00:02:38.940 --> 00:02:41.350
comparing numbers of different
bases, your first thing that

00:02:41.350 --> 00:02:43.840
should register in your brain
is how can I get these to be

00:02:43.840 --> 00:02:44.600
in the same base?

00:02:44.600 --> 00:02:47.020
And it should be pretty obvious
here, because 25 is

00:02:47.020 --> 00:02:49.750
the same thing as 5 squared, so
let's rewrite it that way.

00:02:49.750 --> 00:02:55.170
This statement is equivalent to
saying 5 squared to the n

00:02:55.170 --> 00:02:58.080
is greater than 5 to the 12th.

00:02:58.080 --> 00:03:02.290
And 5 squared to the n, that's
the same thing as 5 to the 2n,

00:03:02.290 --> 00:03:07.340
is greater than 5 to the 12th.

00:03:07.340 --> 00:03:13.240
And this right here is going to
be true when 2n is greater

00:03:13.240 --> 00:03:15.810
than 12, when the exponent up
here is greater than the

00:03:15.810 --> 00:03:16.870
exponent up here.

00:03:16.870 --> 00:03:18.020
That's when it's going
to be true.

00:03:18.020 --> 00:03:21.925
You divide both sides by 2, and
you get is greater than 6.

00:03:24.570 --> 00:03:27.830
So what is the smallest integer,
n, for which 25 to

00:03:27.830 --> 00:03:31.410
the n is greater than
5 to the 12th?

00:03:31.410 --> 00:03:33.850
So n has to be greater than 6,
so we can't pick choice A,

00:03:33.850 --> 00:03:36.550
which is 6, because if we pick
6, then we would have

00:03:36.550 --> 00:03:38.960
5 to the 12th here.

00:03:38.960 --> 00:03:41.110
And then it would be an
inequality, but this is

00:03:41.110 --> 00:03:42.120
greater than.

00:03:42.120 --> 00:03:45.100
So the smallest integer, n, now
has to be greater than 6,

00:03:45.100 --> 00:03:46.280
so n has to be 7.

00:03:46.280 --> 00:03:47.530
So that's choice B.

00:03:50.070 --> 00:03:50.850
And don't get tricked there.

00:03:50.850 --> 00:03:53.660
When you get 6 as your answer,
you might want to pick A, but

00:03:53.660 --> 00:03:55.100
n has to be greater than 6.

00:03:55.100 --> 00:03:56.420
So it's essentially the
smallest integer

00:03:56.420 --> 00:04:00.220
greater than 6, is 7.

00:04:00.220 --> 00:04:03.940
Problem 108.

00:04:03.940 --> 00:04:08.370
If x and y are different prime
numbers, each greater than 2,

00:04:08.370 --> 00:04:11.740
which of the following
must be true?

00:04:11.740 --> 00:04:15.290
Each greater than 2, they're
both prime numbers.

00:04:15.290 --> 00:04:28.330
So statement number 1: x
plus y cannot equal 91.

00:04:28.330 --> 00:04:29.370
Let me think about that.

00:04:29.370 --> 00:04:33.420
If x are y are different prime
numbers, each greater than 2,

00:04:33.420 --> 00:04:36.602
which of the following
must be true?

00:04:36.602 --> 00:04:40.822
x plus y cannot equal 91.

00:04:40.822 --> 00:04:42.900
Let me think about that
a little bit.

00:04:42.900 --> 00:04:47.310
That means that I can't have two
prime numbers that, when I

00:04:47.310 --> 00:04:50.640
add them together, equal 91.

00:04:50.640 --> 00:04:52.000
Well, let's think about it.

00:04:52.000 --> 00:04:55.730
They're greater than 2, and
then they're prime.

00:04:55.730 --> 00:04:58.500
So what do we know about all
prime numbers greater than 2?

00:04:58.500 --> 00:05:00.730
Well, we know that
they're both odd.

00:05:00.730 --> 00:05:02.880
All prime numbers are odd.

00:05:02.880 --> 00:05:04.830
2 is the only even
prime number.

00:05:04.830 --> 00:05:07.500
So both of these
have to be odd.

00:05:07.500 --> 00:05:14.080
So if I add an odd plus
an odd, what do I get?

00:05:14.080 --> 00:05:15.900
One way to think about it, this
might be second nature to

00:05:15.900 --> 00:05:18.530
you at this point, but an odd
number can be written in the

00:05:18.530 --> 00:05:22.500
form, 2 times some
integer plus 1.

00:05:22.500 --> 00:05:25.900
It's some even integer, some
even number plus 1.

00:05:25.900 --> 00:05:28.190
So I could write the first
odd number like that.

00:05:28.190 --> 00:05:31.930
Maybe I'll say this is x.

00:05:31.930 --> 00:05:33.050
I'm just saying that x is odd.

00:05:33.050 --> 00:05:35.370
I'm not saying that x is prime,
but if you're larger

00:05:35.370 --> 00:05:37.690
than 2 and you're prime,
you're going to be odd.

00:05:37.690 --> 00:05:39.110
Otherwise, you're
divisible by 2.

00:05:39.110 --> 00:05:42.250
And let's say y is equal
to 2 times some other

00:05:42.250 --> 00:05:43.810
integer, plus 1.

00:05:43.810 --> 00:05:48.910
So if you add two odd numbers
together, you get 2 times k,

00:05:48.910 --> 00:05:55.470
plus m, plus 2, which
is an even number.

00:05:55.470 --> 00:06:00.780
When you add two odd numbers,
you get an even number.

00:06:00.780 --> 00:06:04.380
So if you add two prime numbers
greater than 2, you're

00:06:04.380 --> 00:06:08.110
essentially adding two odd
numbers greater than 2, so

00:06:08.110 --> 00:06:10.110
they have to be equal
to an even number.

00:06:10.110 --> 00:06:12.370
So they can't be equal to 91.

00:06:12.370 --> 00:06:15.180
So statement 1 must be true.

00:06:15.180 --> 00:06:18.220
Statement 2, let's
think about it.

00:06:18.220 --> 00:06:27.880
Statement 2 tells us,
x minus y is even.

00:06:27.880 --> 00:06:30.500
So once again, x and
y are definitely

00:06:30.500 --> 00:06:32.070
going to be odd numbers.

00:06:32.070 --> 00:06:34.360
When you add them, you
get an even number.

00:06:34.360 --> 00:06:35.440
What happens when you
subtract them?

00:06:35.440 --> 00:06:41.690
If I were to subtract y from
x, what would I get here?

00:06:41.690 --> 00:06:50.810
If I were to subtract it, I
would get 2 times k, minus m,

00:06:50.810 --> 00:06:52.780
and then the 1 minus 1.

00:06:52.780 --> 00:06:57.030
But this k minus m is going
to still be an integer.

00:06:57.030 --> 00:06:58.370
These are two integers,
so this is still

00:06:58.370 --> 00:06:58.910
going to be an integer.

00:06:58.910 --> 00:07:02.340
And they're different prime
numbers, so k and m are going

00:07:02.340 --> 00:07:04.820
to be different, so this isn't
going to be equal 0.

00:07:04.820 --> 00:07:08.200
So the difference has got
to be an even integer.

