MTEL Math Practice Test: 41-45

WEBVTT
Kind: captions
Language: en

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Problem 41.

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Use the histogram below
to answer the

00:00:04.940 --> 00:00:06.350
question that follows.

00:00:06.350 --> 00:00:07.790
Here's our histogram.

00:00:07.790 --> 00:00:10.470
Two six-sided number cubes.

00:00:10.470 --> 00:00:13.590
They sound suspiciously
like regular dice.

00:00:13.590 --> 00:00:16.500
Two six-sided number
cubes are rolled

00:00:16.500 --> 00:00:18.420
simultaneously 10 times.

00:00:18.420 --> 00:00:20.600
So it's like you're rolling
two dice 10 times.

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The sums are recorded in the
histogram shown above.

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So when you look at what's on
top of your dice, and you add

00:00:26.670 --> 00:00:28.910
them up, and then they recorded
the frequency of the

00:00:28.910 --> 00:00:32.119
different sums. Which of the
following statements can be

00:00:32.119 --> 00:00:34.190
inferred from the histogram?

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So the first one says the
mean is less than

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the median by 1/10.

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The mean-- OK, so they're
comparing the mean and the

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median in every one
of these choices.

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So we can actually look at this
histogram and figure out

00:00:44.240 --> 00:00:46.720
all of the results
that happened.

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So the sum, we got
3 three times.

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So we got three 3's, 3, 3, 3.

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We got one 4.

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We got three 5's, right?

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5 had a frequency of
3, so 5, 5, 5.

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And then we had no 6's.

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And then we had three
7's, 7, 7, 7.

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So the median is easy
to figure out.

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The median is literally just
the middle number.

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So how many do we have here?

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We have 1, 2, 3, 4,
5, 6, 7, 8, 9, 10.

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So the middle number is going--
or the middle two

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numbers, because it's an even,
and if they're not the same

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number, we're going to have
to take the average.

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So what's the middle number?

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If we have 4 on this side, and
4 on this side, the middle

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numbers-- right, because we have
an even data center, 5.

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So the median here
is equal to 5.

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Now, the mean is going to take
a little bit of more

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mathematics.

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We're going to have to sum up
all of our results, and then

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divide by the total of data
points there are.

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Well, actually I should have
known that there are 10 data

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points, we're going
to divide by 10.

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So what's the sum?

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It's 3 times 3.

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So this is 9 plus 4, plus 5
times 3-- you have 5 times 3

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here, which is 15-- plus
7 times 3, which is 21.

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So 9 plus 4 is 13,
plus 15, plus 21.

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This is 13 plus 15
is 28, plus 21.

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28 plus 21 is 49.

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And if we have to divide that by
10, we get an average-- so

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if you divide it by 10, because
we have 10 data

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points, the average is going
to be equal to 4.9.

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So our average is 4.9
Our median is 5.

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So our average is 1/10
less than our median.

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Or when I say average,
I should say mean.

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But it tends to mean, you know,
when people say average,

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they tend to mean the mean.

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Our mean, where we just added
up all the data points and

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divided by 10, is 1/10
less than the median.

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So let's see, the mean is less
than the median by 1/10.

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That looks right.

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Our mean is less than
the median by 1/10.

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Choice A.

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Problem 42.

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Use the graph below to answer
the question that follows.

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Distribution of test scores.

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All right.

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The graph above shows the
distribution of scores on a

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test with possible scores of
10, 20, 30, 40, 50, and 60.

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Those are the only
possible scores.

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Maybe there were only six
questions on the exam.

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Well, yeah, sure.

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The minimum passing
score was 40.

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So this is the minimum
passing score.

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You have to get at least
a 40 to pass.

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20 girls and 20 boys took the
test. The percentage of girls

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passing the test was how much
greater than the percentage of

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boys passing the test?

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All right.

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So let's see how many girls
passed the test. So this

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dotted graph is the girls.

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So 5 got a 40, so
those passed.

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They just barely passed.

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So you had those 5
girls get a 40.

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Then you had another
6 girls get a 50.

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So you have another 6 girls get
a 50, and you had another

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4 girls get a 60, plus 4.

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So that's how many
girls passed.

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So that's 5 plus 6 is 11, plus
4 is 15 girls passed.

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There were a total of 20 girls,
so 15 out of 20, or,

00:04:26.070 --> 00:04:27.800
you could think of
that as 75%.

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If you divide 15 by 20, that's
the same thing as 3/4, or 75%

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of girls passed.

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Now let's do the boys.

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And I'll do it in stereotypical

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colors, the boys in blue.

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Once again, they have to get
at least a 40 to pass.

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So 6 boys just barely
passed, got the 40.

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Then 3 boys got a 50.

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They passed.

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Plus 2 boys got a
60, they passed.

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So total of 6 plus 3, plus
2 is 11 boys passed.

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11 out of 20 is the same
thing as what?

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That's 55%.

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If you multiply the numerator
and the denominator by 5, you

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get 55 over 100.

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So this is equal to 55%
of the boys passed.

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Now, what did they ask us?

00:05:19.970 --> 00:05:21.110
I forgot already.

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The percentage of girls passing
the test was how much

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greater than the percentage
of boys passing the test?

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So 75% is how much
greater than 55%?

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Well 75% minus 55%,
that's 20%.

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The girls' percentage is 20%
larger than the boys'

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percentage.

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The percentage of girls passing
the test was how much

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greater than the percentage of
boys passing the test. Right.

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It's 20 percentage
points greater.

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Next question.

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43.

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Use the table below to answer
the question-- oh, I think

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I've learned that.

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A marketing company conducted
a survey to determine

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milkshake flavor preferences
among five

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different age groups.

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These are the different
age groups.

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This is the different options,
I guess, on milkshakes.

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Based on the data in the table,
what is the probability

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that a randomly chosen
35-year-old customer will

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prefer a strawberry-flavored
milkshake?

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So a randomly chosen
35-year-old, that puts us in

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this bucket, right here,
30- to 39-year-olds.

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So if they can be described by
the data that we've collected

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already, the probability is
the same ratio of the data

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sample we've collected that
actually liked-- what was

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it?-- strawberry-flavored
milkshakes.

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So in that age group, how many
total people did we survey?

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We surveyed 8 plus 14 is 22.

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22 plus 12 is 34.

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We surveyed 34 people in this
age group, and how many of

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them liked strawberry
milkshakes?

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Well 12 of them did.

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12 out of 34 30- to 39-year-olds
liked strawberry

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milkshakes.

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So, the probability, if we use
that as our data, that a

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random 30- to 39-year-old will
like strawberry milkshakes is

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12 over 34, or if we reduce
this, that's 6 over 17.

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Just divide the numerator and
the denominator by 2.

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And that is D.

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Good thing that that was one
of the answer choices, D.

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Problem 44.

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A child has a set of blocks,
4 of which are square.

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5 are round.

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And 6 are triangular.

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The child randomly picks a block
from the set and gives

00:07:50.030 --> 00:07:51.250
it to his sister.

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The child then randomly picks
one more block, so he didn't

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put that block back in.

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That's very important--
that first block.

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What is the probability that the
first block was round and

00:08:03.490 --> 00:08:05.920
the second block
was triangular?

00:08:05.920 --> 00:08:07.650
So the probability the
first block is round.

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How may total blocks do we
have to start off with?

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We have 15, right?

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4 plus 6 is 10, plus 5 is 15.

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So we started off
with 15 blocks.

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And what is the probability that
the first block is round?

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Well 5 of them are round,
so there's a 5 over 15

00:08:22.780 --> 00:08:23.920
probability.

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So this is the probability
that first is

00:08:29.610 --> 00:08:32.130
round, right there.

00:08:32.130 --> 00:08:34.419
And now, what is the probably
that the second block is

00:08:34.419 --> 00:08:35.340
triangular?

00:08:35.340 --> 00:08:39.780
Now, remember, we no longer have
15 blocks in the pile.

00:08:39.780 --> 00:08:41.730
Now, he's taken one
of the blocks and

00:08:41.730 --> 00:08:42.659
given it to his sister.

00:08:42.659 --> 00:08:47.400
So now we're choosing
out of 14 blocks.

00:08:47.400 --> 00:08:50.690
And we're also-- and this is an
and-- so we're assuming--

00:08:50.690 --> 00:08:52.770
so what is the probability
that the second block is

00:08:52.770 --> 00:08:56.810
triangular given that the first
block was a circle?

00:08:56.810 --> 00:09:00.520
So we assume that we still have
our full contingency of 6

00:09:00.520 --> 00:09:03.800
triangles there, that that
wasn't picked our first time.

00:09:03.800 --> 00:09:07.430
So the probability that the
second block is triangular--

00:09:07.430 --> 00:09:11.560
there are 6 triangles, there are
6 triangles out of the 14

00:09:11.560 --> 00:09:12.730
blocks left.

00:09:12.730 --> 00:09:15.340
So our answer is going to be 5
over 15, which is the same

00:09:15.340 --> 00:09:21.450
thing as 1/3, times 6 over 14 is
the same thing as 3 over 7,

00:09:21.450 --> 00:09:24.870
which is the same thing
as 1 over 7.

00:09:24.870 --> 00:09:28.780
And that is choice C.

00:09:28.780 --> 00:09:31.710
Problem 45.

00:09:31.710 --> 00:09:33.940
Use the spinner below
to answer the

00:09:33.940 --> 00:09:35.180
question that follows.

00:09:35.180 --> 00:09:37.200
OK, there's our spinner.

00:09:37.200 --> 00:09:40.560
The host of a party tells her
guests that every time the

00:09:40.560 --> 00:09:45.350
spinner above lands on the
section labeled fruit basket--

00:09:45.350 --> 00:09:48.420
that's the section labeled fruit
basket-- a guest will

00:09:48.420 --> 00:09:51.440
win a large basket of fruit.

00:09:51.440 --> 00:09:56.260
If the 180 guests at the party
each spin the spinner once,

00:09:56.260 --> 00:10:00.290
what is the best estimate of
the number of fruit baskets

00:10:00.290 --> 00:10:03.100
that the host will
be giving away?

00:10:03.100 --> 00:10:06.230
So let's see, we have,
1, 2, 3, 4, 5.

00:10:06.230 --> 00:10:09.005
They look about equal, so
you have 5 equal pies.

00:10:09.005 --> 00:10:13.330
So it looks like there's a 0.2,
or 20% chance, of getting

00:10:13.330 --> 00:10:14.130
fruit baskets.

00:10:14.130 --> 00:10:18.110
So each person can almost
pretend like they get 0.2 of a

00:10:18.110 --> 00:10:19.260
fruit basket.

00:10:19.260 --> 00:10:22.460
So they have 0.2, or maybe I'd
say they have a 1/5 chance,

00:10:22.460 --> 00:10:25.620
and then you have 180 of those
people, so times 180.

00:10:25.620 --> 00:10:27.660
I could have multiplied
this times 180.

00:10:27.660 --> 00:10:30.550
This would be the expected
value, the expected number of

00:10:30.550 --> 00:10:32.750
fruit baskets I'm going
to give away.

00:10:32.750 --> 00:10:34.820
So what's 1/5 times 180?

00:10:34.820 --> 00:10:37.360
Or well, how many times
does 5 go into 180?

00:10:37.360 --> 00:10:39.980
5 goes into 18 three times.

00:10:39.980 --> 00:10:41.910
3 times 5 is 15.

00:10:41.910 --> 00:10:43.680
18 minus 15 is 3.

00:10:43.680 --> 00:10:45.960
30, 36 times.

00:10:45.960 --> 00:10:52.400
So I would expect to give
away 36 baskets.


MTEL Math Practice Test: 28-30

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.510
We're on problem 28.

00:00:02.510 --> 00:00:07.600
Four pumps begin draining
a 5,400 gallon pool.

00:00:07.600 --> 00:00:10.580
At the same time, two pumps
begin draining a

00:00:10.580 --> 00:00:12.790
4,000 gallon pool.

00:00:12.790 --> 00:00:17.650
Assuming that all of the pumps
drain at the same rate, how

00:00:17.650 --> 00:00:20.880
many gallons are left in the
smaller pool-- so the 4,000

00:00:20.880 --> 00:00:22.370
gallon pool-- when the
larger pool is

00:00:22.370 --> 00:00:23.690
finished being drained.

00:00:23.690 --> 00:00:26.270
And notice they didn't tell us
how fast the pumps work, so

00:00:26.270 --> 00:00:27.750
let's just assume that.

00:00:27.750 --> 00:00:29.990
Let's say that each
pump drains at

00:00:29.990 --> 00:00:35.030
x gallons per minute.

00:00:35.030 --> 00:00:38.400
So four pumps will drain at 4x
gallons per minute, two pumps

00:00:38.400 --> 00:00:40.520
will drain at 2x gallons
a minute.

00:00:40.520 --> 00:00:42.730
So how many minutes does
it take to drain the

00:00:42.730 --> 00:00:45.070
5,400 gallon pool?

00:00:45.070 --> 00:00:49.210
Well, it will be 5,400 gallons--
I'll just write g's,

00:00:49.210 --> 00:00:54.210
or I'll just say gallons--
divided by x gallons, or

00:00:54.210 --> 00:00:56.030
actually 4x because we
have four pumps,

00:00:56.030 --> 00:01:00.340
4x gallons per minute.

00:01:00.340 --> 00:01:01.380
And what is this equal to?

00:01:01.380 --> 00:01:05.730
If we divide the numerator and
the denominator by 2--

00:01:05.730 --> 00:01:06.980
actually, let's just
divide by 4.

00:01:06.980 --> 00:01:13.870
4 goes into 5,400, 4 goes into
5 one time, 1 times 4 is 4,

00:01:13.870 --> 00:01:15.320
bring down the 1.

00:01:15.320 --> 00:01:17.820
4 goes into 14 three times.

00:01:17.820 --> 00:01:20.170
3 times 4 is 12.

00:01:20.170 --> 00:01:23.300
20, 4 goes into 20 5 times.

