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## GMAT

### Course: GMAT > Unit 1

Lesson 1: Problem solving- GMAT: Math 1
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# GMAT: Math 25

129-132, pgs. 169-170. Created by Sal Khan.

## Want to join the conversation?

- For #130, I need someone to explain to me in a different way why choice E, |x - -1| is <= 4, is the correct answer. The video begins around4:30. More specifically, why does Hal use a negative minus 1 which turns into a plus positive 1?(2 votes)
- I don't have the printed material to refer to so just going by the video.

This refers to the Modulus of a number. Look up Modulus on the internet. It refers to an unsigned number or a number separate from its (+) or (-) signature.

Now (-) x (-) gives you a (+) so that takes care of the x+1

The Modulus |x+1| is now less than or equal to 4 because at this point we are only talking about an absolute value of x and disregarding its positive or negative signature.(2 votes)

- Q 129: In the 11th edition book, the answer uses 284 and 11.4. I can't understand why. Does it have something to do with that fact you stated that it never approaches 11.5 and it's always 11.4999..?(1 vote)
- At5:19In Q 130 , why do we take midpoint i.e -1 and secondly why do we subtract -1 from it?(1 vote)
- Wait so it this kind of like percents?(1 vote)
- what is the square root of 20x to the 6 power over 25 to the simplest form(1 vote)
- hmm, so you mean √((20x⁶)/25) ? or do you mean (√(20x⁶)/25) ?

for √((20x⁶)/25)

Let's use this handy equivalent rule: √(A/B) = √A/ √B

√((20x⁶)/25) = √(20x⁶)/`(√25)`

= √(20x⁶)/`5`

Now we can use this other handy equivalent rule: √AC = √A ∙ √C

so, √(20x⁶) =√20 ∙ √x⁶

The square root of a power (base with an exponent) is going to be that exponent divided by 2 . Because x³ ∙ x³ = x⁶, then √x⁶ = x³

Making progress!

What we have left is (x³ ∙ √20)/5

All we need is √20)

For that we will use √DE = √D ∙ √E

√20 = √(4 ∙ 5) = √4 ∙ √5 = 2√5

At last, putting them all together:

√((20x⁶)/25) = 2/5 x³√5

Likewise, if you meant

(√(20x⁶)/25)

Then the answer, simplified, is

2/25 x³√5

Sorry, hope you weren't in a hurry for that!(1 vote)