00:07:11.810 --> 00:07:14.500
It might be a negative even
integer, because we don't know

00:07:14.500 --> 00:07:18.180
that x is necessarily larger
than y, but we know that it

00:07:18.180 --> 00:07:21.480
has to be an even integer.

00:07:21.480 --> 00:07:24.050
So statement 2 is
also correct.

00:07:24.050 --> 00:07:25.300
Now statement 3.

00:07:29.154 --> 00:07:37.300
x divided by y is
not an integer.

00:07:37.300 --> 00:07:41.600
Well, we know that x is prime,
which means that it only has

00:07:41.600 --> 00:07:42.980
two factors, 1 and itself.

00:07:42.980 --> 00:07:44.560
And we know that x isn't 2.

00:07:44.560 --> 00:07:47.110
We know that y is another
prime number.

00:07:47.110 --> 00:07:51.530
Now, if this statement right
here-- if x divided by y-- if

00:07:51.530 --> 00:07:54.890
that were an integer, then
that means that y divides

00:07:54.890 --> 00:07:56.430
evenly into x.

00:07:56.430 --> 00:07:59.050
Which would mean that
y is a factor of x.

00:07:59.050 --> 00:08:02.380
Which would mean that x has
a factor other than 1 and

00:08:02.380 --> 00:08:04.980
itself, which would mean
that x is not prime.

00:08:04.980 --> 00:08:08.430
But they tell us it x is a prime
number, and y isn't just

00:08:08.430 --> 00:08:10.240
the number 1.

00:08:10.240 --> 00:08:13.280
They're both prime numbers and
they're each greater than 2,

00:08:13.280 --> 00:08:15.050
so y is definitely not 1.

00:08:15.050 --> 00:08:24.440
So in order for this to be an
integer, x could not be prime.

00:08:24.440 --> 00:08:29.260
Since x is prime, and y is some
number greater than 2, we

00:08:29.260 --> 00:08:30.990
know that this is
not an integer.

00:08:30.990 --> 00:08:33.679
So all three of these statements
have to be true.

00:08:33.679 --> 00:08:41.620
And that's statement
E, answer E.

00:08:41.620 --> 00:08:42.380
Problem 109.

00:08:42.380 --> 00:08:44.100
Let me do it in a
different color.

00:08:47.790 --> 00:08:52.100
All of the following have the
same value except-- OK, so

00:08:52.100 --> 00:08:54.460
we're just going to try to do
the math real fast here.

00:08:54.460 --> 00:09:01.300
1 plus 2, plus 3, plus 4, plus
5, all of that over 3.

00:09:01.300 --> 00:09:06.970
1 plus 2 is 3, 3 plus 3 is 6,
6 plus 4 is 10, 10 plus 5 is

00:09:06.970 --> 00:09:10.170
15, divided by 3, this
is equal to 5.

00:09:10.170 --> 00:09:18.740
B, 1/3, times 1, plus 1,
plus 1, plus 1, plus 1.

00:09:18.740 --> 00:09:23.886
That's 1/3 times-- this is 5,
so this is equal to 5/3.

00:09:23.886 --> 00:09:26.010
So after we see the next answer
we should know where

00:09:26.010 --> 00:09:27.070
we're going.

00:09:27.070 --> 00:09:31.180
So the next answer, we have 1/3
plus itself five times.

00:09:31.180 --> 00:09:36.350
1/3 plus 1/3, plus 1/3,
plus 1/3, plus 1/3.

00:09:36.350 --> 00:09:40.210
Well, that's five 1/3s, so
that's also equal to 5/3.

00:09:40.210 --> 00:09:45.510
So we should immediately be able
to recognize that this is

00:09:45.510 --> 00:09:46.630
different than the other two.

00:09:46.630 --> 00:09:49.100
And assuming there's only one
answer-- they say all the

00:09:49.100 --> 00:09:51.720
following have the same value
except-- I think we've done

00:09:51.720 --> 00:09:53.895
enough math, because we
know that B and C are

00:09:53.895 --> 00:09:55.570
the the same value.

00:09:55.570 --> 00:09:57.553
And A is already different,
so A is probably

00:09:57.553 --> 00:09:58.340
going to be our choice.

00:09:58.340 --> 00:09:59.805
And if we kept going,
D and E are going to

00:09:59.805 --> 00:10:00.300
have the same value.

00:10:00.300 --> 00:10:02.050
And just eyeballing
it, it does look

00:10:02.050 --> 00:10:04.270
like that's the case.

00:10:04.270 --> 00:10:06.880
It definitely looks like that's
the case, because if

00:10:06.880 --> 00:10:10.700
you look at the other two
problems, you still get 5/3.

00:10:10.700 --> 00:10:13.110
And E looks confusing, but the
fact that A is already

00:10:13.110 --> 00:10:16.860
different than B and C tells
you that A is the answer.

00:10:16.860 --> 00:10:19.520
And I'll see you in
the next video.


GMAT: Math 19

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:02.800
Problem 100.

00:00:02.800 --> 00:00:07.190
They draw this graph, which I
guess I should also draw.

00:00:07.190 --> 00:00:08.060
So they draw it pretty big.

00:00:08.060 --> 00:00:11.810
So I'll draw it pretty
big as well.

00:00:11.810 --> 00:00:15.530
They have a line that goes from
the center to there, it's

00:00:15.530 --> 00:00:17.000
a pie graph.

00:00:17.000 --> 00:00:20.650
Then they call that category
like that, and another

00:00:20.650 --> 00:00:24.160
category that looks something
like that, and then they have

00:00:24.160 --> 00:00:28.600
a final category that looks
something like that.

00:00:28.600 --> 00:00:35.010
They say that this is
the highway trust

00:00:35.010 --> 00:00:38.390
fund, which is 72%.

00:00:38.390 --> 00:00:41.056
They say x% is a tire tax.

00:00:44.140 --> 00:00:47.480
This is 12%, which
is a gas tax.

00:00:50.160 --> 00:00:57.760
Then you have x plus 4%, which
is a tax on trucks.

00:00:57.760 --> 00:01:01.560
According to the graph above,
what percent of the funds for

00:01:01.560 --> 00:01:04.239
highway maintenance-- oh, and
this title says sources of

00:01:04.239 --> 00:01:05.920
funds for highway maintenance.

00:01:05.920 --> 00:01:07.640
So that's what this
whole thing is.

00:01:07.640 --> 00:01:10.850
According to the graph above,
what percent of the funds for

00:01:10.850 --> 00:01:15.220
highway maintenance came
from the tax on tires?

00:01:15.220 --> 00:01:17.080
So we need to solve for x.

00:01:17.080 --> 00:01:19.785
So let's think about it.

00:01:19.785 --> 00:01:26.590
x plus 12 plus x plus 4, that's
this, plus 72 have to

00:01:26.590 --> 00:01:29.860
equal 100%.

00:01:29.860 --> 00:01:35.780
We have two x's, get those out
of the way, and then what is

00:01:35.780 --> 00:01:37.710
this equal to?

00:01:37.710 --> 00:01:42.020
12 plus 4 is 16.

00:01:42.020 --> 00:01:45.430
16 plus 72-- let me
write that down.