00:01:23.300 --> 00:01:24.940
So it goes 1,350 times.

00:01:24.940 --> 00:01:35.110
So this is equal to 1,350 over
x minutes to drain the 5,400

00:01:35.110 --> 00:01:36.660
gallon pool.

00:01:36.660 --> 00:01:41.410
Now, how much has drained out
of the 4,000 gallon pool at

00:01:41.410 --> 00:01:42.330
that point?

00:01:42.330 --> 00:01:43.940
Well, we're going
to be draining

00:01:43.940 --> 00:01:48.700
2x gallons per minute.

00:01:48.700 --> 00:01:50.690
And how long are we going
to be doing that for?

00:01:50.690 --> 00:01:52.710
Well we're going to do it
until-- or we're going to be

00:01:52.710 --> 00:01:55.850
watching this experiment-- until
this pool is completely

00:01:55.850 --> 00:01:58.220
drained and we've figured out
this is how long it takes to

00:01:58.220 --> 00:02:01.320
drain it: 1,350 over
x minutes.

00:02:01.320 --> 00:02:04.660
So, if we're going to be pumping
out at 2x gallons per

00:02:04.660 --> 00:02:07.280
minute, in the smaller pool,
because we have two pumps, so

00:02:07.280 --> 00:02:13.880
2x gallons per minute, times
this many minutes, times 1,350

00:02:13.880 --> 00:02:16.730
over x minutes.

00:02:16.730 --> 00:02:19.240
This is going to tell us how
many gallons have pumped out

00:02:19.240 --> 00:02:21.160
of the 4,000 gallon pool.

00:02:21.160 --> 00:02:23.950
So the x's cancel out-- you have
an x in the numerator, an

00:02:23.950 --> 00:02:25.150
x in the denominator.

00:02:25.150 --> 00:02:27.375
The minutes cancel out, so our
answer will be in gallons.

00:02:27.375 --> 00:02:30.680
So it's 2 times 1,350.

00:02:30.680 --> 00:02:33.290
So what do we get?

00:02:33.290 --> 00:02:39.240
We get 2,700 gallons.

00:02:39.240 --> 00:02:44.140
That's how much has been pumped
out of the 4,000 gallon

00:02:44.140 --> 00:02:44.860
pool, right?

00:02:44.860 --> 00:02:47.190
This is how much has
been pumped out.

00:02:47.190 --> 00:02:50.530
2x times 1,350 over
x is 2,700.

00:02:50.530 --> 00:02:52.890
That's how much has been pumped
out, but they're asking

00:02:52.890 --> 00:02:56.710
us, how many gallons are left
in the smaller pool?

00:02:56.710 --> 00:03:01.880
So if we started with 4,000, and
we subtract 2,700, that's

00:03:01.880 --> 00:03:05.970
going to be 1,300 gallons
are left in the pool.

00:03:05.970 --> 00:03:08.620
That is choice A.

00:03:08.620 --> 00:03:11.070
Problem 29.

00:03:11.070 --> 00:03:14.850
Use the graph below to answer
the question that follows.

00:03:14.850 --> 00:03:17.780
Switch colors, ease
the monotony.

00:03:17.780 --> 00:03:21.850
The graph above shows the
distance, d, in miles, and the

00:03:21.850 --> 00:03:27.380
time, t, in minutes for six
bus routes around a city.

00:03:27.380 --> 00:03:30.220
Which of the following equations
best models a

00:03:30.220 --> 00:03:34.490
relationship between d and
t for these bus routes?

00:03:34.490 --> 00:03:36.330
So these are different
bus routes.

00:03:36.330 --> 00:03:37.600
It's almost a line.

00:03:37.600 --> 00:03:39.866
I mean we can kind of-- let me
do it in a light color so I

00:03:39.866 --> 00:03:42.400
don't mess up the graph-- but we
can see, you know, I see a

00:03:42.400 --> 00:03:47.010
line that looks something like
that, right there, if I had to

00:03:47.010 --> 00:03:48.570
fit a line into there.

00:03:48.570 --> 00:03:50.190
Let's see, which of the
following equations best

00:03:50.190 --> 00:03:51.600
models the relationship?

00:03:51.600 --> 00:03:54.630
So t is equal to d, what
would that look like?

00:03:54.630 --> 00:03:58.990
That line-- I'll do it in this
light green color-- t is equal

00:03:58.990 --> 00:04:02.990
to d, so when t is equal to
10, d is equal to 10.

00:04:02.990 --> 00:04:08.050
When t is equal to 20, d is
equal to 20. t is equal to 30,

00:04:08.050 --> 00:04:08.790
d is equal to 30.

00:04:08.790 --> 00:04:12.490
So that's this line
right here.

00:04:12.490 --> 00:04:16.540
That doesn't seem to model these
black dots too well, so

00:04:16.540 --> 00:04:18.570
it's not going to
be that answer.

00:04:18.570 --> 00:04:19.750
Let me do another one.

00:04:19.750 --> 00:04:22.280
t is equal to d plus 10.

00:04:22.280 --> 00:04:27.340
So if we start at 10, so
when d is 0, t will

00:04:27.340 --> 00:04:28.970
be 10, right there.

00:04:28.970 --> 00:04:34.290
When d is 10, d will be 20,
right? d plus 10, 20.

00:04:34.290 --> 00:04:37.500
When d is 20, t will be 30.

00:04:37.500 --> 00:04:39.260
It's going to look like this.

00:04:39.260 --> 00:04:41.140
This is a little, it's an
improvement, but it still

00:04:41.140 --> 00:04:42.510
doesn't look that great to me.

00:04:42.510 --> 00:04:45.290
So, let's just, yeah, it's
probably not that.

00:04:45.290 --> 00:04:47.200
t is equal to 2d.

00:04:47.200 --> 00:04:50.640
Well, 2d, when you're
0, you're at 0.

00:04:50.640 --> 00:04:52.910
When t is equal to 10, you're
going to be at 20.

00:04:52.910 --> 00:04:55.570
When t is equal to 20, you're
going to be at 40, so it's

00:04:55.570 --> 00:04:58.190
going to go like, something
like this.

00:04:58.190 --> 00:05:01.940
It's going to look something--
40 right here-- it's going to

00:05:01.940 --> 00:05:03.320
look something like this.

00:05:03.320 --> 00:05:04.090
Which is close.

00:05:04.090 --> 00:05:05.730
It seems to have the
right slope.

00:05:05.730 --> 00:05:07.630
Let's look at the last choice.

00:05:07.630 --> 00:05:08.870
I'm guessing the last
choice is going to

00:05:08.870 --> 00:05:10.040
get us right there.

00:05:10.040 --> 00:05:11.960
t is equal to 2d plus 10.

00:05:11.960 --> 00:05:13.580
Let me do this in blue, because
I have a feeling this

00:05:13.580 --> 00:05:14.840
is the right answer.

00:05:14.840 --> 00:05:19.220
So we start at 10 when
d is 0, t is 10.

00:05:19.220 --> 00:05:21.280
And then we go up by 2d.

00:05:21.280 --> 00:05:25.810
Or another you can think of
it, when d is 10, we get 2

00:05:25.810 --> 00:05:27.310
times 10 plus 10.

00:05:27.310 --> 00:05:29.860
We're going to go to 30.

00:05:29.860 --> 00:05:34.010
When d is 20, 2 times
20 plus 10 is 50.

00:05:34.010 --> 00:05:35.850
We're going to be at 50.

00:05:35.850 --> 00:05:40.030
So the line looks something
like this.

00:05:40.030 --> 00:05:42.280
It will look like that,
which is a pretty

00:05:42.280 --> 00:05:43.190
good fit of the curve.

00:05:43.190 --> 00:05:45.960
It's definitely the best out
of the four choices.

00:05:45.960 --> 00:05:48.280
So I'll go with D.

00:05:48.280 --> 00:05:49.450
Next question.

00:05:49.450 --> 00:05:49.540
Let's see.

00:05:49.540 --> 00:05:51.010
Are there any questions
on this page?

00:05:51.010 --> 00:05:51.380
No.

00:05:51.380 --> 00:05:53.740
Next page.

00:05:53.740 --> 00:05:55.026
Next question.

00:05:55.026 --> 00:05:56.700
Let's see.

00:05:56.700 --> 00:05:59.910
Use the graph below to answer
the question that follows.

00:05:59.910 --> 00:06:01.170
OK, they've drawn us
a little line.

00:06:01.170 --> 00:06:03.470
They've drawn some points
on the line.

00:06:03.470 --> 00:06:08.050
The graph above represents the
equation wx plus 4y is equal

00:06:08.050 --> 00:06:09.560
to minus 12.

00:06:09.560 --> 00:06:12.510
What is the value of w?

00:06:12.510 --> 00:06:14.600
So they're esentially, they gave
us two points, so we can

00:06:14.600 --> 00:06:18.100
easily figure out, actually,
the equation of this line.

00:06:18.100 --> 00:06:19.530
Maybe we do not have
to go that far.

00:06:19.530 --> 00:06:24.240
So if we can just write this
line, y is equal to mx plus b

00:06:24.240 --> 00:06:26.180
form, and maybe if we just
figure out the slope of the

00:06:26.180 --> 00:06:27.800
line, that'll be enough.

00:06:27.800 --> 00:06:32.960
So they said that the line
is wx plus 4y is

00:06:32.960 --> 00:06:34.710
equal to minus 12.

00:06:34.710 --> 00:06:37.500
If we subtract-- I just want to
put it into the standard mx

00:06:37.500 --> 00:06:38.890
plus b form.

00:06:38.890 --> 00:06:44.260
So if we subtract wx from both
sides, we get 4y is equal to

00:06:44.260 --> 00:06:49.180
minus wx minus 12, and then
divide both sides by 4, you

00:06:49.180 --> 00:06:56.100
get y is equal to minus
w over 4x minus 3.

00:06:56.100 --> 00:06:58.140
So this is the equation of the
line, and we have to figure

00:06:58.140 --> 00:06:59.520
out what w is.

00:06:59.520 --> 00:07:04.310
This right here, minus w over
4 is the slope of the line.

00:07:04.310 --> 00:07:05.900
So let's figure out
what the slope is.

00:07:05.900 --> 00:07:08.160
Slope is just change in x.

00:07:08.160 --> 00:07:13.830
Slope is equal to change in x,
over-- sorry-- is equal to

00:07:13.830 --> 00:07:17.130
change in y, change in
y over change in x.

00:07:17.130 --> 00:07:18.270
Rise over run.

00:07:18.270 --> 00:07:20.780
How much do you change in y
when you change a certain

00:07:20.780 --> 00:07:21.660
amount in x?

00:07:21.660 --> 00:07:24.760
So in this case, it's going to
be equal to-- let's take this

00:07:24.760 --> 00:07:26.100
as our first point.

00:07:26.100 --> 00:07:27.240
So we take the y's.

00:07:27.240 --> 00:07:35.420
2 minus minus 3 over-- right,
that's the change in y-- 2

00:07:35.420 --> 00:07:38.550
minus minus 3, right?

00:07:38.550 --> 00:07:41.380
And then, the second one, we use
the 2 as the first point

00:07:41.380 --> 00:07:44.160
here, so I use the minus 4
as the first point to the

00:07:44.160 --> 00:07:44.480
denomintor.

00:07:44.480 --> 00:07:49.460
Minus 4 minus this
x value, is 0.

00:07:49.460 --> 00:07:52.950
So the slope here is equal
to 2 minus minus 3.

00:07:52.950 --> 00:07:55.530
That's the same thing as
2 plus 3, which is just

00:07:55.530 --> 00:07:59.350
5 over minus 4.

00:07:59.350 --> 00:08:00.690
So this is the slope.

00:08:00.690 --> 00:08:02.870
So 5 over minus 4 has to
be the same thing as

00:08:02.870 --> 00:08:05.210
minus w over 4.

00:08:05.210 --> 00:08:09.520
You could write minus 5/4 is
equal to minus w over 4.

00:08:09.520 --> 00:08:11.100
You could pattern match
that immediately.

00:08:11.100 --> 00:08:13.440
The w has got to
be equal to 5.

00:08:13.440 --> 00:08:15.180
You could also solve for
w, but let's just

00:08:15.180 --> 00:08:16.175
do the pattern matching.

00:08:16.175 --> 00:08:17.840
That makes it easier.


MTEL Math Practice Test: 24-27

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:02.900
We're on problem 24.

00:00:02.900 --> 00:00:06.650
Use the solution procedure below
to answer the question

00:00:06.650 --> 00:00:07.960
that follows.

00:00:07.960 --> 00:00:10.600
Which of the following is a
major flaw in the procedure

00:00:10.600 --> 00:00:11.360
shown above?

00:00:11.360 --> 00:00:13.430
So let's look for some flaws.

00:00:13.430 --> 00:00:16.070
So the way I always like to do
it is just think about how I

00:00:16.070 --> 00:00:16.790
would have done it.

00:00:16.790 --> 00:00:22.850
So if I started with minus 3x
plus 25 is equal to 4, the

00:00:22.850 --> 00:00:25.990
first thing I would do is
subtract 25 from both sides of

00:00:25.990 --> 00:00:27.250
the equation.

00:00:27.250 --> 00:00:32.030
So I would take a minus 25 from
the left-hand side, and

00:00:32.030 --> 00:00:34.860
then a minus 25 from the
right-hand side.

00:00:34.860 --> 00:00:37.840
And it looks like that's what
the person did-- took a minus

00:00:37.840 --> 00:00:43.320
25 there and they took a minus
the 25, just like that.

00:00:43.320 --> 00:00:45.160
And then he said, OK,
well that's the same

00:00:45.160 --> 00:00:48.040
thing as minus 21.

00:00:48.040 --> 00:00:52.080
So then you would get minus 3x,
on the left-hand side, is

00:00:52.080 --> 00:00:54.290
equal to minus 21.