## Video transcript

We're on problem 129. On a recent trip, Cindy drove
her car 290 miles, rounded to the nearest 10 miles, and used
12 gallons of gasoline, rounded to the nearest
gallons. The actual number of miles per
gallon that Cindy's car got on this trip must have
been between? So they want us to kind of say
these are the rounded numbers, now what were the actual numbers
between, and then from that what's the range
of miles per gallon? So if this is rounded to the
nearest 10, then her miles had to have been greater than or
equal to 285, because anything 285 or more would have
been rounded to 290. But then less than 295 because
anything less than 295 would be would be rounded down,
but 295 itself would be rounded up to 300. And her gallons had to have been
greater than or equal to 11.5 gallons, by the
same logic, and less than 12.5 gallons. So what was the lowest possible
miles per gallon? So the lowest possible miles
per gallon would have the lowest miles and the
highest gallons. You want to minimize the
numerator and maximize the denominator. So lowest possible miles
would be 285. And the highest possible
gallons, at least it'll approach 12.5. So 12.499999 is the possible
number of gallons, so it'll approach this. But it can never be exactly 12.5
gallons, so her miles per gallon are going to be
greater than this. It would've been greater than
equal to this with like 12.499999, but since 12.5 can
never get there, we're going to be greater than this in terms
of miles per gallon. And then the highest possible
miles per gallon is the highest possible miles,
which is 295. But we're never going to be able
to go twice quite there. We're always going to
be less than 295. And then the lowest number of
gallons-- going the highest mileage with the lowest possible
gallons, you want to minimize the demonimator--
divided by 11.5. Let's see if that is a choice. I don't see that exactly, but
choice D looks interesting because choice D says-- so they
say the actual number of miles per gallon that Cindy's
car got on this trip must have been between. Choice D looks close to what we
have because we have this. Choice D says between
284 divided by 12.5. That's going to be even less
than that, because you have the 12.5 but the numerator's
less. So that's going to be even lower
than the range that we have. And then on choice D,
295 divided 11.4. This number is larger than this
number, because you have the same numerator and you have
a smaller denominator. So you're dividing by
a smaller number. So this number's larger. So if her miles per gallon is
between these two numbers, and these two numbers are
between these two numbers, then D is right. Her miles per gallon need to be
between 284 divided by 12.5 and 295 divided by 11.4. So it's choice D. Problem 130. Which of the following
inequalities is an algebraic expression for the shaded part
of the number line above? So this number line, let's
try to draw it. That's the line. And I'm not going to draw all
of the dots on it, but it essentially starts at positive
3 and it goes backwards. Let's switch colors. It starts essentially from
minus 5 to positive 3. That's all of what
this includes. This includes minus 5 to
positive 3, then you have 2, 1, 0, and then minus 5. So you could say that x is
less than or equal to 3 because it's filled in at
3, and greater than or equal to minus 5. And that's not one
of the choices. They want it in terms
of absolute value. So how we can view this, this is
a range around some number. Whenever you're dealing without
absolute value, you're essentially saying the distance
from some number. So this is a range
of minus 5 and 3. This has a length of 8,
so they're both 4 away from minus 1. Think about it. You could go 4 backwards
or 4 forwards. So one way to think about it
is the difference between x and minus 1, and that might
be a negative or positive difference depending on whether
we're positive or negative, depending on whether
we're above negative 1 or below negative 1. But we just want the absolute
difference. We just want the distance
away from negative 1. And that's given by this, the
difference between any x and negative 1. And we took the absolute value
in case x is less than negative 1. The distance is going to be
less than or equal to 4, because if we go above negative
1 we're at most at 3, which is 4 more than
negative 1. If we go below negative 1, we're
at minus 5 which is 4 less than negative 1. So it's less than
or equal to 4. So if you simplify this, you
get x plus 1, the absolute value is less than
or equal to 4. And that is choice E. And this is a useful skill. You have probably taken the SAT
already, but in general, a lot of standardized tests think
of absolute value as a distance in 1 dimension. So the distance along
the number line. So they're saying all of these
points, they're within some range from 1 point. And you can say if you take the
midpoint of this, you're at negative 1. And every point here's within
4 of negative 1. So the distance between every
point on this line, every x that satisfies this and negative
1, the distance is less than or equal to 4. Anyway, I kind of just did the
problem twice for you. Problem 131. A factory has 500 workers,
15% of whom are women. So 15% times 500 is
equal to women. If 50 additional workers are
to be hired and all of the present workers remain, how many
of the additional workers must be women in order to raise
the percent of women employees to 20%? So how many women workers
do we have right now? 5 times 15, we have
75 workers. 15 times 500. We have 75 women right now. If we wanted to find the
proportion of women, it's 75 over 500, which is
once again 15. It's 15%. We're going to add a total of 50
workers to this population. And they want to know how many
of the additional workers must be women in order to raise
the percent of women population to 20%. So we're adding 50
total workers. So how many of those have to be
women in order for the new proportion-- this is going to
be the new number of women, the number of women that are
added to the 75 that were already there-- divided by the
500 workers who were there, plus the 50, some of whom
might be women. So this now has to equal 20%. We get 75 plus w over
550 is equal to 0.2. So we get 75 plus the number of
women that need to be added is equal to 0.02 times 550. We multiply both sides
times 550. That equals 110. And then subtract 75
from both sides. Women has to be equal
to 110 minus 75. And that's equal to-- 25 plus
10-- that's equal to 35. 35 of the 50 employees
have to be women. And that's what they asked. How many of the additional
workers must be women? So that's choice E. Next question, 132. At a small snack shop, the
average arithmetic mean revenue was $400 per day
over a 10 day period. So the average over
10 days was $400. During this period, if the
average daily revenue was $360 for the first 6 days-- so the
average of 6 days was $360-- what was the average daily
revenue for the last 4 days? So we've done a lot of
problems like this. Hopefully this is a bit of
second nature if you've watched the solution
to the other ones. But what's the average
of all of them? It's the average of
the first 6 days. We can almost assume that the
first 6 days had exactly $360 in revenue. So their average times 6. So this would be the total
revenue of the first 6 days plus what's the total revenue
of the next 4 days? It's going to be the average
of the next 4 days times 4. That would be the total
of those 4. We're not saying that they all
have to be exactly the average, but when you sum them
all up they're equal to the average times 4. Now if you divide all of those,
this is a total of the revenue all 10 days. If you take that sum and divide
it by 10, you get the average for the 10 days. And that is equal to $400. Let's simplify this
a little bit. Let's see what we could do. So we get 360 times 6 plus the
average for 4 days times 4 is equal to 10 times 400,
which is 4,000. Before I get too involved in
multiplication, let's divide both sides of this
equation by 4. So 360 divided by 4, that's 90,
times 6 plus this divided by 4 plus A4 is equal
to 1,000. 90 times 6 is 540 plus
the average of the 4 days is equal to 1,000. So the average of the 4 days is
equal to 1,000 minus 540. Which is what? That's $460 per day. And that's choice D. And I'm out of time. See you in the next video.