00:01:45.430 --> 00:01:46.450
I always make careless
mistakes.

00:01:46.450 --> 00:01:47.380
That's 16.

00:01:47.380 --> 00:01:49.670
16 plus 72 is 88.

00:01:49.670 --> 00:01:54.300
2x plus 88 is equal to 100.

00:01:54.300 --> 00:01:57.620
Then you have 2x is equal
to, subtract 88

00:01:57.620 --> 00:01:59.170
from both sides, 12.

00:01:59.170 --> 00:02:00.540
x is equal to 6.

00:02:00.540 --> 00:02:02.990
That's exactly what we're
looking for, the tire tax.

00:02:02.990 --> 00:02:04.660
So that's 6%.

00:02:04.660 --> 00:02:05.910
That's choice B.

00:02:09.270 --> 00:02:12.410
Drawing it was harder
than solving it.

00:02:12.410 --> 00:02:15.270
Problem 101.

00:02:15.270 --> 00:02:19.640
A poll reveals that the average
arithmetic mean income

00:02:19.640 --> 00:02:24.900
of 10 households is $25,000.

00:02:24.900 --> 00:02:29.630
So we could say, average
of 10 is equal to 25K.

00:02:29.630 --> 00:02:34.570
If 6 of the households have
income of $30,000 each, what

00:02:34.570 --> 00:02:39.400
is the average of the
other 4 households?

00:02:39.400 --> 00:02:43.090
So if we were to average all of
them, we could say it would

00:02:43.090 --> 00:02:49.050
be 6 times the households that
are 30,000, so 6 times 30K,

00:02:49.050 --> 00:02:53.400
plus 4 times the average
of the other 4.

00:02:53.400 --> 00:02:55.840
So let's say that's what we're
trying to solve for, 4 times

00:02:55.840 --> 00:02:57.820
the average of the other
4 households.

00:02:57.820 --> 00:03:00.280
This would be the sum of
all of the households.

00:03:00.280 --> 00:03:03.520
If we were to divide that by
10, we're going to get the

00:03:03.520 --> 00:03:06.190
total average of all
10, which is 25.

00:03:06.190 --> 00:03:07.910
Hopefully that makes sense.

00:03:07.910 --> 00:03:11.010
You have 6 households making
30K, so the sum of those

00:03:11.010 --> 00:03:13.390
households is going
to be 6 times 30.

00:03:13.390 --> 00:03:16.440
Then the sum of the other 4
households is going to be 4

00:03:16.440 --> 00:03:17.850
times their average.

00:03:17.850 --> 00:03:19.440
That's what they're asking
us for, what

00:03:19.440 --> 00:03:20.690
is the average income?

00:03:23.450 --> 00:03:25.700
If you take the sum of all the
households, all 10, you divide

00:03:25.700 --> 00:03:27.310
it by 10, you get
their average.

00:03:27.310 --> 00:03:31.030
So you get 6 times 30-- I don't
want to make this look

00:03:31.030 --> 00:03:32.340
like a variable,
that's 30,000.

00:03:32.340 --> 00:03:34.120
We know we're dealing
in thousands.

00:03:34.120 --> 00:03:40.220
So 6 times 30 is 180 plus 4x
is equal to 10 times 25, is

00:03:40.220 --> 00:03:43.930
equal to 250.

00:03:43.930 --> 00:03:47.770
4x is equal to, what's
250 minus 180?

00:03:47.770 --> 00:03:52.200
That's 50, and you have another
20, it's equal to 70.

00:03:52.200 --> 00:03:57.910
Then you have x is equal to,
let's see, 4 goes into 70,

00:03:57.910 --> 00:04:02.240
130, goes into it 7,
7 times 4 is 28.

00:04:02.240 --> 00:04:06.070
Goes into 20 5 times,
and the decimal.

00:04:06.070 --> 00:04:12.260
So 17.5, and we're dealing with
thousands, so $17,500.

00:04:12.260 --> 00:04:14.970
That's choice C.

00:04:14.970 --> 00:04:21.200
Problem 102.

00:04:21.200 --> 00:04:34.350
If T is equal to 5/9 times K
minus 32, and T is equal to

00:04:34.350 --> 00:04:37.020
290, then what is K equal to?

00:04:37.020 --> 00:04:39.660
This is kind of one of
those speed problems.

00:04:39.660 --> 00:04:47.260
So we have 290 is equal to
5/9 times K minus 32.

00:04:47.260 --> 00:04:53.700
So multiply both sides by 9/5,
you get 9/5 times 290 is equal

00:04:53.700 --> 00:04:56.380
to K minus 32.

00:04:56.380 --> 00:05:01.070
So what's 290 divided by 5?

00:05:01.070 --> 00:05:07.130
5 goes into 29 5 times,
25, 40, 58 times.

00:05:07.130 --> 00:05:13.450
So it's 9 times 58 is
equal to K minus 32.

00:05:13.450 --> 00:05:19.750
58 times 9, 8 times 9 is 72.

00:05:19.750 --> 00:05:24.950
5 times 9 is 45 plus 7 is 52.

00:05:24.950 --> 00:05:26.210
522.

00:05:26.210 --> 00:05:32.130
So 522 is equal to K minus 32.

00:05:32.130 --> 00:05:44.020
So K is equal to 522 plus 32,
which is equal to 554, Which

00:05:44.020 --> 00:05:44.580
is choice D.

00:05:44.580 --> 00:05:47.100
After doing all that math, I
was afraid that I'd made a

00:05:47.100 --> 00:05:47.880
careless mistake.

00:05:47.880 --> 00:05:50.300
But it's reassuring
to see that 554

00:05:50.300 --> 00:05:51.660
is one of the choices.

00:05:51.660 --> 00:05:57.160
Problem 103.

00:05:57.160 --> 00:06:01.020
The water from one outlet
flowing at a constant rate can

00:06:01.020 --> 00:06:04.620
fill a swimming pool
in 9 hours.

00:06:04.620 --> 00:06:11.610
So you could say 1 pool per
9 hours from 1 outlet.

00:06:11.610 --> 00:06:14.240
The water from a second outlet
flowing at a constant rate can

00:06:14.240 --> 00:06:15.820
fill the same pool in 5 hours.

00:06:15.820 --> 00:06:22.190
So 1 pool in 5 hours, that's
the second one.

00:06:22.190 --> 00:06:25.630
If both outlets are used at the
same time, approximately

00:06:25.630 --> 00:06:29.850
what is the number of hours
required to fill the pool?

00:06:29.850 --> 00:06:34.750
So essentially, we can just
add their two rates.

00:06:34.750 --> 00:06:41.870
This one does 1/9 pool per hour,
plus 1/5 pool per hour.

00:06:41.870 --> 00:06:43.570
I just rewrote those two.

00:06:43.570 --> 00:06:45.295
So what's 1/9 plus 1/5?

00:06:45.295 --> 00:06:47.500
Let's find a common
denominator.

00:06:47.500 --> 00:06:51.120
45 I think is the first
common denominator.

00:06:51.120 --> 00:06:53.710
1/9 is 5 over 45.

00:06:53.710 --> 00:06:55.810
1/5 is 9 over 45.

00:06:55.810 --> 00:07:06.840
So together, they fill 5 plus
9 is 14/45 pool per hour.