00:00:54.290 --> 00:00:56.630
I have a sense that I already
know what the major flaw is.

00:00:56.630 --> 00:00:59.200
He's doing two steps, and he's
not writing a nice clean

00:00:59.200 --> 00:01:00.940
equality, and he's trying
to skip steps.

00:01:00.940 --> 00:01:04.310
And my guess is that he's
going to get confused.

00:01:04.310 --> 00:01:06.610
So then you end up with
minus 3x on the

00:01:06.610 --> 00:01:07.910
left-hand side, right?

00:01:07.910 --> 00:01:09.380
Because these cancel out.

00:01:09.380 --> 00:01:13.770
Minus 3x is equal to minus 21.

00:01:13.770 --> 00:01:15.660
So you get that.

00:01:15.660 --> 00:01:17.600
So you get that right there.

00:01:17.600 --> 00:01:20.070
So you get minus 3x is equal
to minus 21, and then you

00:01:20.070 --> 00:01:24.750
divide both sides by minus 3,
and you get x is equal to 7.

00:01:24.750 --> 00:01:29.340
So the guy got the right answer,
however he or she may

00:01:29.340 --> 00:01:29.870
have done it.

00:01:29.870 --> 00:01:31.920
But this is a little confusing,
because then

00:01:31.920 --> 00:01:35.410
divides by 3, but you know, this
divide by minus 3, which

00:01:35.410 --> 00:01:37.510
you do have to do, you
have to do to both

00:01:37.510 --> 00:01:38.670
sides of this equation.

00:01:38.670 --> 00:01:40.870
I just don't see where
he or she did it.

00:01:40.870 --> 00:01:41.940
Maybe, you know, somehow
they just

00:01:41.940 --> 00:01:43.830
divided by minus 3 randomly.

00:01:43.830 --> 00:01:44.400
I don't know why.

00:01:44.400 --> 00:01:47.090
They didn't really understand
the notion of

00:01:47.090 --> 00:01:48.660
inequality on an equation.

00:01:48.660 --> 00:01:50.210
You have to do the same
thing to both

00:01:50.210 --> 00:01:52.060
sides of this equation.

00:01:52.060 --> 00:01:54.840
So let's see what
the choices are.

00:01:54.840 --> 00:01:58.230
The concept of the opposite
number is confused with

00:01:58.230 --> 00:01:58.950
subtraction.

00:01:58.950 --> 00:01:59.630
No, not really.

00:01:59.630 --> 00:02:01.960
The equal sign is used to
connect expressions

00:02:01.960 --> 00:02:03.560
that are not equal.

00:02:03.560 --> 00:02:09.030
That sounds about right, because
this expression, that

00:02:09.030 --> 00:02:12.210
expression, and this
are equal.

00:02:12.210 --> 00:02:16.280
We got minus 3x plus 25, minus
25, which is just minus 3x.

00:02:16.280 --> 00:02:20.550
That does indeed equal 4
minus 25 or minus 21.

00:02:20.550 --> 00:02:24.090
But that does not equal minus
21 divided by minus 3.

00:02:24.090 --> 00:02:26.020
So this looks like my answer.

00:02:26.020 --> 00:02:27.490
Let's look at the
other choices.

00:02:27.490 --> 00:02:28.990
The solution contains
an error in the

00:02:28.990 --> 00:02:31.120
arithmetic of signed numbers.

00:02:31.120 --> 00:02:36.510
No, the solution is
arithmetically correct.

00:02:36.510 --> 00:02:38.640
D, the order of operations
between subtraction and

00:02:38.640 --> 00:02:40.210
division is reversed.

00:02:40.210 --> 00:02:41.400
I really don't see that.

00:02:41.400 --> 00:02:43.440
So, my choice would be B.

00:02:43.440 --> 00:02:47.460
They don't-- the equal sign--
this, right here, does not

00:02:47.460 --> 00:02:50.990
equal this, right over there.

00:02:50.990 --> 00:02:52.370
Next problem.

00:02:52.370 --> 00:02:53.370
25.

00:02:53.370 --> 00:02:55.570
Use the diagram below
to answer the

00:02:55.570 --> 00:02:56.355
question that follows.

00:02:56.355 --> 00:02:57.740
So let's see, they're extending

00:02:57.740 --> 00:02:58.970
by 1 in each direction.

00:02:58.970 --> 00:03:01.450
They're kind of creating
this triangle.

00:03:01.450 --> 00:03:05.540
If the pattern continues, how
many more small squares are in

00:03:05.540 --> 00:03:10.050
figure 100 than are
in figure 99?

00:03:10.050 --> 00:03:12.980
So let's think, every time we
increment it, how may squares

00:03:12.980 --> 00:03:13.550
are we adding?

00:03:13.550 --> 00:03:15.875
When we go from 1 to 2,
how many did we add?

00:03:15.875 --> 00:03:17.790
We added 1, 2 squares.

00:03:17.790 --> 00:03:19.680
When we went from 2 to 3,
how many did we add?

00:03:19.680 --> 00:03:22.030
We added 1, 2, 3 squares.

00:03:22.030 --> 00:03:24.450
When we went from 3 to 4,
how many did we add?

00:03:24.450 --> 00:03:27.560
We added 1, 2, 3, 4 squares.

00:03:27.560 --> 00:03:30.330
Those are the 4 that are
kind of layered over

00:03:30.330 --> 00:03:31.580
the 3, right there.

00:03:33.990 --> 00:03:36.430
So when you go from
99 to 100, we're

00:03:36.430 --> 00:03:39.210
going to add 100 squares.

00:03:39.210 --> 00:03:40.540
We're going to add
100 squares.

00:03:40.540 --> 00:03:49.832
When we go from n to n plus 1,
we're adding n plus 1 squares.

00:03:49.832 --> 00:03:53.960
So you go from 99 to 100,
you add 100 squares.

00:03:53.960 --> 00:03:56.820
Or there are 100 more squares
in the 100 figure

00:03:56.820 --> 00:03:58.700
than in the 99 figure.

00:03:58.700 --> 00:03:59.950
Next problem.

00:04:02.710 --> 00:04:04.170
All right.

00:04:04.170 --> 00:04:05.890
Problem 26.

00:04:05.890 --> 00:04:08.860
Use the table below to answer
the question that follows.

00:04:08.860 --> 00:04:12.486
We have a x-axis, and then
we have a y-axis.

00:04:12.486 --> 00:04:14.820
And what do they
want us to do?

00:04:14.820 --> 00:04:18.940
Each number in the table above
represents a value of w that

00:04:18.940 --> 00:04:23.040
is determined by the values
of integers x and y.

00:04:23.040 --> 00:04:30.340
For example, when x equals 2, y
equals 1, w is equal to 21.

00:04:30.340 --> 00:04:35.210
If the pattern continues, what
is the value of w when x is

00:04:35.210 --> 00:04:43.320
equal to 20, and y
is equal to 8?

00:04:43.320 --> 00:04:47.800
So let's see if we can get
a sense of what w is.

00:04:47.800 --> 00:04:51.160
So w, if you remember your
functions, w is a function

00:04:51.160 --> 00:04:52.820
of-- I can write it
like that-- is a

00:04:52.820 --> 00:04:54.550
function of x and y.

00:04:54.550 --> 00:04:55.255
That might confuse you.

00:04:55.255 --> 00:04:56.480
It may or may not.

00:04:56.480 --> 00:04:58.730
We just have to figure out
what this function is.

00:04:58.730 --> 00:05:01.240
So if you look at this pattern
every time we look here, we

00:05:01.240 --> 00:05:04.750
go-- so this is 0 to 8.

00:05:04.750 --> 00:05:07.660
And then we go from 8 to 16.

00:05:07.660 --> 00:05:12.600
So, this is, we're adding 8
every time we go to the right.

00:05:12.600 --> 00:05:15.160
We're adding 8, right?

00:05:15.160 --> 00:05:16.180
It looks like we are.

00:05:16.180 --> 00:05:17.920
And then as we go up,
what are we doing?

00:05:17.920 --> 00:05:20.890
So this is plus 8.

00:05:20.890 --> 00:05:25.310
So for example, this right
here is 4 times 8.

00:05:25.310 --> 00:05:27.670
This is 3 times 8.

00:05:27.670 --> 00:05:31.850
So it looks like w is
equal to 8 times x.

00:05:31.850 --> 00:05:34.260
But then as you go
up, what happens?

00:05:34.260 --> 00:05:38.490
As you go up, we're adding
5 every time.

00:05:38.490 --> 00:05:39.710
Is that throughout
the whole table?

00:05:39.710 --> 00:05:40.150
Yeah, sure.

00:05:40.150 --> 00:05:42.550
37 plus 5 is 42.

00:05:42.550 --> 00:05:44.410
We add 5 every time.

00:05:44.410 --> 00:05:47.060
And we start here,
just 8 times x.

00:05:47.060 --> 00:05:51.110
And then when y is 0, and then
we add 1 times 5, 2 times 5.

00:05:51.110 --> 00:05:54.520
Looks like it's 8x plus 5y.

00:05:54.520 --> 00:05:56.760
That's what w is.

00:05:56.760 --> 00:05:57.040
Right?

00:05:57.040 --> 00:05:58.190
And let's verify it.

00:05:58.190 --> 00:06:03.690
We pick the point 3, 2, so that
would be w is equal to 8

00:06:03.690 --> 00:06:08.800
times 3, plus 5 times 2.

00:06:08.800 --> 00:06:14.790
So w is equal to 24 plus 10 is
equal to 34, which is, indeed,

00:06:14.790 --> 00:06:15.830
what it equals.

00:06:15.830 --> 00:06:18.470
So we found our function
of x and y.

00:06:18.470 --> 00:06:19.240
And what are they asking?

00:06:19.240 --> 00:06:22.370
If the pattern continues, what
is the value when x is equal

00:06:22.370 --> 00:06:24.420
to 20 and y is equal to 8?

00:06:24.420 --> 00:06:29.600
Well, in that situation, w is
equal to 8 times x, which is

00:06:29.600 --> 00:06:36.620
20, plus 5 times y, which is 8,
which is equal to 160 plus

00:06:36.620 --> 00:06:40.290
40, which is equal to 200.

00:06:40.290 --> 00:06:42.430
And that is choice B.

00:06:42.430 --> 00:06:45.690
Next problem.

00:06:45.690 --> 00:06:48.070
Problem 27.

00:06:48.070 --> 00:06:54.340
The function r of x gives the
remainder when a whole number

00:06:54.340 --> 00:06:58.600
x is divided by 10.

00:06:58.600 --> 00:07:03.030
Which of the following graphs
represents r of x?

00:07:03.030 --> 00:07:05.860
So when you take something on
the x-axis and divide it by

00:07:05.860 --> 00:07:07.720
10, r of x should be
its remainder.

00:07:07.720 --> 00:07:10.350
So we take 5 divided by 10.

00:07:10.350 --> 00:07:11.610
The remainder is 5.

00:07:11.610 --> 00:07:14.680
So we should plot out to
be a 5 right there.

00:07:14.680 --> 00:07:19.620
When you take 10 divided by 10,
your answer is-- well, the

00:07:19.620 --> 00:07:21.480
answer is 1, and your
remainder is 0.

00:07:21.480 --> 00:07:22.860
So that looks right.

00:07:22.860 --> 00:07:25.760
And then 15, the
remainder is 5.

00:07:25.760 --> 00:07:26.690
So those look right.

00:07:26.690 --> 00:07:31.760
Let's see, when you plot, when
you have 1, 1 divided by 5, 1

00:07:31.760 --> 00:07:34.980
divided by-- sorry-- 1 divided
by 10, the remainder is 1.

00:07:34.980 --> 00:07:35.930
That looks right.

00:07:35.930 --> 00:07:38.600
This graph A looks
pretty good.

00:07:38.600 --> 00:07:41.180
If you take, you know,
20 divided by 10, the

00:07:41.180 --> 00:07:41.960
remainder is 0.

00:07:41.960 --> 00:07:45.010
If you take 30 divided by
10, the remainder is 0.

00:07:45.010 --> 00:07:47.180
If you take 29 divided
by 10, the

00:07:47.180 --> 00:07:49.260
remainder is 9, right there.

00:07:49.260 --> 00:07:52.230
19 divided by 10, the
remainder is 9.

00:07:52.230 --> 00:07:53.970
9 divided by 10, the
remainder is 9.

00:07:53.970 --> 00:07:55.110
This one looks pretty good.

00:07:55.110 --> 00:07:56.940
Let's see what the
other ones do.

00:07:56.940 --> 00:08:02.250
So here, let's see, when I take
10 divided by 10, the

00:08:02.250 --> 00:08:03.580
remainder should be 0.

00:08:03.580 --> 00:08:05.610
This should be a 0 right here,
but they have it the

00:08:05.610 --> 00:08:07.080
remainder being 10.

00:08:07.080 --> 00:08:09.590
So this is wrong, right
off the bat.

00:08:09.590 --> 00:08:13.220
And then here, the remainder
when a whole number x is

00:08:13.220 --> 00:08:17.290
divided by-- this is a whole
number, right, whole number x.

00:08:17.290 --> 00:08:19.970
Here, they're doing it for
essentially all of the real

00:08:19.970 --> 00:08:21.050
numbers, right?

00:08:21.050 --> 00:08:23.330
This would be true if they
said any real number, but

00:08:23.330 --> 00:08:25.820
they're including fractions,
because they're filling in the

00:08:25.820 --> 00:08:26.780
gaps right here.

00:08:26.780 --> 00:08:29.970
So C and D are not dealing
with whole numbers.

00:08:29.970 --> 00:08:34.289
So our answer is definitely A.


MTEL Math Practice Test: 20-23

WEBVTT
Kind: captions
Language: en

00:00:00.310 --> 00:00:01.150
Problem 20.

00:00:01.150 --> 00:00:05.610
The expression 5 to the minus
8 times 7 to the minus 9 is

00:00:05.610 --> 00:00:08.160
equal to which of
the following?

00:00:08.160 --> 00:00:10.000
And here we see a bunch
of answers.

00:00:10.000 --> 00:00:12.090
Let's see, where they
have a 35 there.

00:00:12.090 --> 00:00:16.210
So I suspect that they want to
have us somehow multiply the 5

00:00:16.210 --> 00:00:17.690
times the 7.

00:00:17.690 --> 00:00:20.890
So let me write this in terms
that I have an easier time

00:00:20.890 --> 00:00:21.570
understanding it.

00:00:21.570 --> 00:00:25.660
So, 5 to the minus 8 times
7 to the minus 9.

00:00:25.660 --> 00:00:31.070
That's the same thing as 1
over 5 to the 8 times 1

00:00:31.070 --> 00:00:32.830
over 7 to the 9.

00:00:32.830 --> 00:00:35.690
The negative in the exponent
just means one over the base

00:00:35.690 --> 00:00:38.310
to the positive exponent, that's
all the negative means.

00:00:38.310 --> 00:00:43.450
That's equal to 1 over 5 to
the 8 times 7 to the 9.

00:00:43.450 --> 00:00:44.920
And that's not one
of our answers.

00:00:44.920 --> 00:00:47.230
Somehow we have to multiply
the 5 and the 7.

00:00:47.230 --> 00:00:49.580
And the way we can do that is
by having the same exponent.

00:00:49.580 --> 00:00:54.070
So let me rewrite 7 to the 9.

00:00:54.070 --> 00:01:00.080
I can rewrite that as 7 to
the eighth power times 7.

00:01:00.080 --> 00:01:07.330
So then, our whole expression
becomes 1 over 5 to the 8

00:01:07.330 --> 00:01:11.660
times 7 to the 8 times 7.

00:01:11.660 --> 00:01:13.210
All right, I just
rewrote this.

00:01:13.210 --> 00:01:16.380
I rewrote 7 to the ninth as
7 to the eighth times 7.

00:01:16.380 --> 00:01:19.100
But now if I group this, just
put parentheses around it, I

00:01:19.100 --> 00:01:21.850
can do that because
multiplication is associative.

00:01:21.850 --> 00:01:23.420
It doesn't matter what
order I do it.

00:01:23.420 --> 00:01:28.160
This is the same thing
as 5 times 7 to the

00:01:28.160 --> 00:01:30.390
eighth power, right?

00:01:30.390 --> 00:01:33.875
a to the x times b to the x is
the same thing as ab to the x,

00:01:33.875 --> 00:01:36.630
so the same thing as 5 times
7 to the eighth power.

00:01:36.630 --> 00:01:41.160
So this is equal 1 over-- 5
times 7 to the eighth power is

00:01:41.160 --> 00:01:45.800
35 to the eighth
power times 7.

00:01:45.800 --> 00:01:48.220
This piece, right here, is 35 to
the eighth power, and then

00:01:48.220 --> 00:01:49.300
you have your 7.

00:01:49.300 --> 00:01:52.540
And then that is choice
B, right there.

00:01:52.540 --> 00:01:56.430
They just switched the order
of the 7 and the 35 to the

00:01:56.430 --> 00:01:58.060
eighth power.

00:01:58.060 --> 00:01:59.290
21.

00:01:59.290 --> 00:02:01.430
Use the diagram below
to answer the

00:02:01.430 --> 00:02:03.270
question that follows.

00:02:03.270 --> 00:02:06.530
OK, they just have numbers
listed in order and

00:02:06.530 --> 00:02:08.460
have five per row.

00:02:08.460 --> 00:02:10.870
Which of the following algebraic
equations could best

00:02:10.870 --> 00:02:16.960
be used to explain why, for any
3 by 3 cross, like the one

00:02:16.960 --> 00:02:19.520
shown above, the sum of the
numbers in the vertical

00:02:19.520 --> 00:02:22.510
rectangle-- so this is the
vertical rectangle, right

00:02:22.510 --> 00:02:26.120
there-- is equal to the
sum of the numbers in

00:02:26.120 --> 00:02:28.310
the horizontal rectangle.

00:02:28.310 --> 00:02:33.650
So the horizontal rectangle
is that right there.

00:02:33.650 --> 00:02:35.260
So let's look at this
a little bit.

00:02:35.260 --> 00:02:37.410
What's the sum in the
vertical rectangle?

00:02:37.410 --> 00:02:40.870
If we call this number x, let
me just draw it generally.

00:02:40.870 --> 00:02:44.590
Let me draw as best I can.

00:02:44.590 --> 00:02:46.310
If that middle entry
is x, what's the

00:02:46.310 --> 00:02:47.870
entry right above that?

00:02:47.870 --> 00:02:49.770
Well no matter where you go,
it's going to be 5 less,

00:02:49.770 --> 00:02:51.420
that's because we have
5 per row, right?

00:02:51.420 --> 00:02:53.350
14 minus 5 is 9.

00:02:53.350 --> 00:02:55.210
12 minus 5 is 7.

00:02:55.210 --> 00:02:56.330
8 minus 5 is 3.

00:02:56.330 --> 00:02:58.950
So this is going to
be x minus 5.

00:02:58.950 --> 00:03:02.040
By the same logic, if you go
down a row, you're adding 5.

00:03:02.040 --> 00:03:04.470
14 plus 5 is 19.

00:03:04.470 --> 00:03:05.910
15 plus 5 is 20.

00:03:05.910 --> 00:03:09.010
So this is going
to be x plus 5.

00:03:09.010 --> 00:03:11.620
So the sum of the entries in the
vertical column, the sum

00:03:11.620 --> 00:03:18.040
of the numbers in the vertical
rectangle are x minus 5 plus x

00:03:18.040 --> 00:03:21.200
plus x plus 5.

00:03:21.200 --> 00:03:23.270
So that's the sum in the
vertical column.

00:03:23.270 --> 00:03:27.420
And then in the horizontal
column, if this is x, what's

00:03:27.420 --> 00:03:29.090
this going to be?

00:03:29.090 --> 00:03:30.280
It's just one below that.

00:03:30.280 --> 00:03:33.220
It's x minus 1, and then
to the right of it

00:03:33.220 --> 00:03:35.170
is just x plus 1.

00:03:35.170 --> 00:03:38.640
So the sum in the horizontal
column is going to be x minus

00:03:38.640 --> 00:03:42.980
1 plus x plus x plus 1.

00:03:45.730 --> 00:03:48.460
Now let's see if any of these
choices look like what I just

00:03:48.460 --> 00:03:49.570
wrote down.

00:03:49.570 --> 00:03:51.000
A bunch of 9's.

00:03:51.000 --> 00:03:53.620
Maybe, so that's simplified.

00:03:53.620 --> 00:03:54.440
Here you go.

00:03:54.440 --> 00:03:55.500
This looks very similar.

00:03:55.500 --> 00:03:57.910
Actually this is exactly what
I wrote up here if you just

00:03:57.910 --> 00:04:00.010
switch the left- and the
right-hand side.

00:04:00.010 --> 00:04:01.710
And you can do that because
there's equality.

00:04:01.710 --> 00:04:06.180
They wrote this here, and
they wrote this here.

00:04:06.180 --> 00:04:09.660
So this is what I could use to
prove that the sum of the

00:04:09.660 --> 00:04:11.995
vertical rectangle is equal to
the sum of the horizontal.

00:04:11.995 --> 00:04:14.720
If I wanted to keep going, you'd
notice that these 5's

00:04:14.720 --> 00:04:18.000
would cancel out, these 1's
would cancel out, you'd get 3x

00:04:18.000 --> 00:04:18.829
is equal to 3x.

00:04:18.829 --> 00:04:22.310
And that would be, that would
complete your proof.

00:04:22.310 --> 00:04:23.560
Next problem.

00:04:27.720 --> 00:04:31.970
A landscaper bought some
decorative cement blocks from

00:04:31.970 --> 00:04:33.110
a landscaping store.

00:04:33.110 --> 00:04:37.960
The supplier charged 5%
sales tax, and the

00:04:37.960 --> 00:04:40.150
total came to $315.

00:04:40.150 --> 00:04:42.760
All right, 315 after 5%.

00:04:42.760 --> 00:04:44.720
Without the tax, the landscaper
could have bought

00:04:44.720 --> 00:04:49.930
six more blocks for the same
total cost. How many blocks

00:04:49.930 --> 00:04:51.480
did the landscaper buy?

00:04:51.480 --> 00:04:54.900
If p represents the price
of one block.

00:04:54.900 --> 00:04:57.030
OK, p is the price one block.

00:04:57.030 --> 00:05:00.560
In which of the following
equations does x represent the

00:05:00.560 --> 00:05:01.520
answer to the problem.

00:05:01.520 --> 00:05:03.970
But remember, the answer to
the problem is, how many

00:05:03.970 --> 00:05:06.170
blocks did the landscaper buy?

00:05:06.170 --> 00:05:10.970
So x equals blocks.

00:05:10.970 --> 00:05:13.900
x is equal to blocks bought.

00:05:13.900 --> 00:05:17.570
Now, this first equation says
that after paying 5% sales

00:05:17.570 --> 00:05:22.160
tax-- so before tax, he bought
x blocks, he payed p for each

00:05:22.160 --> 00:05:22.910
of those blocks.

00:05:22.910 --> 00:05:24.580
That's what they tell
us right here.

00:05:24.580 --> 00:05:26.890
So this is the pretax price.

00:05:26.890 --> 00:05:29.940
p per blocks times x blocks is
how much he pays, but he has

00:05:29.940 --> 00:05:31.350
to pay sales tax on that.

00:05:31.350 --> 00:05:34.480
He has to pay 5% of that on
top of that as sales tax.

00:05:34.480 --> 00:05:39.130
So plus 0.05-- that's the same
thing as 5%-- times whatever

00:05:39.130 --> 00:05:41.170
the pretax value is.

00:05:41.170 --> 00:05:46.840
And that's going to
be equal to $315.

00:05:46.840 --> 00:05:48.910
And then this, we could just
add the coefficients.

00:05:48.910 --> 00:05:50.520
This is a 1 out here, right?

00:05:50.520 --> 00:05:56.620
1 plus 0.05 is 1.05
times price times

00:05:56.620 --> 00:06:00.610
quantity is equal to 315.

00:06:00.610 --> 00:06:05.300
And let's see if we have
anything like that over here

00:06:05.300 --> 00:06:08.430
in our answers.

00:06:08.430 --> 00:06:11.050
Which equation does x represent
the answer to the

00:06:11.050 --> 00:06:12.740
problem above?

00:06:12.740 --> 00:06:21.050
0.95 px, price, he bought six
more without the tax.

00:06:21.050 --> 00:06:23.030
So another way you can think
about it, if you didn't have

00:06:23.030 --> 00:06:27.240
tax, $315 would have been
the price without tax.

00:06:27.240 --> 00:06:28.680
So let's see if that simplifies,
because I don't

00:06:28.680 --> 00:06:31.610
see exactly what I wrote here
from that first equation.

00:06:31.610 --> 00:06:35.980
So without the tax, he could
have bought x plus 6, if there

00:06:35.980 --> 00:06:38.900
was no tax.

00:06:38.900 --> 00:06:43.290
So he would have bought x plus
6 for p per block, and then

00:06:43.290 --> 00:06:45.500
that is going to be
equal to 315.

00:06:45.500 --> 00:06:48.190
This is the situation
with no tax.

00:06:48.190 --> 00:06:52.470
If there was no tax, for $315 he
could have bought six more

00:06:52.470 --> 00:06:53.690
than he originally bought.

00:06:53.690 --> 00:06:58.460
So this is what the second,
this-- let me graph it.

00:06:58.460 --> 00:06:59.580
Without the tax.

00:06:59.580 --> 00:07:02.390
So this first equation came
from the first sentence.

00:07:02.390 --> 00:07:07.710
The second equation comes from
this red sentence, right here.

00:07:07.710 --> 00:07:11.680
Let's see if we see
this anywhere.

00:07:11.680 --> 00:07:17.770
p times x plus 6 is
equal to 315.

00:07:17.770 --> 00:07:19.400
I don't see that
anywhere here.

00:07:19.400 --> 00:07:21.850
But we can see-- we can do
something interesting.

00:07:21.850 --> 00:07:23.770
We can use all of the
information from the problem.

00:07:23.770 --> 00:07:27.410
Remember, the first problem, it
said, the supplier charged

00:07:27.410 --> 00:07:30.450
5% sales tax, and the
total came to 315.

00:07:30.450 --> 00:07:33.930
That gave us this equation,
right here.

00:07:33.930 --> 00:07:38.660
The second one said, without
the tax, for $315 he could

00:07:38.660 --> 00:07:40.230
have bought six more blocks.

00:07:40.230 --> 00:07:41.760
So we get this one,
right here.

00:07:41.760 --> 00:07:45.180
Notice, both of these
equations equal 315.

00:07:45.180 --> 00:07:47.460
So we could set this
equal to that.

00:07:47.460 --> 00:07:55.760
So we could say p times x
plus 6 is equal to 1.05

00:07:55.760 --> 00:07:58.020
times price times x.

00:07:58.020 --> 00:08:00.810
And let's see if that looks like
anything we have here.

00:08:00.810 --> 00:08:09.710
So we have, let's see, p times
x plus 6 is equal to 1.05px.

00:08:09.710 --> 00:08:14.420
So the answer is
C, right there.

00:08:14.420 --> 00:08:17.170
We're just saying, both of these
things equal 315, so we

00:08:17.170 --> 00:08:18.950
set them equal to each other.

00:08:18.950 --> 00:08:21.880
Problem 23.

00:08:21.880 --> 00:08:25.810
A store that sells handcrafted
items-- let me pick a nice

00:08:25.810 --> 00:08:29.600
highlighting color-- a store
that sells handcrafted items

00:08:29.600 --> 00:08:35.370
takes $3 per item plus
40% of the sale price

00:08:35.370 --> 00:08:37.280
for each item sold.