00:07:06.840 --> 00:07:11.100
If you want to fill exactly
1 pool, so it's just like

00:07:11.100 --> 00:07:14.100
production is equal to
rate times time.

00:07:14.100 --> 00:07:17.450
So now your production is, you
want to fill 1 pool, and you

00:07:17.450 --> 00:07:21.150
can fill it at 14 over
45 pools per

00:07:21.150 --> 00:07:25.750
hour, what is the time?

00:07:25.750 --> 00:07:30.190
Well the time is going to be
equal to 1/1 pool divided by

00:07:30.190 --> 00:07:40.400
14/45 pools per hour, which is
equal to, 45/14-- 1 over a

00:07:40.400 --> 00:07:41.850
fraction is just the
inverse of that

00:07:41.850 --> 00:07:46.350
fraction-- so 45/14 hours.

00:07:46.350 --> 00:07:50.553
How many times does
14 going into 45?

00:07:50.553 --> 00:07:54.190
14 goes into 45,
is it 3 times?

00:07:54.190 --> 00:07:58.935
3 times 14 is 12, is 42.

00:07:58.935 --> 00:08:03.030
Then you have another 30.

00:08:03.030 --> 00:08:05.320
Well, I already see one of the
choices, 3 point something.

00:08:05.320 --> 00:08:08.310
But 14 goes into 30 2 times.

00:08:08.310 --> 00:08:10.650
2 times 14 is 28.

00:08:10.650 --> 00:08:13.250
Let's see if one of our choices
is already-- yes,

00:08:13.250 --> 00:08:15.090
choice D is already close
enough to what we did.

00:08:15.090 --> 00:08:16.570
We don't have to do
any more math.

00:08:16.570 --> 00:08:18.150
So we get 3.2 hours.

00:08:18.150 --> 00:08:20.900
If we'd kept dividing,
we'd get 3.21.

00:08:20.900 --> 00:08:22.940
That's choice D.

00:08:22.940 --> 00:08:24.200
It's just a little confusing.

00:08:24.200 --> 00:08:25.780
The important thing
is to always get

00:08:25.780 --> 00:08:26.910
it in terms of rate.

00:08:26.910 --> 00:08:29.700
Add the rate, and just remember
distance is equal to

00:08:29.700 --> 00:08:32.400
rate times time, or production
is equal to rate times time.

00:08:32.400 --> 00:08:36.360
Don't let all these fractions
daunt you too much.

00:08:36.360 --> 00:08:37.602
Next problem.

00:08:37.602 --> 00:08:39.659
I'm almost out of space.

00:08:39.659 --> 00:08:42.590
Problem 104.

00:08:42.590 --> 00:08:47.550
Diana bought a stereo for $530,
which was the retail

00:08:47.550 --> 00:08:50.890
price, plus a 6% sales tax.

00:08:50.890 --> 00:08:55.470
So that is equal to 1.06 times
the retail, 6% percent more

00:08:55.470 --> 00:08:56.570
than the retail.

00:08:56.570 --> 00:08:59.280
How much could she have saved
if she had bought the stereo

00:08:59.280 --> 00:09:02.890
at the same retail price in a
neighboring state, where she

00:09:02.890 --> 00:09:05.250
would have paid a
sales tax of 5%?

00:09:07.980 --> 00:09:12.880
So essentially we want to
figure out how much she

00:09:12.880 --> 00:09:14.760
would've saved.

00:09:14.760 --> 00:09:19.460
So let's figure out what
the retail price was.

00:09:19.460 --> 00:09:24.950
So retail is equal to
530 divided by 1.06.

00:09:24.950 --> 00:09:32.890
1.06 goes into 530, it's a
decimal, add two 0's, shift

00:09:32.890 --> 00:09:36.300
the decimal over to
the right twice.

00:09:36.300 --> 00:09:41.270
106 goes into it
5 times, right?

00:09:41.270 --> 00:09:47.700
106 goes into 530 5 times.

00:09:47.700 --> 00:09:56.330
5 times 106, it's 530.

00:09:56.330 --> 00:09:58.220
OK, so the math here
was simple.

00:09:58.220 --> 00:09:59.280
so you have two 0's.

00:09:59.280 --> 00:10:07.310
So the retail price is $500.

00:10:07.310 --> 00:10:10.280
So she paid $30 in
tax, fair enough.

00:10:10.280 --> 00:10:15.270
If she went to the neighboring
state and paid 5%, how much

00:10:15.270 --> 00:10:16.980
would she have paid in taxes?

00:10:16.980 --> 00:10:24.800
So times 0.05, 5 times 5, well 5
times 500 is 2,500, plus two

00:10:24.800 --> 00:10:27.000
decimals, so it's $25 in tax.

00:10:27.000 --> 00:10:27.980
So in the neighboring
state, she

00:10:27.980 --> 00:10:30.250
would've paid $25 in taxes.

00:10:30.250 --> 00:10:32.350
She paid $30 in taxes.

00:10:32.350 --> 00:10:34.420
So she would have saved $5.

00:10:34.420 --> 00:10:36.855
So the choice is D.

00:10:41.170 --> 00:10:43.440
Let's just try to fit one more
problem in here, just so we

00:10:43.440 --> 00:10:44.690
finish this page.

00:10:46.720 --> 00:10:49.720
They say, if a square mirror has
a 20 inch diagonal, what

00:10:49.720 --> 00:10:53.300
was the approximate perimeter
of the mirror in inches?

00:10:53.300 --> 00:10:54.550
Let's try to do that.

00:10:56.920 --> 00:11:00.060
That's not what I want to do.

00:11:00.060 --> 00:11:02.945
I have a square mirror,
it has a diagonal.

00:11:07.610 --> 00:11:11.340
That diagonal, they're telling
us, is 20 inches.

00:11:11.340 --> 00:11:15.330
It's a square, so all the
sides are the same.

00:11:15.330 --> 00:11:16.360
They want to know
its perimeter.

00:11:16.360 --> 00:11:18.430
So they want to know essentially
what 4x is equal

00:11:18.430 --> 00:11:21.190
to if we add up all the sides.

00:11:21.190 --> 00:11:22.710
So what is x?

00:11:22.710 --> 00:11:24.260
Well we just use Pythagorean
theorem.

00:11:24.260 --> 00:11:25.440
This is a right angle.

00:11:25.440 --> 00:11:31.010
So we know that x squared plus
x squared is equal to 20

00:11:31.010 --> 00:11:32.710
squared, is equal to
the hypotenuse

00:11:32.710 --> 00:11:34.520
squared, is equal to 400.

00:11:34.520 --> 00:11:37.670
2x squared is equal to 400.

00:11:37.670 --> 00:11:41.065
x squared is equal to 200.

00:11:41.065 --> 00:11:45.680
x is equal to the square
root of 200.

00:11:45.680 --> 00:11:46.540
They want to know
the perimeter.

00:11:46.540 --> 00:11:48.410
Well, it's going to
be 4 times that.

00:11:48.410 --> 00:11:51.820
So the perimeter is going to be
equal to 4 times the square

00:11:51.820 --> 00:11:54.570
root of 200.