00:08:37.280 --> 00:08:42.520
OK, the rest of the money--
takes $3 per item plus 40% of

00:08:42.520 --> 00:08:45.860
the sale price for each item
sold-- the rest of the money

00:08:45.860 --> 00:08:48.440
from the item sales goes
to the craft person.

00:08:48.440 --> 00:08:49.740
That's nice of them.

00:08:49.740 --> 00:08:54.010
All items sold cost
$5 or more.

00:08:54.010 --> 00:09:00.320
OK, if p represents the sale
price of one item, which of

00:09:00.320 --> 00:09:02.670
the following expressions
represents the amount of money

00:09:02.670 --> 00:09:06.530
the craft person gets
for each item sold?

00:09:06.530 --> 00:09:08.560
So you start off with p.

00:09:08.560 --> 00:09:10.540
That's how much the
customer gives.

00:09:10.540 --> 00:09:15.010
And then the store takes $3.

00:09:15.010 --> 00:09:19.790
So p minus 3, plus 40%
of the sale price.

00:09:19.790 --> 00:09:22.480
So not 40% of after the
$3, it says plus

00:09:22.480 --> 00:09:24.110
40% of the sale price.

00:09:24.110 --> 00:09:27.940
So the store is going to take
another 40%, right?

00:09:27.940 --> 00:09:32.150
0.4, or 0.40, times
the sales price.

00:09:32.150 --> 00:09:35.110
So the store takes $3 and then
40% of the sales price.

00:09:35.110 --> 00:09:37.580
And this is what the
craft person gets.

00:09:37.580 --> 00:09:39.600
So this is what, you know,
this is our answer.

00:09:39.600 --> 00:09:41.550
This is what craft
person gets.

00:09:41.550 --> 00:09:42.780
So what is this?

00:09:42.780 --> 00:09:52.980
This is p-- switch colors-- p
minus 0.4p is 0.6p, then you

00:09:52.980 --> 00:09:54.080
have the minus 3.

00:09:54.080 --> 00:09:55.635
That's what the craft
person gets.

00:09:55.635 --> 00:09:57.590
And if you look here, you
don't see that answer.

00:09:57.590 --> 00:09:58.980
They wrote it all
as fractions.

00:09:58.980 --> 00:10:03.080
0.6 is the same thing
as 6/10 p minus 3.

00:10:03.080 --> 00:10:06.750
6/10 is the same thing
as 3/5 p minus 3.

00:10:06.750 --> 00:10:10.080
So it's 3/5 p minus 3.

00:10:10.080 --> 00:10:12.360
Choice D.


MTEL Math Practice Test: 12 -15

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:02.000
Problem 12.

00:00:02.000 --> 00:00:05.160
The prime factorization
of a natural number.

00:00:05.160 --> 00:00:07.020
You can kind of view that
as positive integers.

00:00:07.020 --> 00:00:09.750
Sometimes they'll throw a 0 in
the natural number definition,

00:00:09.750 --> 00:00:11.770
depending on what context
you're working in.

00:00:11.770 --> 00:00:14.780
But the prime factorization of
a natural number n can be

00:00:14.780 --> 00:00:20.130
written as n is equal to pr
squared, where p and r are

00:00:20.130 --> 00:00:21.610
distinct prime numbers.

00:00:21.610 --> 00:00:22.580
Fair enough.

00:00:22.580 --> 00:00:26.420
How many factors does n have,
including 1 and itself?

00:00:26.420 --> 00:00:28.710
Well, they're saying that this
is some natural number that

00:00:28.710 --> 00:00:29.620
can be written this way.

00:00:29.620 --> 00:00:31.060
Let's just pick one example.

00:00:31.060 --> 00:00:34.320
So if we just set p-- I'm just
going to pick some prime

00:00:34.320 --> 00:00:39.180
numbers here-- p is equal to 2
and r is equal to 3, then n is

00:00:39.180 --> 00:00:43.710
equal to p times r squared,
times 3 squared, which is

00:00:43.710 --> 00:00:46.480
equal to 2 times 9, which
is equal to 18.

00:00:46.480 --> 00:00:50.320
This is just one instance--
one n that satisfies these

00:00:50.320 --> 00:00:50.770
conditions.

00:00:50.770 --> 00:00:53.930
I could have picked 5 and 7, or
I could have picked 3 and

00:00:53.930 --> 00:00:56.330
2, but all of them should have
the same answer, because

00:00:56.330 --> 00:00:58.200
that's what this question
is implying.

00:00:58.200 --> 00:00:59.630
So this is a possible n.

00:00:59.630 --> 00:01:01.990
And then how many factors
does it have?

00:01:01.990 --> 00:01:03.930
Let's factor it out.

00:01:03.930 --> 00:01:12.770
It's got 1 and 18,
2 and 9, 3 and 6.

00:01:12.770 --> 00:01:17.710
So it's got 1, 2, 3,
4, 5, 6 factors.

00:01:17.710 --> 00:01:19.540
So the answer is D.

00:01:19.540 --> 00:01:21.540
And you could try it out with
other numbers, if you like.

00:01:21.540 --> 00:01:25.060
Make p is equal to-- make
p equal to 3, and

00:01:25.060 --> 00:01:26.660
make r equal to 2.

00:01:26.660 --> 00:01:28.520
Then you'd have 3 times
2, times 2.

00:01:28.520 --> 00:01:31.950
You'd have the number 12, and
you'd see its factors are 1

00:01:31.950 --> 00:01:36.780
and 12, 2 and 6, and 3 and 4.

00:01:36.780 --> 00:01:40.550
So you'll see the same thing,
exactly six factors.

00:01:40.550 --> 00:01:41.970
Problem 13.

00:01:41.970 --> 00:01:46.790
Given pn is equal to 150, where
p is a prime number and

00:01:46.790 --> 00:01:51.520
n is a natural number, which of
the following must be true?

00:01:51.520 --> 00:01:52.860
So, let's just think about it.

00:01:52.860 --> 00:01:55.110
What are all the possible
values for p?

00:01:55.110 --> 00:01:56.620
To do that, we'd have to
figure out the prime

00:01:56.620 --> 00:01:59.540
factorization of 150.

00:01:59.540 --> 00:02:02.020
I'll do that in black.

00:02:02.020 --> 00:02:05.900
So 150, let's do its prime
factorization.

00:02:05.900 --> 00:02:08.680
That's 2 times 75.

00:02:08.680 --> 00:02:12.260
75 is 3 times 25.

00:02:12.260 --> 00:02:15.630
25 is 5 times 5.

00:02:15.630 --> 00:02:22.670
So 150 is equal to 2 times
3, times 5, times 5.

00:02:22.670 --> 00:02:24.980
That's its prime
factorization.

00:02:24.980 --> 00:02:27.730
So p could be any of
these numbers.

00:02:27.730 --> 00:02:34.380
So for example, p could be-- if
p was 2, then n would be 3

00:02:34.380 --> 00:02:36.620
times 5, times 5.

00:02:36.620 --> 00:02:43.890
If p were 3, then n would
be 2 times 5, times 5.

00:02:43.890 --> 00:02:50.910
If p were 5, then n would be
2 times 3, times 5, right?

00:02:50.910 --> 00:02:53.190
Let me write this, make this
a little bit clearer.

00:02:53.190 --> 00:02:57.390
These are my p's, because p
has to be one of the prime

00:02:57.390 --> 00:03:01.060
factors of 150, and then n is
whatever's left over when you

00:03:01.060 --> 00:03:03.000
divide by that p, right?

00:03:03.000 --> 00:03:04.580
In each of these cases.

00:03:04.580 --> 00:03:05.700
So let's look at the choices.

00:03:05.700 --> 00:03:07.520
Which of the following
must be true?

00:03:07.520 --> 00:03:11.380
p is a factor of either
10 or 15.

00:03:11.380 --> 00:03:14.060
Let's see, 2 is a
factor of 10.

00:03:14.060 --> 00:03:16.130
3 is a factor of 15.

00:03:16.130 --> 00:03:18.350
5 is a factor of
both 10 or 15.

00:03:18.350 --> 00:03:20.190
So that actually looks like
the correct answer.

00:03:20.190 --> 00:03:22.080
Let's write that, let's
square that right now.

00:03:22.080 --> 00:03:24.340
Let's look at the other choices,
just to make sure we

00:03:24.340 --> 00:03:25.890
haven't missed anything.

00:03:25.890 --> 00:03:27.740
10 is a factor of n.

00:03:27.740 --> 00:03:30.720
Well, let's see, over here--
well, no, 10 is not a

00:03:30.720 --> 00:03:31.630
factor of this n.

00:03:31.630 --> 00:03:34.290
3 times 5, times 5 is 75.

00:03:34.290 --> 00:03:37.710
10 is not a factor of this
number, so it's not going-- 10

00:03:37.710 --> 00:03:39.670
is a factor of these two
choices, but this is a

00:03:39.670 --> 00:03:40.720
possible n right here.

00:03:40.720 --> 00:03:42.610
75 could be n.

00:03:42.610 --> 00:03:44.600
So this is not the
correct answer.

00:03:44.600 --> 00:03:48.420
n is a factor of either
10 or 15.

00:03:48.420 --> 00:03:49.190
n is a factor.

00:03:49.190 --> 00:03:50.640
Remember these are our
n's right here.

00:03:50.640 --> 00:03:52.380
These are the potential n's.

00:03:52.380 --> 00:03:55.260
This is not a factor of--
this is neither a

00:03:55.260 --> 00:03:56.680
factor of 10 nor 15.

00:03:56.680 --> 00:04:01.360
10 and 15 are smaller numbers
than all of these over here.

00:04:01.360 --> 00:04:03.330
So this is not the answer.

00:04:03.330 --> 00:04:06.210
And then 15 is a factor of n.

00:04:06.210 --> 00:04:07.790
15 is a factor of n.

00:04:07.790 --> 00:04:09.810
So in order to have 15 to
be a factor, you have

00:04:09.810 --> 00:04:10.875
to have a 3 in there.

00:04:10.875 --> 00:04:12.500
Now you look at this
choice, right here.

00:04:12.500 --> 00:04:13.950
2 times 5, times 5.

00:04:13.950 --> 00:04:15.830
This is 50.

00:04:15.830 --> 00:04:17.890
This n right here is 50.

00:04:17.890 --> 00:04:19.760
15 is not a factor of 50.

00:04:19.760 --> 00:04:22.970
So our first inclination
was correct.

00:04:22.970 --> 00:04:27.320
p is a factor of either
10 or 15.

00:04:27.320 --> 00:04:28.570
Next problem.

00:04:32.130 --> 00:04:33.550
All right.

00:04:33.550 --> 00:04:38.820
The greatest common factor
of n and 540 is 36.

00:04:38.820 --> 00:04:40.440
Which of the following
could be the prime

00:04:40.440 --> 00:04:42.940
factorization of n?

00:04:42.940 --> 00:04:44.480
So let's just think about
this a little bit.

00:04:44.480 --> 00:04:47.580
The greatest common factor of
n and 540, so the largest

00:04:47.580 --> 00:04:50.750
number that goes into
both 540 and this

00:04:50.750 --> 00:04:53.470
mystery number n is 36.

00:04:53.470 --> 00:04:55.150
Which of the following
could be the prime

00:04:55.150 --> 00:04:56.930
factorization of n?

00:04:56.930 --> 00:04:59.000
Let's just take the prime
factorization of everything,

00:04:59.000 --> 00:05:02.880
just so we can kind of go down
to the most digestible parts.

00:05:02.880 --> 00:05:10.980
So the prime factorization of
540, let's see, 2 times 70.

00:05:10.980 --> 00:05:17.050
This is the same thing as 3
times 90, and 90 is 3 times

00:05:17.050 --> 00:05:23.500
30, and 30 is 3 times
10, 10 is 2 times 5.

00:05:23.500 --> 00:05:28.580
So the prime factorization of
540 is equal to-- there's two

00:05:28.580 --> 00:05:32.460
2's here-- so it's equal to
2 squared times 3 to the

00:05:32.460 --> 00:05:37.190
third, times 5.

00:05:37.190 --> 00:05:40.070
That's 540's prime
factorization.

00:05:40.070 --> 00:05:43.940
Now, the greatest common factor
of n and 540 is 36.

00:05:43.940 --> 00:05:46.710
36 is equal to what?

00:05:46.710 --> 00:05:51.300
That's-- I'll do it right
here-- 36 is 2 times 18.

00:05:51.300 --> 00:05:55.990
That's 2 times 9, and
that is 3 times 3.

00:05:55.990 --> 00:06:02.120
So 36 is equal to 2 squared
times 3 squared.

00:06:02.120 --> 00:06:10.330
So, n has to be some multiple
of this that does not

00:06:10.330 --> 00:06:12.370
still go into 540.

00:06:12.370 --> 00:06:13.280
Let me show you what I mean.

00:06:13.280 --> 00:06:16.850
Let's see which of these
could be possible prime

00:06:16.850 --> 00:06:18.610
factorizations of n.

00:06:18.610 --> 00:06:20.440
So A is 2 times 3 squared.

00:06:20.440 --> 00:06:23.730
Well, that doesn't work because
remember, n, this

00:06:23.730 --> 00:06:28.890
right here, 36 has to be
divisible into our n.

00:06:28.890 --> 00:06:32.160
36, or 2 squared times 3 squared
is not divisible into

00:06:32.160 --> 00:06:34.340
this, because we don't have
a 2 squared here.

00:06:34.340 --> 00:06:35.870
So A is not our choice.

00:06:35.870 --> 00:06:38.720
I'll do it in red when
I knock out choices.

00:06:38.720 --> 00:06:41.300
So A is not our answer.

00:06:41.300 --> 00:06:43.030
2 squared times 3 squared.

00:06:43.030 --> 00:06:47.810
This is tempting, because 36
definitely goes into 2 squared

00:06:47.810 --> 00:06:49.840
times 3 to the third.

00:06:49.840 --> 00:06:52.420
Sorry, this is 2 squared
times 3 to the third.

00:06:52.420 --> 00:06:54.560
2 squared times 3 squared
definitely goes into that.

00:06:54.560 --> 00:06:56.220
In fact, it goes into
it three times.

00:06:56.220 --> 00:06:58.730
This is 36 times 3 right
here, right?

00:06:58.730 --> 00:07:01.740
Because the difference between
this number and this number is

00:07:01.740 --> 00:07:04.680
the power of 3, and this one
has a higher power of 3.

00:07:04.680 --> 00:07:07.320
So 36 definitely goes
into this number.

00:07:07.320 --> 00:07:09.300
But we have to be
very careful.

00:07:09.300 --> 00:07:14.170
This number, this number right
here, also goes into 540.

00:07:14.170 --> 00:07:20.950
If n was this number right here,
then n would go into

00:07:20.950 --> 00:07:23.810
540, because this is 2 squared
times 3 to the third, 2

00:07:23.810 --> 00:07:25.950
squared times 3 to the third.

00:07:25.950 --> 00:07:29.680
Then the greatest common factor
of n and 540 would be

00:07:29.680 --> 00:07:32.110
this, instead of this.

00:07:32.110 --> 00:07:35.470
So this can't be our answer
because if this was our n,

00:07:35.470 --> 00:07:40.050
then n would be the greatest
common factor of n and 540.

00:07:40.050 --> 00:07:42.640
So this can't be our choice,
because this is actually a

00:07:42.640 --> 00:07:45.980
larger number that goes
into 540, than

00:07:45.980 --> 00:07:47.290
this one, right here.

00:07:47.290 --> 00:07:50.100
Let's look at choice C.

00:07:50.100 --> 00:07:54.160
2 to the fourth times
3 squared, times 7.

00:07:54.160 --> 00:07:57.460
This looks good, because
36 definitely

00:07:57.460 --> 00:07:58.660
goes into this number.

00:07:58.660 --> 00:08:00.665
2 squared definitely goes
into 2 to the fourth.

00:08:00.665 --> 00:08:03.070
It goes into it four times.

00:08:03.070 --> 00:08:04.980
3 squared definitely goes
into 3 squared.

00:08:04.980 --> 00:08:07.180
And then we're multiplying
it by 7.

00:08:07.180 --> 00:08:09.600
And what's good about this is
we're multiplying it by a

00:08:09.600 --> 00:08:14.470
number or a factor that's
not in 540.

00:08:14.470 --> 00:08:17.530
So this looks like our best
choice right here.

00:08:17.530 --> 00:08:19.420
This looks like our
best choice.

00:08:19.420 --> 00:08:21.450
Let's look at the last choice,
just to make sure we haven't

00:08:21.450 --> 00:08:22.750
missed something.

00:08:22.750 --> 00:08:24.710
2 to the fourth times
3 to the fifth.

00:08:24.710 --> 00:08:28.470
This part, right here, is
definitely a multiple of our

00:08:28.470 --> 00:08:31.380
greatest common factor, so
it looks good so far.

00:08:31.380 --> 00:08:35.740
But then we do it times 5.

00:08:35.740 --> 00:08:40.500
If this was our n, if this,
right here, was our n, then

00:08:40.500 --> 00:08:43.409
the greatest common factor
of this number and 540

00:08:43.409 --> 00:08:45.100
would not be 36.

00:08:45.100 --> 00:08:47.920
It would be-- let's see, we
would have 2 to the squared,

00:08:47.920 --> 00:08:50.050
because that's the largest power
of 2 that goes into both

00:08:50.050 --> 00:08:55.050
of them, it would be 2 squared
times 3 to the third, times 3

00:08:55.050 --> 00:08:57.780
the third right-- because that's
the largest power of 3

00:08:57.780 --> 00:09:01.340
that goes into both-- times 5.

00:09:01.340 --> 00:09:04.490
The greatest common factor of
this number and 540 is this,

00:09:04.490 --> 00:09:06.320
which just happens to be 540.

00:09:06.320 --> 00:09:08.720
So it can't be this
choice right here.

00:09:08.720 --> 00:09:11.130
And you get to the same
argument on choice B.

00:09:11.130 --> 00:09:15.150
The greatest common factor
of choice B and 540 is 2

00:09:15.150 --> 00:09:19.660
squared-- because 2 squared goes
into both numbers-- times

00:09:19.660 --> 00:09:22.890
3 to the third, which is a
very different number.

00:09:22.890 --> 00:09:24.850
Or not a very different number,
it's 3 times 36.

00:09:24.850 --> 00:09:27.830
So that's the other rationale
for why B is not

00:09:27.830 --> 00:09:30.120
a legitimate answer.

00:09:30.120 --> 00:09:32.670
Next problem, 15.

00:09:32.670 --> 00:09:34.520
Scroll down a little bit.

00:09:34.520 --> 00:09:36.230
A shipping container.

00:09:36.230 --> 00:09:41.130
A shipping container measures 8
feet by 12 feet by 24 feet.

00:09:41.130 --> 00:09:41.970
Let me draw that.

00:09:41.970 --> 00:09:47.370
So we have 8 feet by 12
feet, and there's 24

00:09:47.370 --> 00:09:49.500
feet, just like that.

00:09:49.500 --> 00:09:51.100
So it maybe looks something
like that.

00:09:51.100 --> 00:09:53.150
That's actually what shipping
containers really do look

00:09:53.150 --> 00:09:55.890
like, looks something
like that.

00:09:55.890 --> 00:09:59.150
The container is to be filled
with identical cube-shaped

00:09:59.150 --> 00:10:01.630
boxes, each having
sides measuring a

00:10:01.630 --> 00:10:03.130
whole number of feet.

00:10:03.130 --> 00:10:05.080
Which of the following
expressions represents the

00:10:05.080 --> 00:10:08.150
smallest number of such
identical boxes that could be

00:10:08.150 --> 00:10:14.160
packed into the container with
no empty space remaining?

00:10:14.160 --> 00:10:17.030
They say the smallest number
of identical boxes, which

00:10:17.030 --> 00:10:19.910
means we want to get the biggest
possible, we need the

00:10:19.910 --> 00:10:23.860
biggest possible boxes that
can be fit-- cubic boxes--

00:10:23.860 --> 00:10:26.510
that can be fit into this
container with no space.

00:10:26.510 --> 00:10:29.330
So the dimensions of those boxes
have to be the greatest

00:10:29.330 --> 00:10:32.610
common factor of
8, 12, and 24.

00:10:32.610 --> 00:10:33.970
So we could write this
right there.

00:10:33.970 --> 00:10:37.110
Dimensions or, let me say the
box side, has to be equal to

00:10:37.110 --> 00:10:42.630
the greatest common factor
of 8, 12, and 24.

00:10:42.630 --> 00:10:45.590
So what's the biggest number the
goes into 8, 12, and 24?

00:10:45.590 --> 00:10:46.660
You could do this
from inspection.

00:10:46.660 --> 00:10:47.750
It's 4.

00:10:47.750 --> 00:10:49.040
And you can verify that.

00:10:49.040 --> 00:10:52.420
If the boxes were 4 on a side,
then in this dimension you'd

00:10:52.420 --> 00:10:55.415
fit 8 divided by 4 of
them, or 2 of them.

00:10:55.415 --> 00:10:58.840
So you'd have one, two boxes
in that dimension.

00:10:58.840 --> 00:11:00.760
In this dimension, over
here, you would have

00:11:00.760 --> 00:11:02.310
three of them, right?

00:11:02.310 --> 00:11:03.750
12 divided by 4.

00:11:03.750 --> 00:11:06.040
You'd have three boxes
in that dimension.

00:11:06.040 --> 00:11:09.980
And in that dimension, you would
have-- let me draw these

00:11:09.980 --> 00:11:12.550
just like that-- you
would have six.

00:11:12.550 --> 00:11:15.770
1, 2, 3, 4, 5.

00:11:15.770 --> 00:11:17.670
1, 2, 3, 4, 5, 6.

00:11:17.670 --> 00:11:21.200
You would have 6, or
24 divided by 4.

00:11:21.200 --> 00:11:23.260
So if you want to know the
total number of boxes you

00:11:23.260 --> 00:11:26.130
fit-- I'm just looking at the
choices, they didn't actually

00:11:26.130 --> 00:11:29.250
do the math-- they say, well,
in this dimension, I have 8

00:11:29.250 --> 00:11:31.600
divided by 4 boxes.

00:11:31.600 --> 00:11:38.230
In this dimension, I have
12 divided by 4 boxes.

00:11:38.230 --> 00:11:43.960
And in this dimension, I have
24 divided by 4 boxes.

00:11:43.960 --> 00:11:45.630
So that's how many
boxes we can fit.

00:11:45.630 --> 00:11:48.710
We could multiply it out, but
this is actually one of our

00:11:48.710 --> 00:11:51.160
choices, right there.

00:11:51.160 --> 00:11:54.770
So the whole key is just to
realize, well, look, if the

00:11:54.770 --> 00:11:57.500
boxes have to be cubes, and
there's no remaining spaces,

00:11:57.500 --> 00:12:01.840
the dimension of the boxes has
to be divisible into all three

00:12:01.840 --> 00:12:04.520
of these-- all three
of these-- numbers.

00:12:04.520 --> 00:12:06.730
And I want the biggest possible
dimension because we

00:12:06.730 --> 00:12:09.370
want the smallest number of
boxes, and that's where we got

00:12:09.370 --> 00:12:11.550
the greatest common
factor of 4.

00:12:11.550 --> 00:12:15.520
And then you say, OK, 4, and if
we have boxes of dimension

00:12:15.520 --> 00:12:18.140
4 feet, and in this direction
we're only going to be able to

00:12:18.140 --> 00:12:22.340
fit 8 divided by 4, this
direction 12 divided by 4, and

00:12:22.340 --> 00:12:25.330
this direction 24
divided by 4.


MTEL Math Practice Test: 16-19

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:01.720
Problem 16.

00:00:01.720 --> 00:00:03.920
Use the diagram below
to answer the

00:00:03.920 --> 00:00:05.690
question that follows.

00:00:05.690 --> 00:00:08.390
This diagram, which
is n by 2n.

00:00:08.390 --> 00:00:11.130
Maybe they want us to do some
type of area or something.

00:00:11.130 --> 00:00:13.630
They shaded in this triangle,
which is about 1/2 of it.

00:00:13.630 --> 00:00:14.920
It's exactly 1/2 of the area.

00:00:14.920 --> 00:00:16.350
Let's see what they
want to ask us.

00:00:16.350 --> 00:00:19.545
The diagram above could best
be used to derive a formula

00:00:19.545 --> 00:00:21.920
for which of the following
quantities?

00:00:21.920 --> 00:00:23.220
Interesting.

00:00:23.220 --> 00:00:25.860
All right, the sum
of the first n

00:00:25.860 --> 00:00:27.700
consecutive odd integers.

00:00:27.700 --> 00:00:29.065
Now does that make any sense?

00:00:29.065 --> 00:00:32.490
Let's see, we have n here, and
they say the sum of the first

00:00:32.490 --> 00:00:35.580
n consecutive odd integers.

00:00:35.580 --> 00:00:39.940
Let's see, if we look at the
shaded area, if we look at the

00:00:39.940 --> 00:00:44.120
height here-- that height here
is 1-- the height of this next

00:00:44.120 --> 00:00:45.970
column, right here, is 3.

00:00:45.970 --> 00:00:48.310
The height of the next
column is 5.

00:00:48.310 --> 00:00:52.410
Every column is incrementing
by 2, so these are the odd

00:00:52.410 --> 00:00:54.620
integers, and there are going
to be n of them, right?

00:00:54.620 --> 00:00:58.010
One, two, three, four, all
the way to n of them.

00:00:58.010 --> 00:00:59.770
And so if we wanted the sum
of them, we would have

00:00:59.770 --> 00:01:02.950
essentially just to figure out
the area of the shaded region,

00:01:02.950 --> 00:01:04.870
so we could use the
diagram for that.

00:01:04.870 --> 00:01:09.060
So, the area of the shaded
region is 2n times n divided

00:01:09.060 --> 00:01:11.780
by 2, which is equal
to n squared.

00:01:11.780 --> 00:01:14.500
Which is a neat outcome by
itself, that the sum of the

00:01:14.500 --> 00:01:17.610
first n consecutive odd
integers is n squared.

00:01:17.610 --> 00:01:19.210
So that's going to
be our choice, A.

00:01:19.210 --> 00:01:21.490
You can actually use
the diagram to

00:01:21.490 --> 00:01:24.060
figure out that formula.

00:01:24.060 --> 00:01:25.140
The other ones don't
make sense.

00:01:25.140 --> 00:01:29.150
The product of the first n
consecutive even integers.

00:01:29.150 --> 00:01:30.720
I don't see where you
get even integers.

00:01:30.720 --> 00:01:32.550
There's a 1 and a 3 and a 5.

00:01:32.550 --> 00:01:34.800
So anything with even
doesn't make sense.

00:01:34.800 --> 00:01:39.000
And then choice D tells us the
product of the first n

00:01:39.000 --> 00:01:40.600
consecutive odd integers.

00:01:40.600 --> 00:01:43.030
But I don't see how this is
helping you visualize what

00:01:43.030 --> 00:01:45.610
happens when you multiply
1 times 3 times 5

00:01:45.610 --> 00:01:47.420
times 7, and so forth.

00:01:47.420 --> 00:01:48.850
Next problem.

00:01:48.850 --> 00:01:50.390
Problem 9.

00:01:53.060 --> 00:01:55.460
Use the diagram below
to answer the

00:01:55.460 --> 00:01:57.340
question that follows.

00:01:57.340 --> 00:02:00.020
OK, it looks like some type
of lattice multiplication.