00:11:54.570 --> 00:11:56.990
Now this becomes an
approximation problem.

00:11:56.990 --> 00:11:58.350
So let's think about that.

00:11:58.350 --> 00:12:05.960
The square root of
200 is about 15.

00:12:05.960 --> 00:12:09.670
15 times 15 is 225.

00:12:09.670 --> 00:12:12.810
Yeah, I think 15 would be a
pretty good approximation.

00:12:12.810 --> 00:12:14.330
14 would also work.

00:12:14.330 --> 00:12:17.875
14, when you square
it, is 196.

00:12:17.875 --> 00:12:24.690
14 times 14, 4 times 16 is 56.

00:12:24.690 --> 00:12:26.980
Then you have 140.

00:12:26.980 --> 00:12:30.910
So 14 is about as close as
you're going to get to 200.

00:12:30.910 --> 00:12:33.620
It's going to be someplace
between 14 and 15.

00:12:33.620 --> 00:12:38.850
So 14 times 4 is 40.

00:12:38.850 --> 00:12:44.010
So 4 times 4 is 16, it's 56.

00:12:44.010 --> 00:12:45.680
So it's going to be something
higher than 56.

00:12:45.680 --> 00:12:47.850
If we pick 14, it's 56.

00:12:47.850 --> 00:12:49.840
If we pick 15, it's 60.

00:12:49.840 --> 00:12:52.600
So someplace between 56 and 60
is going to be the perimeter

00:12:52.600 --> 00:12:53.860
of that mirror.

00:12:53.860 --> 00:12:55.900
So it's one of the
choices B, 60.

00:12:55.900 --> 00:12:57.360
So that's going to be
the closest one.

00:12:57.360 --> 00:12:59.520
So I'd go with B.

00:12:59.520 --> 00:13:01.670
I'll see you in the
next video.


GMAT: Math 18

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:05.030
We're on problem 96.

00:00:05.030 --> 00:00:17.310
It says 1/2 plus, open brackets,
2/3 times 3/8

00:00:17.310 --> 00:00:23.460
divided by 4 minus 9/16.

00:00:23.460 --> 00:00:25.570
Do you know order of operations,
and can you deal

00:00:25.570 --> 00:00:26.510
with fractions?

00:00:26.510 --> 00:00:31.380
2/3 times 3/8, the 3's cancel
out, and the 2 and the 8

00:00:31.380 --> 00:00:34.610
become 1/4.

00:00:34.610 --> 00:00:43.910
1/4 divided by 4, let me write
this, 1/2 plus 1/4 divided by

00:00:43.910 --> 00:00:50.300
4, that's the same thing as
times 1/4, minus 9 over 16.

00:00:50.300 --> 00:00:57.990
So that is equal to 1/2
plus 1/16 minus 9/16.

00:00:57.990 --> 00:01:00.080
And now it doesn't matter what
order we do it in, but since

00:01:00.080 --> 00:01:01.980
everything's in 1/16ths, so let
me write this in 1/16ths.

00:01:01.980 --> 00:01:08.780
So this is 8/16 plus
1/16 minus 9/16.

00:01:08.780 --> 00:01:12.430
So that's equal to 8 plus 1
minus 9 all over that 16.

00:01:12.430 --> 00:01:15.480
That's 9 minus 9,
that equals 0.

00:01:15.480 --> 00:01:19.960
Choice E, all that
work to get to 0.

00:01:19.960 --> 00:01:25.630
Next problem, 97.

00:01:25.630 --> 00:01:29.560
Water consists of hydrogen and
oxygen, and the approximate

00:01:29.560 --> 00:01:39.890
ratio by mass of hydrogen to
oxygen is equal to 2:16.

00:01:39.890 --> 00:01:46.230
Approximately how many grams
of oxygen are there in 144

00:01:46.230 --> 00:01:48.100
grams of water?

00:01:48.100 --> 00:01:51.000
OK, 144 grams of water, that
means that the oxygen and

00:01:51.000 --> 00:01:59.020
hydrogen is equal to 144.

00:01:59.020 --> 00:02:01.420
They say 144 grams of water,
not when you have

00:02:01.420 --> 00:02:03.120
144 grams of hydrogen.

00:02:03.120 --> 00:02:05.860
So when you add these together,
you have 144.

00:02:05.860 --> 00:02:09.340
And then they tell us that
the ratio of hydrogen

00:02:09.340 --> 00:02:11.450
to oxygen is 2:16.

00:02:11.450 --> 00:02:14.060
Or another way to write it,
you can say hydrogen over

00:02:14.060 --> 00:02:17.230
oxygen is equal to 2/16, which
is another way of saying 1/8.

00:02:20.920 --> 00:02:22.780
And they want to know how many
grams of oxygen, so we want to

00:02:22.780 --> 00:02:26.740
solve for oxygen, so we want
to substitute for hydrogen.

00:02:26.740 --> 00:02:32.720
So we could say hydrogen is
equal to 1/8th the oxygen.

00:02:32.720 --> 00:02:35.940
I just multiplied both
sides times oxygen.

00:02:35.940 --> 00:02:39.630
So if hydrogen is equal to 1/8
oxygen, you have oxygen plus

00:02:39.630 --> 00:02:42.060
1/8 oxygen, let me make
sure that's right.

00:02:42.060 --> 00:02:47.680
Yeah, hydrogen to oxygen
is equal to 144.

00:02:47.680 --> 00:02:51.275
And if you view this as 8/8, 1
oxygen is the same thing as

00:02:51.275 --> 00:02:55.800
8/8, so you get 9/8 oxygen
is equal to 144.

00:02:55.800 --> 00:03:03.430
So you get oxygen is equal
to 144 times 8 over 9.

00:03:03.430 --> 00:03:05.662
Does 9 go into 144?

00:03:05.662 --> 00:03:08.610
I think it should, let me see
how many times does 9-- 3

00:03:08.610 --> 00:03:09.540
definitely does.

00:03:09.540 --> 00:03:18.420
This is 12 times 12, so 3 will
go into 144 48 times.

00:03:18.420 --> 00:03:25.370
9 goes into 144 1,
9, 54, 16 times.

00:03:25.370 --> 00:03:30.290
So this is equal to 16 times
8 over 1, which is what?

00:03:30.290 --> 00:03:35.650
That's equal to 250-- no,
that can't be right.

00:03:39.000 --> 00:03:51.670
No, you can't have more grams of
oxygen than you do-- let me

00:03:51.670 --> 00:03:55.490
see, I've made a mistake
someplace.

00:03:55.490 --> 00:04:01.080
Hydrogen to oxygen is equal
to 2:16, which is 1:8.

00:04:01.080 --> 00:04:06.250
Oxygen plus hydrogen is equal
to 144, you get 9/8.

00:04:06.250 --> 00:04:08.730
Oh, I see my mistake.

00:04:08.730 --> 00:04:10.290
That's good.

00:04:10.290 --> 00:04:11.915
I've been making a lot of
careless mistakes, maybe I

00:04:11.915 --> 00:04:17.450
need to take a rest. 9/8 oxygen
is equal to 144, so

00:04:17.450 --> 00:04:22.570
oxygen is equal to 144 times
9 over 8-- sorry,

00:04:22.570 --> 00:04:24.400
times 8 over 9.

00:04:24.400 --> 00:04:29.010
I was doing it right, so
where was my mistake?

00:04:29.010 --> 00:04:34.700
So 144 times 8 over 9, that's
equal to-- oh, I just made a

00:04:34.700 --> 00:04:36.510
mistake at the very last step.

00:04:36.510 --> 00:04:41.510
9 goes into 144 16 times.

00:04:41.510 --> 00:04:45.620
16 times 8 isn't
256, it's 128.

00:04:45.620 --> 00:04:46.790
That was my mistake.

00:04:46.790 --> 00:04:49.865
So there's 128 grams
of oxygen.

00:04:49.865 --> 00:04:53.290
I just made a mistake right
at that last step.

00:04:53.290 --> 00:04:54.650
All right.

00:04:54.650 --> 00:04:56.010
And that's choice D.

00:04:56.010 --> 00:05:00.960
Question 98.

00:05:00.960 --> 00:05:11.670
If x times 2x plus 1 is equal
to 0, and they also tell us

00:05:11.670 --> 00:05:23.250
that x plus 1/2 times 2x minus
3 is equal to 0, then x is

00:05:23.250 --> 00:05:24.620
equal to what?

00:05:24.620 --> 00:05:26.010
So this is interesting.

00:05:26.010 --> 00:05:29.580
x times 2x plus 1 is equal to 0,
that means that either x or

00:05:29.580 --> 00:05:33.380
2x plus 1 has to
be equal to 0.

00:05:33.380 --> 00:05:36.410
So one of these has
to be equal to 0.

00:05:36.410 --> 00:05:39.310
But then x plus 1/2 times 2x
plus 3, this tell us that one

00:05:39.310 --> 00:05:40.870
of these have to
be equal to 0.

00:05:40.870 --> 00:05:43.860
So if x is equal to 0, then this
has to be the one that's

00:05:43.860 --> 00:05:44.530
equal to 0.

00:05:44.530 --> 00:05:48.170
So either this and that
is equal to zero,

00:05:48.170 --> 00:05:51.460
or-- do you see that?

00:05:51.460 --> 00:05:54.070
Well I guess both of these
could be equal to 0.

00:05:54.070 --> 00:05:55.950
Let me multiply them out.

00:05:55.950 --> 00:05:57.320
Maybe I'm going down
the wrong path.

00:05:57.320 --> 00:06:01.610
If I multiply these out,
I get 2x squared plus

00:06:01.610 --> 00:06:05.306
2 is equal to 0.

00:06:05.306 --> 00:06:08.600
If I multiply these out, I get
x times 2x, which is 2x

00:06:08.600 --> 00:06:14.070
squared, x times minus 3
minus 3x, 1/2 times 2x,

00:06:14.070 --> 00:06:15.400
so it's plus x.

00:06:15.400 --> 00:06:19.310
And then 1/2 times minus
3 minus 3/2 is equal

00:06:19.310 --> 00:06:23.790
to 0, so I get what?

00:06:23.790 --> 00:06:29.675
2x squared minus 2x.

00:06:32.420 --> 00:06:40.290
3x plus x is minus 2x, minus
3/2 is equal to 0.

00:06:40.290 --> 00:06:42.430
Actually I think I made a
mistake, x times 2x plus 1,

00:06:42.430 --> 00:06:43.700
that's not right.

00:06:43.700 --> 00:06:51.440
That's 2x squared plus
x is equal to 0.

00:06:51.440 --> 00:06:54.300
Fair enough.

00:06:54.300 --> 00:06:58.970
We know 2x squared, so we can
just write that 2x squared is

00:06:58.970 --> 00:07:02.340
equal to minus x, and
then substitute this

00:07:02.340 --> 00:07:03.260
information in here.

00:07:03.260 --> 00:07:07.900
So we get minus x, 2x squared
is equal to minus x.

00:07:07.900 --> 00:07:13.710
Minus x minus 2x minus
3/2 is equal to 0.

00:07:13.710 --> 00:07:20.820
So you get minus 3x, let's
add 3/2 to both

00:07:20.820 --> 00:07:23.040
sides, is equal to 3/2.

00:07:23.040 --> 00:07:27.700
And then divide both sides by
minus 3, you get x is equal to

00:07:27.700 --> 00:07:33.960
3/2 times minus 1/3, so that
equal to minus 1/2.

00:07:33.960 --> 00:07:35.420
And that is choice B.

00:07:38.000 --> 00:07:46.470
Next problem, 99.

00:07:46.470 --> 00:07:48.820
On a scale that measures the
intensity of a certain

00:07:48.820 --> 00:07:56.650
phenomenon, a reading of n
plus 1 corresponds to an

00:07:56.650 --> 00:07:59.550
intensity that is 10 times the
intensity corresponding to a

00:07:59.550 --> 00:08:01.620
reading of n.

00:08:01.620 --> 00:08:04.540
So n plus 1 is equal
to 10 times the

00:08:04.540 --> 00:08:06.580
reading of just an n.

00:08:06.580 --> 00:08:09.800
On that scale, the intensity
corresponding to a reading of

00:08:09.800 --> 00:08:15.530
8 is how many times as great as
the intensity corresponding

00:08:15.530 --> 00:08:18.180
to reading of 3?

00:08:18.180 --> 00:08:21.580
So if you call a reading
of 3, I don't know,

00:08:21.580 --> 00:08:22.620
let's call that a 1.

00:08:22.620 --> 00:08:27.430
A reading of 4 is going to be
10 times the reading of 3.

00:08:27.430 --> 00:08:30.390
A reading of 5 is going to
be 10 times that, so 100.

00:08:30.390 --> 00:08:34.760
A reading of 6 is going to be
10 times that, or 1,000.

00:08:34.760 --> 00:08:38.470
A reading of 7 is going to be
10 times that, or 10,000.

00:08:38.470 --> 00:08:43.100
And then a reading of 8 would be
10 times that, so it would

00:08:43.100 --> 00:08:48.610
be 100,000.

00:08:48.610 --> 00:08:50.510
Which is not one
of the choices.

00:08:50.510 --> 00:08:52.450
Let me make sure I'm
reading that.

00:08:52.450 --> 00:08:53.790
On a scale that measures the
intensity of certain

00:08:53.790 --> 00:08:56.690
phenomenon, a reading of n
plus 1 corresponds to an

00:08:56.690 --> 00:08:59.670
intensity that is 10 times the
intensity corresponding to a

00:08:59.670 --> 00:09:01.750
reading of n.

00:09:01.750 --> 00:09:04.970
So a reading of 4 is 10 times
the intensity of a reading of

00:09:04.970 --> 00:09:08.290
3, and a reading of 5
is 10 times that.

00:09:08.290 --> 00:09:08.820
Fine.

00:09:08.820 --> 00:09:11.690
On that scale, the intensity
corresponding to a reading of

00:09:11.690 --> 00:09:18.440
8 is how many times as great as
the intensity corresponding

00:09:18.440 --> 00:09:20.250
to a reading of 3?

00:09:22.880 --> 00:09:26.220
A reading of 4 would be 10 times
as great, a reading of 5

00:09:26.220 --> 00:09:30.320
would be 100 times as great, a
reading of 6 would be 1,000

00:09:30.320 --> 00:09:31.570
times as great.