00:02:00.020 --> 00:02:04.540
The diagram above demonstrates
how the lattice multiplication

00:02:04.540 --> 00:02:07.410
algorithm is used to multiply--
I was right, it's

00:02:07.410 --> 00:02:12.100
lattice multiplication--
multiply 231 by 25 to get the

00:02:12.100 --> 00:02:15.480
product 5,775.

00:02:15.480 --> 00:02:19.760
What value does the circled
digit represent?

00:02:19.760 --> 00:02:22.280
So, if lattice multiplication
looks like something

00:02:22.280 --> 00:02:25.150
completely foreign to you, you
just have to-- well, I'll give

00:02:25.150 --> 00:02:26.440
you a quick primer on it.

00:02:26.440 --> 00:02:28.540
All you're doing, each of
these boxes, you're just

00:02:28.540 --> 00:02:30.220
multiplying the corresponding
digit.

00:02:30.220 --> 00:02:32.200
So 1 times 2 is 2.

00:02:32.200 --> 00:02:33.220
0, 2.

00:02:33.220 --> 00:02:35.370
3 times 2 is 0, 6.

00:02:35.370 --> 00:02:37.450
2 times 2 is 0, 4.

00:02:37.450 --> 00:02:39.700
1 times 5 is 0, 5.

00:02:39.700 --> 00:02:43.610
3 times 5 is 15, 1 5.

00:02:43.610 --> 00:02:46.180
2 times 5 is 10.

00:02:46.180 --> 00:02:50.000
And then, you add up the core,
the different places.

00:02:50.000 --> 00:02:52.640
So this is the 1's diagonal,
right here.

00:02:52.640 --> 00:02:53.610
You have just a 5 there.

00:02:53.610 --> 00:02:54.320
You get a 5.

00:02:54.320 --> 00:02:56.780
This is our 10's diagonal,
right here.

00:02:56.780 --> 00:02:59.810
So it's 20 plus 0
plus 50 is 70.

00:02:59.810 --> 00:03:04.290
And then this is your 100's
diagonal, 600 plus 100 is

00:03:04.290 --> 00:03:05.560
equal to 700.

00:03:05.560 --> 00:03:06.930
So I just answered
your question.

00:03:06.930 --> 00:03:11.000
They said, what does the circled
digit represent?

00:03:11.000 --> 00:03:14.690
Well the circled digit is in
this hundreds diagonal, so it

00:03:14.690 --> 00:03:16.960
represents 100.

00:03:16.960 --> 00:03:19.460
Now, if that still is kind of
unfamiliar and you're like, oh

00:03:19.460 --> 00:03:20.720
gee, how was I supposed to know

00:03:20.720 --> 00:03:22.460
about this 100's diagonal?

00:03:22.460 --> 00:03:24.290
You could think about
it this way.

00:03:24.290 --> 00:03:27.720
This 15 right here is the
byproduct of multiplying that

00:03:27.720 --> 00:03:29.810
3 and that 5.

00:03:29.810 --> 00:03:30.070
Right?

00:03:30.070 --> 00:03:31.900
3 times 5 is 15.

00:03:31.900 --> 00:03:33.410
And then they circled the 1.

00:03:33.410 --> 00:03:35.070
But what is this, really?

00:03:35.070 --> 00:03:36.710
This is really a 30.

00:03:36.710 --> 00:03:37.110
Right?

00:03:37.110 --> 00:03:38.790
This is 231.

00:03:38.790 --> 00:03:41.290
So this is really 30 times 5.

00:03:41.290 --> 00:03:43.320
This is just a regular 5.

00:03:43.320 --> 00:03:46.850
30 times 5 is equal to 150.

00:03:46.850 --> 00:03:51.450
So this 15 is really
representing 150, or this 1 is

00:03:51.450 --> 00:03:53.540
really representing 100.

00:03:53.540 --> 00:03:55.780
And that makes sense because
it's in that 100's diagonal.

00:03:55.780 --> 00:04:00.490
So once again, our answer
would be C, 100.

00:04:00.490 --> 00:04:03.210
Problem 18.

00:04:03.210 --> 00:04:06.600
Use the samples of a student's
work below to answer the

00:04:06.600 --> 00:04:08.950
question that follows.

00:04:08.950 --> 00:04:11.280
Which of the following
statements best describes the

00:04:11.280 --> 00:04:14.360
mathematical validity of the
algorithm that the student

00:04:14.360 --> 00:04:15.820
appears to be using?

00:04:15.820 --> 00:04:17.190
So let's see what the
student's doing.

00:04:17.190 --> 00:04:20.399
9/16 divided by 3/4.

00:04:20.399 --> 00:04:23.530
He or she is just dividing the
numerators and then dividing

00:04:23.530 --> 00:04:25.100
the denominators.

00:04:25.100 --> 00:04:27.380
Now that's not the standard--
the traditional way-- that,

00:04:27.380 --> 00:04:30.970
you know, most of us are taught
of doing division of

00:04:30.970 --> 00:04:31.680
fractions, right?

00:04:31.680 --> 00:04:36.750
The standard way, if we have
a/b divided by c/d-- or at

00:04:36.750 --> 00:04:39.860
least the way I learned to do
it-- I would say that this is

00:04:39.860 --> 00:04:46.060
equal to a/b times d/c, times
the reciprocal of this.

00:04:46.060 --> 00:04:50.110
And I would have gotten that
it's equal to ad over bc.

00:04:50.110 --> 00:04:51.760
This is the standard.

00:04:51.760 --> 00:04:53.790
This is the standard
algorithm.

00:04:53.790 --> 00:04:57.880
Now let's see, this guy over
here, if he takes a/b and he

00:04:57.880 --> 00:05:02.420
divides it by c/d, what he's
getting essentially for the

00:05:02.420 --> 00:05:06.320
new numerator, he takes
a/c, and for the new

00:05:06.320 --> 00:05:09.820
denominator he takes b/d.

00:05:09.820 --> 00:05:13.690
And if you were to simplify
this-- a/c divided by b/d--

00:05:13.690 --> 00:05:21.060
this is the same thing as a/c
times d/b, which is equal to

00:05:21.060 --> 00:05:23.620
ad over bc.

00:05:23.620 --> 00:05:26.710
So even though this looks very
bizarre and actually very

00:05:26.710 --> 00:05:28.720
naive, because he's just
dividing the numerator and

00:05:28.720 --> 00:05:30.840
dividing the denominator,
it works.

00:05:30.840 --> 00:05:33.720
This'll work for any numbers
that you put in there.

00:05:33.720 --> 00:05:36.150
And so, let's see,
A isn't right.

00:05:36.150 --> 00:05:38.710
It is definitely, you know,
try doing this the

00:05:38.710 --> 00:05:39.130
way you know it.

00:05:39.130 --> 00:05:40.980
You'll get these same answers.

00:05:40.980 --> 00:05:41.740
So A isn't it.

00:05:41.740 --> 00:05:43.120
It's definitely, well,
here you go, D.

00:05:43.120 --> 00:05:46.850
It is valid for all rational
numbers, as unintuitive as

00:05:46.850 --> 00:05:47.360
that might be.

00:05:47.360 --> 00:05:49.670
But it looks like this kid, oh,
he just said oh, I'll just

00:05:49.670 --> 00:05:52.210
divide the numerators and divide
the denominators just

00:05:52.210 --> 00:05:54.570
because multiplying fractions,
you kind of do the same thing.

00:05:54.570 --> 00:05:56.260
But it just happens to work.

00:05:56.260 --> 00:05:57.790
In fact, this is probably
how they should

00:05:57.790 --> 00:05:58.710
teach it, or maybe.

00:05:58.710 --> 00:06:01.140
Arguably how they might
want to teach it.

00:06:01.140 --> 00:06:03.220
Problem 19.

00:06:03.220 --> 00:06:05.250
Use the inequality below
to answer the

00:06:05.250 --> 00:06:06.370
question that follows.

00:06:06.370 --> 00:06:07.940
OK, maybe it's an inequality.

00:06:07.940 --> 00:06:10.160
Which of the following
inequalities describes all

00:06:10.160 --> 00:06:12.280
possible values of x?

00:06:12.280 --> 00:06:13.200
So let me write.

00:06:13.200 --> 00:06:20.710
So they're writing 3/x
is greater than 4/x.

00:06:23.410 --> 00:06:28.520
So, if we assume-- well x can't
be equal to 0, because

00:06:28.520 --> 00:06:29.500
that would just be undefined.

00:06:29.500 --> 00:06:34.040
So let's just write that down.
x cannot be equal to 0.

00:06:34.040 --> 00:06:39.580
So to solve this equation, we
can multiply both sides by x.

00:06:39.580 --> 00:06:41.770
We have to be very careful,
though, because if x is

00:06:41.770 --> 00:06:43.990
negative, and you multiply both
sides by it, you'd have

00:06:43.990 --> 00:06:45.290
to switch this inequality.

00:06:45.290 --> 00:06:54.130
But let's just assume
x is greater than 0.

00:06:54.130 --> 00:06:57.250
If x is greater than 0, then
when you multiply both sides

00:06:57.250 --> 00:07:00.940
of these, of this equation, by
x, you don't have to switch

00:07:00.940 --> 00:07:05.920
the inequality, and then you
get 3-- let me put it in a

00:07:05.920 --> 00:07:09.760
different color-- if you
multiply both sides of this

00:07:09.760 --> 00:07:14.030
equation by x, you get 3 is
greater than 4, which is

00:07:14.030 --> 00:07:15.940
definitely, this is false.

00:07:15.940 --> 00:07:16.965
This cannot be true.

00:07:16.965 --> 00:07:19.735
So x cannot be greater than 0.

00:07:19.735 --> 00:07:22.570
So let's see what happens
when x is less than 0.

00:07:22.570 --> 00:07:26.580
If x is less than 0, then when
you multiply both sides of

00:07:26.580 --> 00:07:30.340
this equation by x, you have to
switch the inequality sign.

00:07:30.340 --> 00:07:34.350
So then you would get
3 is less than 4.

00:07:34.350 --> 00:07:36.310
Which is definitely
true, right?

00:07:36.310 --> 00:07:39.490
What I did is just multiplied
both sides of this by x.

00:07:39.490 --> 00:07:41.680
When you multiply both sides
of an inequality by a

00:07:41.680 --> 00:07:43.050
negative, you have
to swap this.

00:07:43.050 --> 00:07:44.890
That's why the greater than
became a less than.

00:07:44.890 --> 00:07:45.800
These canceled out.

00:07:45.800 --> 00:07:48.760
So you got 3 is less than
4, which is true.

00:07:48.760 --> 00:07:55.040
So any x less than 0
will satisfy this

00:07:55.040 --> 00:07:56.950
equation, right here.

00:07:56.950 --> 00:08:01.710
So then you could try it out
as, you know, minus 3/4 is

00:08:01.710 --> 00:08:05.020
greater than minus 4/4, right?

00:08:05.020 --> 00:08:08.720
You could try it out with any
numbers that you might like.


MTEL Math Practice Test: 9-11

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:01.850
We're on problem nine.

00:00:01.850 --> 00:00:03.780
Use the problem below
to answer the

00:00:03.780 --> 00:00:05.570
question that follows.

00:00:05.570 --> 00:00:09.730
Given that 100 milliliters is
equal to approximately 0.4

00:00:09.730 --> 00:00:13.190
cups, 205 milliliters
is equal to

00:00:13.190 --> 00:00:15.760
approximately how many cups?

00:00:15.760 --> 00:00:17.280
So we could set up a
little ratio here.

00:00:17.280 --> 00:00:26.670
We could say 100 milliliters
is to 0.4 cups as 200

00:00:26.670 --> 00:00:30.940
milliliters-- I'll write
the cups here--

00:00:30.940 --> 00:00:33.250
is to how many cups?

00:00:33.250 --> 00:00:34.430
Just like that.

00:00:34.430 --> 00:00:36.920
And so we just have
to solve for x.

00:00:36.920 --> 00:00:39.480
To solve for x, we could
essentially cross multiply,

00:00:39.480 --> 00:00:41.700
which essentially you're just
multiplying both sides of the

00:00:41.700 --> 00:00:43.910
equation by both denominators.

00:00:43.910 --> 00:00:47.170
So we get 100x.

00:00:47.170 --> 00:00:52.110
100 times x is equal
to 200 times 0.4.

00:00:52.110 --> 00:00:57.960
It's not 200, it's 205 to be
exact, 205 times 0.4-- is

00:00:57.960 --> 00:01:02.410
equal to 205 times 0.4.

00:01:02.410 --> 00:01:04.440
And then we want to divide
both sides by 100.

00:01:04.440 --> 00:01:13.050
So x is equal to 205/100
times 0.4.

00:01:13.050 --> 00:01:15.430
Let me write it a little
bit different,

00:01:15.430 --> 00:01:17.950
0.4, just like that.

00:01:17.950 --> 00:01:19.990
We could divide either by 100,
but let's see if any of the

00:01:19.990 --> 00:01:23.730
choices look like what we have
there, something times 0.4.

00:01:23.730 --> 00:01:25.390
Well this one has subtraction
in it, so

00:01:25.390 --> 00:01:26.300
that's not our answer.

00:01:26.300 --> 00:01:27.520
This one, also has

00:01:27.520 --> 00:01:29.360
subtraction, is not our answer.

00:01:29.360 --> 00:01:37.350
105% of 0.4-- that's the same
thing as 1.05 times 0.4.

00:01:37.350 --> 00:01:44.270
205% times 0.4-- that's the same
thing is 2.05 times 0.4,

00:01:44.270 --> 00:01:46.440
which is exactly what
we have here.

00:01:46.440 --> 00:01:49.850
And just to verify, 205 divided
by 100-- I'll do it

00:01:49.850 --> 00:01:54.200
right here-- 100
goes into 205.

00:01:54.200 --> 00:01:57.120
100 goes into 200 2 times.

00:01:57.120 --> 00:01:59.290
2 times 100 is 200.

00:01:59.290 --> 00:02:01.530
Let me put a little decimal
right there.