00:09:36.330 --> 00:09:40.240
Oh yeah, 100,000, which
is 10 to the fifth.

00:09:40.240 --> 00:09:41.360
10 to the fifth power.

00:09:41.360 --> 00:09:45.140
Five 0's, so that is choice C.

00:09:45.140 --> 00:09:47.640
Confusing myself unnecessarily,

00:09:47.640 --> 00:09:49.090
and I'm out of time.

00:09:49.090 --> 00:09:50.070
See you in the next video.

00:09:50.070 --> 00:09:52.990
Let me stop this one.


GMAT: Math 17

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.010
Problem 92.

00:00:03.010 --> 00:00:06.810
In a certain furniture store,
each week Nancy earns a salary

00:00:06.810 --> 00:00:13.840
of $240, plus 5% of the amount
of total sales that exceeds

00:00:13.840 --> 00:00:15.590
$800 for the week.

00:00:15.590 --> 00:00:21.990
Plus 5% of whatever is greater
than 800 for the week.

00:00:21.990 --> 00:00:25.480
If Nancy earned a total of $450
in a week, what were her

00:00:25.480 --> 00:00:27.210
total sales that week?

00:00:27.210 --> 00:00:32.189
So she earns $450, so what
was her 5% bonus part?

00:00:32.189 --> 00:00:38.800
So 450 minus 240, she
got $210 bonus.

00:00:38.800 --> 00:00:44.440
So 210 represents 5% of
her sales above 800.

00:00:44.440 --> 00:00:48.160
210 is 5% of what number?

00:00:48.160 --> 00:00:54.626
So you could say 210 is equal to
0.05 times x, and x will be

00:00:54.626 --> 00:00:56.850
her sales above 800.

00:00:56.850 --> 00:01:04.530
So x is equal to 210 divided
by 0.05, so 0.05 goes into

00:01:04.530 --> 00:01:09.380
210, add some decimal points.

00:01:09.380 --> 00:01:12.790
That's the same thing as 5 going
into, put the decimal

00:01:12.790 --> 00:01:15.100
point there, 21,000.

00:01:15.100 --> 00:01:21.570
So 5 goes into 21 4 times, 4
times 5 is 20, bring down 1.

00:01:21.570 --> 00:01:24.830
5 goes into 10 2 times,
2 times 5 is 10.

00:01:24.830 --> 00:01:26.010
We're done.

00:01:26.010 --> 00:01:28.790
So her sales were 4,200, but
that's not her total sales.

00:01:28.790 --> 00:01:32.130
Remember, she only gets 5%
on her sales above 800.

00:01:32.130 --> 00:01:36.170
So she sold $4,200 more
than $800 that week.

00:01:36.170 --> 00:01:38.640
So her total sales are going
to be 4,200 plus

00:01:38.640 --> 00:01:43.640
800, which is $5,000.

00:01:43.640 --> 00:01:46.670
And that's choice E.

00:01:46.670 --> 00:01:54.180
Next question, 93.

00:01:54.180 --> 00:01:55.420
They give us two lists.

00:01:55.420 --> 00:02:02.580
So that's list one, and they
give it 3, 6, 8, and 19, and

00:02:02.580 --> 00:02:11.860
they give us list two, they say
it's x, 3, 6, 8, and 19.

00:02:11.860 --> 00:02:14.720
It's actually the same list,
except they have an x here.

00:02:14.720 --> 00:02:18.830
If the median of the numbers in
list one above is equal to

00:02:18.830 --> 00:02:23.340
the median of the numbers in
list two above, what is the

00:02:23.340 --> 00:02:25.260
value of x?

00:02:25.260 --> 00:02:28.320
So what's the median
of this list?

00:02:28.320 --> 00:02:30.350
Since we have an even number of
numbers, this would be the

00:02:30.350 --> 00:02:32.810
average of the middle two
numbers, so what's the middle

00:02:32.810 --> 00:02:35.440
two numbers right here?

00:02:35.440 --> 00:02:39.470
The middle two numbers are 6 and
8, the average is 6 plus 8

00:02:39.470 --> 00:02:40.620
divided by 2, it's 7.

00:02:40.620 --> 00:02:41.870
So the median is 7.

00:02:45.310 --> 00:02:49.250
So they're telling us that the
median of this list is also

00:02:49.250 --> 00:02:52.750
equal to 7, so where can
I put x so that the

00:02:52.750 --> 00:02:54.070
median doesn't change.

00:02:54.070 --> 00:02:55.640
Well what if x is 7?

00:02:55.640 --> 00:03:02.140
If x is 7, then this list
becomes 3, 6, 7, 8, 19, and

00:03:02.140 --> 00:03:03.940
now the middle number, since
we have odd numbers, is

00:03:03.940 --> 00:03:04.690
actually the middle number.

00:03:04.690 --> 00:03:06.530
We don't have to do any of
this averaging business.

00:03:06.530 --> 00:03:09.740
So now the median is actually
7, if we place 7 there.

00:03:09.740 --> 00:03:12.110
So x equals 7.

00:03:12.110 --> 00:03:14.320
Choice B.

00:03:14.320 --> 00:03:17.140
94.

00:03:17.140 --> 00:03:20.960
In a certain city, 60% of the
registered voters are

00:03:20.960 --> 00:03:22.406
Democrat, and the rest
are Republican.

00:03:28.480 --> 00:03:34.320
So 40% of voters.

00:03:34.320 --> 00:03:37.990
In a mayoral race, if 75% of the
registered voters who are

00:03:37.990 --> 00:03:40.960
Democrats, and 20% of the
registered voters who are

00:03:40.960 --> 00:03:43.480
Republicans are expected to
vote for candidate A, what

00:03:43.480 --> 00:03:45.520
percentage of the registered
voters are expected to vote

00:03:45.520 --> 00:03:47.290
for candidate A?

00:03:47.290 --> 00:03:53.740
In a mayoral race, if 75% of
Democrats, so 0.75 Democrats

00:03:53.740 --> 00:03:59.290
are voting for candidate A,
and 20% of Republicans, so

00:03:59.290 --> 00:04:03.290
plus 0.2 times the Republicans,
are expected to

00:04:03.290 --> 00:04:05.350
vote for candidate A, what
percent of the voters are

00:04:05.350 --> 00:04:07.290
voting for candidate A?

00:04:07.290 --> 00:04:11.030
So let's just express this in
terms of the total voters.

00:04:11.030 --> 00:04:13.720
The Democrats are 60% of the
voters, Republicans are 40%

00:04:13.720 --> 00:04:17.220
of, not the Republicans,
of the voters.

00:04:17.220 --> 00:04:20.800
So 0.75 times the Democrats, the
Democrats are 60% of the

00:04:20.800 --> 00:04:22.900
voters, so let's just
substitute.

00:04:22.900 --> 00:04:29.890
0.6 voters for D, so we get
0.75 times 0.6 of all the

00:04:29.890 --> 00:04:33.370
voters, that's the number of
Democrats there are, plus 0.2

00:04:33.370 --> 00:04:36.750
times the Republicans,
Republicans are 0.4 times all

00:04:36.750 --> 00:04:38.580
the voters.

00:04:38.580 --> 00:04:44.280
And so let's see, what's
75 times 6?

00:04:44.280 --> 00:04:47.930
Let me just do the whole
decimals, 0.75 times 0.6.

00:04:47.930 --> 00:04:51.480
6 times 5 is 30.

00:04:51.480 --> 00:04:57.580
6 times 7 is 42 plus 3 is 45.

00:04:57.580 --> 00:04:59.970
And we have one, two, three
numbers behind the decimal

00:04:59.970 --> 00:05:05.860
point, so it's 0.45 of the
voters plus, let's see, 2

00:05:05.860 --> 00:05:09.350
times 4 is 8, and we have two
numbers behind the decimal, so

00:05:09.350 --> 00:05:14.410
it's plus 0.08 times
the voters.

00:05:14.410 --> 00:05:20.230
0.45 plus 0.08 that's 0.53.

00:05:20.230 --> 00:05:24.980
You have 45 plus 8, is 0.53, or
53% of the voters are going

00:05:24.980 --> 00:05:29.760
to vote for candidate
A, that's choice B.

00:05:29.760 --> 00:05:34.640
Problem 95.

00:05:34.640 --> 00:05:38.120
A certain company retirement
plan has a rule of 70

00:05:38.120 --> 00:05:41.010
provision that allows an
employee to retire when the

00:05:41.010 --> 00:05:49.770
employee's age plus years of
employment within the company

00:05:49.770 --> 00:05:51.210
total at least 70.

00:05:51.210 --> 00:05:54.280
So greater than or
equal to 70.

00:05:54.280 --> 00:06:00.770
In what year could a female
employee hired in 1986, this

00:06:00.770 --> 00:06:05.830
is interesting, 1986, on her
32nd birthday, first be

00:06:05.830 --> 00:06:08.665
eligible to retire under
this provision?

00:06:14.280 --> 00:06:16.640
So let's say that y is
equal to the year.

00:06:22.070 --> 00:06:32.780
Her age is going to be the
year minus 1986 plus 32.

00:06:32.780 --> 00:06:38.240
That's her age, because she
starts at 32, so if the year's

00:06:38.240 --> 00:06:41.420
1986, we're just going
to get her age as 32.

00:06:41.420 --> 00:06:44.960
If the year's 1987, we're going
to get 1987 minus 1986,

00:06:44.960 --> 00:06:47.810
which is 1 year, plus 32,
her age will be 33.

00:06:47.810 --> 00:06:49.060
So that's her age.

00:06:57.040 --> 00:06:59.740
Her years of employment
are going to be

00:06:59.740 --> 00:07:05.000
the years minus 1986.

00:07:05.000 --> 00:07:08.590
So a plus e is going to be equal
of this, and so that is

00:07:08.590 --> 00:07:09.430
equal to what?

00:07:09.430 --> 00:07:20.510
2y minus 2 times 1986 plus 32
have to be greater than or

00:07:20.510 --> 00:07:23.580
equal to 70.

00:07:23.580 --> 00:07:25.480
Just so I don't have to multiply
2 times 1986, let's

00:07:25.480 --> 00:07:27.690
divide both sides of this
equation by 2, and I don't

00:07:27.690 --> 00:07:29.700
have to change the inequality,
because that's positive.

00:07:29.700 --> 00:07:37.870
So I get y minus 1986
plus 32 is greater

00:07:37.870 --> 00:07:42.360
than or equal to 70.

00:07:42.360 --> 00:07:48.070
So y is greater than or equal to
70, let me add 1986 to both

00:07:48.070 --> 00:07:54.530
sides, plus 1986, minus 32.

00:07:54.530 --> 00:07:58.500
So what's 1986 minus 32?

00:07:58.500 --> 00:08:06.230
That's 1954, 8 minus 3
is 5, 6 minus 2 is 4.

00:08:06.230 --> 00:08:08.390
What's 70 plus 1954?

00:08:08.390 --> 00:08:20.810
Plus 70, is 4, 5 plus 7 is 12,
1 plus 9 is 10, so 2024 is an

00:08:20.810 --> 00:08:22.770
acceptable year for her.

00:08:22.770 --> 00:08:24.350
2024.

00:08:24.350 --> 00:08:27.480
And that is not one of
the choices, so I

00:08:27.480 --> 00:08:29.340
clearly made a mistake.

00:08:29.340 --> 00:08:32.330
Let me see where I might
have made a mistake.

00:08:32.330 --> 00:08:35.330
So a certain employee retirement
has a rule of 70

00:08:35.330 --> 00:08:37.289
provision that allows an
employee to retire when the

00:08:37.289 --> 00:08:41.470
employee's age, age plus years
of employment within the

00:08:41.470 --> 00:08:46.930
company total at least 70.

00:08:46.930 --> 00:08:49.680
In what year could a female
employee, the fact that she's

00:08:49.680 --> 00:08:52.710
female shouldn't matter, a
female employee hired in 1986

00:08:52.710 --> 00:08:57.500
on her 32nd birthday,
first be eligible to

00:08:57.500 --> 00:09:00.050
retire under this provision?

00:09:00.050 --> 00:09:07.030
OK, her age is going to be the
year, whatever year we're

00:09:07.030 --> 00:09:10.070
talking about, minus
1986 plus 32.

00:09:10.070 --> 00:09:12.610
And then her years of employment
are going to be

00:09:12.610 --> 00:09:21.590
whatever year we're talking
about minus 1986, fair enough.

00:09:21.590 --> 00:09:24.500
So I have a year plus year, and
I have two subtractions of

00:09:24.500 --> 00:09:29.180
my 1986 plus 32 is greater
than or equal to 70.

00:09:29.180 --> 00:09:32.200
I divide both sides by 30--
oh, I see my mistake.

00:09:32.200 --> 00:09:35.090
Here, this stuff right here,
where I divided both sides by

00:09:35.090 --> 00:09:37.880
2, I divided only the
left side by 2.

00:09:37.880 --> 00:09:41.310
That by 2, that by 2-- oh,
I didn't even do that.

00:09:41.310 --> 00:09:45.690
2 times 1986 divided by 2 is
1986, 32 divided by 2 is 16,

00:09:45.690 --> 00:09:48.610
70 divided by 2 is 35.

00:09:48.610 --> 00:09:50.740
OK, so let me rewrite it here.

00:09:53.680 --> 00:09:56.510
That's what happens when you
try to do too much at once.

00:09:56.510 --> 00:10:08.400
So I have y is greater than or
equal to 35 plus 1986 minus

00:10:08.400 --> 00:10:12.170
16, I just added 1986 to
both sides, subtracted

00:10:12.170 --> 00:10:14.100
16 from both sides.

00:10:14.100 --> 00:10:23.230
So this is equal to 1970, and
then we have 35 plus 1970, so

00:10:23.230 --> 00:10:30.240
y has to be greater than
or equal to 2005,

00:10:30.240 --> 00:10:33.230
which is choice C.

00:10:33.230 --> 00:10:37.050
And I'm out of time, sorry for
that careless error, and I'll

00:10:37.050 --> 00:10:39.120
see you in the next video.