00:02:01.530 --> 00:02:04.780
You do the subtraction,
you get 5.

00:02:04.780 --> 00:02:06.410
Bring down a 0.

00:02:06.410 --> 00:02:08.860
100 goes into 50 0 times.

00:02:08.860 --> 00:02:11.090
0 times 100 is 0.

00:02:11.090 --> 00:02:12.260
Do the subtraction.

00:02:12.260 --> 00:02:13.670
Bring down another 0.

00:02:13.670 --> 00:02:16.500
100 goes into 500 5 times.

00:02:16.500 --> 00:02:17.690
5 times 100 is 500.

00:02:17.690 --> 00:02:19.845
It all evens out to 0.

00:02:19.845 --> 00:02:21.710
So you get 2.05.

00:02:21.710 --> 00:02:27.200
So this thing is the same thing
as 2.05 times 0.4, which

00:02:27.200 --> 00:02:31.170
is the same thing
as 205% of 0.4.

00:02:31.170 --> 00:02:32.950
Choice D.

00:02:32.950 --> 00:02:34.200
Next problem.

00:02:37.630 --> 00:02:41.166
All right, we are
on problem 10.

00:02:41.166 --> 00:02:43.280
I'll switch colors.

00:02:43.280 --> 00:02:45.930
Use the number line below
to answer the

00:02:45.930 --> 00:02:46.885
question that follows.

00:02:46.885 --> 00:02:48.550
OK, we have the number line.

00:02:48.550 --> 00:02:50.730
What number is represented
by point P on

00:02:50.730 --> 00:02:52.530
the number line above?

00:02:52.530 --> 00:02:57.370
OK, so they give us two
endpoints, 0.003 and 0.004.

00:02:57.370 --> 00:03:00.270
And how many buckets do
we have between them?

00:03:00.270 --> 00:03:02.490
We're going one increment.

00:03:02.490 --> 00:03:04.870
Oh sorry, I'm on the
wrong layer.

00:03:04.870 --> 00:03:09.110
We go one increment, two
increments, three increments,

00:03:09.110 --> 00:03:12.570
four increments, five
increments, six increments,

00:03:12.570 --> 00:03:16.390
seven increments, eight
increments.

00:03:16.390 --> 00:03:21.890
So over eight increments, we
go a distance of how far?

00:03:21.890 --> 00:03:28.470
Well, the distance we go is
0.004 minus 0.003, which is

00:03:28.470 --> 00:03:30.880
just 0.001.

00:03:30.880 --> 00:03:34.200
So this whole distance,
this whole distance

00:03:34.200 --> 00:03:41.340
right there, is 0.001.

00:03:41.340 --> 00:03:45.170
And so each of these
are 1/8 of 0.001.

00:03:45.170 --> 00:03:45.263
Right?

00:03:45.263 --> 00:03:46.560
Let me do that in
another color.

00:03:46.560 --> 00:03:49.210
All right, we have one,
two, three, four,

00:03:49.210 --> 00:03:50.600
five, six, seven, eight.

00:03:50.600 --> 00:03:53.770
Don't want to make
it any dirtier.

00:03:53.770 --> 00:03:56.640
Each of those are
1/8 of 0.001.

00:03:56.640 --> 00:04:03.310
Or you could say it's 8 divided
by or 0.001 divided by

00:04:03.310 --> 00:04:05.940
8 is how much each of these
increments are.

00:04:05.940 --> 00:04:09.040
And what we're at, the point
in question, is exactly two

00:04:09.040 --> 00:04:12.360
increments past 0.003.

00:04:12.360 --> 00:04:14.330
So let's figure out how
far each increment is.

00:04:14.330 --> 00:04:19.160
It's 0.001, which is its entire
distance, divided by 8.

00:04:19.160 --> 00:04:22.920
And the best way we can do this,
we'll just put a decimal

00:04:22.920 --> 00:04:23.420
right there.

00:04:23.420 --> 00:04:25.910
8 goes into 0, 0 times.

00:04:25.910 --> 00:04:30.930
8 goes into-- 0 times 8
is 0, bring down a 0.

00:04:30.930 --> 00:04:33.180
8 goes into 0 0 times.

00:04:33.180 --> 00:04:36.440
Bring down a 0, 0 there.

00:04:36.440 --> 00:04:38.320
Bring down a 1.

00:04:38.320 --> 00:04:41.100
8 goes into 1 0 times.

00:04:41.100 --> 00:04:43.840
0 times 8 is 0.

00:04:43.840 --> 00:04:44.970
Bring down the 1.

00:04:44.970 --> 00:04:47.100
You could bring down another
0, and now it gets

00:04:47.100 --> 00:04:47.640
interesting.

00:04:47.640 --> 00:04:50.020
8 goes into 10 1 time.

00:04:50.020 --> 00:04:51.980
1 times 8 is 8.

00:04:51.980 --> 00:04:53.620
10 minus 8 is 2.

00:04:53.620 --> 00:04:54.600
Bring down a 0.

00:04:54.600 --> 00:04:56.980
8 goes into 20 2 times.

00:04:56.980 --> 00:05:00.040
2 times 8 is 16.

00:05:00.040 --> 00:05:01.690
20 minus 16 is 4.

00:05:01.690 --> 00:05:02.850
Bring another 0.

00:05:02.850 --> 00:05:06.140
8 goes into 40 5 times.

00:05:06.140 --> 00:05:08.820
So it's 0.000125.

00:05:08.820 --> 00:05:13.380
And you might already know that
1/8 is equal to 0.125.

00:05:13.380 --> 00:05:15.630
That's something you might
just know from heart.

00:05:15.630 --> 00:05:20.050
And so, if you have this divided
by-- this isn't--

00:05:20.050 --> 00:05:23.370
we're not dealing with 1/8,
we're dealing with 0.0018, so

00:05:23.370 --> 00:05:27.270
with this divided by 1,000,
you're going to have three 0's

00:05:27.270 --> 00:05:30.790
and then your 125.

00:05:30.790 --> 00:05:32.360
So that's how much each
increment is.

00:05:32.360 --> 00:05:36.160
And we're going two of these
increments, just like that.

00:05:36.160 --> 00:05:40.160
So if you multiply this times
2, what do you get?

00:05:40.160 --> 00:05:47.150
You say 2 times 125 is 250, but
we have one, two, three,

00:05:47.150 --> 00:05:50.420
four, five, six numbers behind
the decimal point.

00:05:50.420 --> 00:05:53.495
So one, two, three,
four, five, six.

00:05:53.495 --> 00:05:55.410
We have to have six numbers
behind the decimal point in

00:05:55.410 --> 00:05:56.370
our answer.

00:05:56.370 --> 00:06:02.950
So that's how much we
move above 0.003.

00:06:02.950 --> 00:06:08.600
So essentially, this is going to
be 0.000125, another 00125,

00:06:08.600 --> 00:06:11.390
and it's all above 0.003.

00:06:11.390 --> 00:06:17.460
So our answer is going to be
0.003 plus this-- plus 0.003,

00:06:17.460 --> 00:06:23.080
which gets us to 0.00325.

00:06:23.080 --> 00:06:26.010
So that is 0.00325.

00:06:26.010 --> 00:06:28.710
And I just realized, there is
an actual much, much easier

00:06:28.710 --> 00:06:29.350
way to think about it.

00:06:29.350 --> 00:06:32.960
Instead of dividing by 8 and all
of that, I could have just

00:06:32.960 --> 00:06:33.890
said, you know what?

00:06:33.890 --> 00:06:35.700
There are eight increments
here.

00:06:35.700 --> 00:06:38.930
I've made this problem messy,
but there are eight increments

00:06:38.930 --> 00:06:43.540
here-- one, two, three, four,
five, six, seven, eight.

00:06:43.540 --> 00:06:44.720
We've gone two of them.

00:06:44.720 --> 00:06:46.650
So we've gone 1/4
of the distance

00:06:46.650 --> 00:06:49.980
between 0.003 and 0.004.

00:06:49.980 --> 00:06:52.360
And that would have been the
easiest way to do the problem.

00:06:52.360 --> 00:06:55.680
1/4 of the distance is
just another 0.25.

00:06:55.680 --> 00:06:58.190
That would've been way easier to
do the problem, especially

00:06:58.190 --> 00:07:01.080
if you didn't want to
waste a lot of time.

00:07:01.080 --> 00:07:02.500
Problem 11.

00:07:02.500 --> 00:07:06.170
A book distributor is trying
to divide an order of

00:07:06.170 --> 00:07:08.680
textbooks into equally-sized
groups

00:07:08.680 --> 00:07:10.230
for shipping in cartons.

00:07:10.230 --> 00:07:13.080
The textbooks can be divided
into groups of 12, groups of

00:07:13.080 --> 00:07:17.580
15, or groups of 18, with
no books left over.

00:07:17.580 --> 00:07:20.550
So that means that the number of
books we have is divisible

00:07:20.550 --> 00:07:22.820
by these three numbers.

00:07:22.820 --> 00:07:27.710
Which of the following
inequalities is satisfied if N

00:07:27.710 --> 00:07:32.420
is the smallest possible total
number of textbooks?

00:07:32.420 --> 00:07:36.110
So essentially, we know that N
is divisible by these three

00:07:36.110 --> 00:07:38.680
numbers, and we want to find
the least possible N that's

00:07:38.680 --> 00:07:40.820
divisible by these
three numbers.

00:07:40.820 --> 00:07:44.240
Or essentially, the least common
multiple of-- so we

00:07:44.240 --> 00:07:46.230
want to-- let me write
this down.

00:07:46.230 --> 00:07:53.570
N is equal to the least common
multiple of 12, 15, and 18.

00:07:53.570 --> 00:07:56.340
And the easiest way for, at
least for me to think about

00:07:56.340 --> 00:08:00.100
this type of a least common
multiple problem, is to just

00:08:00.100 --> 00:08:03.040
take the factors of each
of these numbers.

00:08:03.040 --> 00:08:05.690
Let's take the prime
factorization of these guys,

00:08:05.690 --> 00:08:06.230
right here.

00:08:06.230 --> 00:08:10.510
So 12, if we want to take the
prime factorization of 12,

00:08:10.510 --> 00:08:11.590
it's-- let's see.

00:08:11.590 --> 00:08:13.720
We could do a little tree here,
but it's pretty easy.

00:08:13.720 --> 00:08:19.140
It's 4 times 3, 4 is the same
thing is 2 times 2 times 3.

00:08:19.140 --> 00:08:20.030
That's 12.

00:08:20.030 --> 00:08:22.470
So if we just have the least
common multiple of just 12, we

00:08:22.470 --> 00:08:24.650
would say, well they have
to have a 2 in it

00:08:24.650 --> 00:08:27.180
times a 2 times a 3.

00:08:27.180 --> 00:08:28.590
So we have other numbers here.

00:08:28.590 --> 00:08:29.800
We have the 15.

00:08:29.800 --> 00:08:31.050
Another color.

00:08:34.250 --> 00:08:35.580
15 is equal to what?

00:08:35.580 --> 00:08:40.270
It's prime factorization
is 3 times 5.

00:08:40.270 --> 00:08:43.730
So our answer right now, we
already have a 3 as a prime,

00:08:43.730 --> 00:08:47.870
so to make this divisible by
15, we just have to include

00:08:47.870 --> 00:08:49.930
the 5 only.

00:08:49.930 --> 00:08:51.540
So 3 times 5.

00:08:51.540 --> 00:08:53.800
So this number, right now,
is now divisible

00:08:53.800 --> 00:08:55.650
by both 12 and 15.

00:08:55.650 --> 00:08:56.610
How do I know?

00:08:56.610 --> 00:09:00.000
Because all of 12's constituent
factors, or prime

00:09:00.000 --> 00:09:01.770
factors, are in this number.

00:09:01.770 --> 00:09:04.880
And all of 15's constituent
numbers are in this number.

00:09:04.880 --> 00:09:06.280
So this, right now, is
the least common

00:09:06.280 --> 00:09:08.290
multiple of 12 and 15.

00:09:08.290 --> 00:09:11.580
And then we can throw
18 into the mix.

00:09:11.580 --> 00:09:15.930
So 18, that's the same thing
as 2 times 9, or 2

00:09:15.930 --> 00:09:18.460
times 3 times 3.

00:09:18.460 --> 00:09:20.750
And our answer, here, we already
have a 2 times 3, so

00:09:20.750 --> 00:09:22.770
we already have these two
constituents, but we

00:09:22.770 --> 00:09:23.650
need another 3.

00:09:23.650 --> 00:09:26.510
So let's multiply
by another 3.

00:09:26.510 --> 00:09:29.710
Now this is going to be the
least common multiple of 12,

00:09:29.710 --> 00:09:33.070
15, and 18, because it has all
of the constituents of all of

00:09:33.070 --> 00:09:33.630
these guys.

00:09:33.630 --> 00:09:37.380
You have one 2, and two 3's.

00:09:37.380 --> 00:09:39.270
A 3 and a 5.

00:09:39.270 --> 00:09:41.750
Two 2's and a 3.

00:09:41.750 --> 00:09:44.685
You have all the prime factors
of all of these guys, and now

00:09:44.685 --> 00:09:46.360
you can just multiply.

00:09:46.360 --> 00:09:53.040
4 times 4-- sorry-- 2 times
2 is 4 times 3.

00:09:53.040 --> 00:09:55.890
Let me do it-- times
3 times 5 times 3.

00:09:55.890 --> 00:09:59.730
4 times 3 is 12 times
5 times 3.

00:09:59.730 --> 00:10:05.960
12 times 5 is 60 times 3,
which is equal to 180.

00:10:05.960 --> 00:10:10.440
So that is the smallest total
number of books, the least

00:10:10.440 --> 00:10:12.070
common multiple of these.

00:10:12.070 --> 00:10:14.210
And that means that's
choice B.

00:10:14.210 --> 00:10:19.500
180 is between 150 and 200.