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## GMAT

### Course: GMAT > Unit 1

Lesson 1: Problem solving- GMAT: Math 1
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# GMAT: Math 30

154-160, pg. 173. Created by Sal Khan.

## Want to join the conversation?

- 04:55

I am confused here, could anyone help, please. It said "the tank contains 10,000 gallons of a SOLUTION", as I understand it, this SOLUTION is a combination of WATER + SALT. And as the salt is 500 gallons, it makes the water 9,500 gallons. If 2,500 gallons of water evaporates, the remaining water will be 9,500 - 2,500 = 7,000 gallons? Which will give a much different answer.

Could anyone please explain? Thank you so much(4 votes) - 154. what did i do wrong? i was okay up to this point.

i got 40/3 = 5x/2,

cross multiply to come up with 20/3 = 5x,

x = 4/3.

x=1 1/3(1 vote)- 40/3 = 5x/2

Cross Multiply and then you come up with:

40*2 = 5x*3

And then:

80 = 15x

You divide 15 to both sides:

80/15 = x

5.3333... = x

Which is:

5 and 1/3(3 votes)

- 154. 8/3 = x/2

why cant i now cross multiply to come up with 4/3 = x?(1 vote)- 8/3 = x/2

You cross multiply:

8*2 = 3*x

Which is equal to:

16 = 3x

Divide both sides by 3:

5.3333 = x

Which is equal to:

5 and 1/3 = x(3 votes)

- 154. why cant i say : 8/.6 = x/.4 then cross multiply .6x=.32 then divide and get 5.3 which is 5 hours and 30 minutes ?(2 votes)
- 154. I got a alternate way to solve it with a similar reasoning.

In 10000 Liters 5% is Sodium Chloride ie. 500 kgs

2500 liters evaporate so we are left with 7500 liters. Also we know the salt doesn't change

So 500 = x% of 7500 [500 is the salt which stays the same even after the water evaporates]

500 = x/100 x 7500

x=500x100 /7500

= 6.6666 or 6.67%(1 vote) - 155. At2:21, could I divide both sides by X instead of 3? Then I wouldn't have to move the three back over?(1 vote)
- 160. Another easier option with less Math involved.

There are two circles as in the figure and the Radius of the larger one is *(8+3) while that of the smaller one is 3. We need to find the area of circular path.

R1 = 11 and r2 = 8

Set the formula as the

Area = pie (R square - r square)

= pie (11 square - 8 square)

remember a square - b square = (a+b) (a-b)

so the equation above is pie times (11+8) (11-8) = pie (19)(3) = 57 pie

saves us a lot of multiplication and squares if we use this formula.(0 votes)- You are right. The difference of squares does work here for circles. The problem with using it for all paths is that it won't work for all paths(Like for example it won't work for a hexagonal path to take the difference of squares with the radii of the 2 hexagons).(1 vote)

## Video transcript

We're on problem 154. An empty pool being filled with
water at a constant rate takes 8 hours to fill to
3/5 of its capacity. So you could say you can fill
3/5 of a pool every 8 hours, which is just another way of
saying that you can fill, let's see, 3/5 divided by 8 is
3 over-- what's 5 times 8? 3/40 pools per hour. I have a feeling that they're
just going to ask us something along just the rate
of the fill. So then they say, how much
more time will it take to finish filling the pool? Oh, well, this actually is
a little bit easier. So if it takes 8 hours to fill
the pool to 3/5 of capacity, how many hours will it take to
finish filling the pool? So how many hours will it take
to fill it essentially what? What's left? 2/5, right? To fill it 2/5 more. So this is going to be the same
thing as 8/3 is equal to x/2, And you could do that by
dividing both sides of this equation by 5, right? And you get a 5 in the
denominator in both places, which cancels out. And this is easy, 8/3 as x/2 so
you have 16 is equal to 3x. And then you get x is equal
to 5 1/3 hours. That's 5 hours and 20 minutes,
because a third of an hour is 20 minutes. And that is choice B. Problem 155. A positive number x is
multiplied by 2 and this product is then divided by 3. And they told us
x was positive. If the positive square root
of the result of these two operations equals x, what
is the value of x? Let's solve this. Let's square both sides of this,
so we get 2x/3 is equal to x squared. And then multiply both sides by
3, you get 2x is equal to 3x squared. And they tell us that
x is positive. Well, first of all, that tells
us that x does not equal 0. They tell us that x
is greater than 0. So let's see, let's divide
both sides of this equation by x. Well, there's a couple of ways
we could think about it. Maybe the easiest way-- yeah,
let's just divide both sides of this equation by x, and we
can do that because we know that x cannot equal 0 and you
get 2 is equal to 3x. And then divide both sides by 3,
you get x is equal to 2/3, and that's choice D. I was thinking about saying,
oh, well, let's subtract 2x from both sides. You get 3x squared minus
2x is equal to 0. You could factor out an x, and
then you get 3x minus 2 times x is equal to 0. And that would tell you that
either 3x minus 2 is equal to 0, which would give you that
answer, or x is equal to 0. but? X can't equal 0, because they
told us that x is positive. But anyway, it gets you
the same answer, 2/3. Next question. Problem 156. A tank contains 10,000 gallons
of a solution that is 5% sodium chloride by volume. What is that? NaCl? This isn't chemistry. Let's see, 2,500 gallons of
water evaporate from the tank. The remaining solution will be
approximately what percent sodium chloride? So they're essentially saying,
well, if water evaporates, the salt in the solution doesn't
go away, right? The salt doesn't evaporate
with the water. So if we're 5% sodium chloride,
what's 5% of 10,000? 5% of 100 is 5, 5% of 1,000 is
50, 5% of 10,000 is 500. I guess you could
say there's 500 gallons of sodium chloride. If we lose 2,500 gallons of the
water, now the percentage of sodium chloride is still
500, but it's going to be 500 over what? We don't have 10,00 gallons
of water anymore. We have 10,000 gallons of water
minus 2,500 gallons of water, so we have 7,500
gallons of water. Let's see, that's equal to 5/75,
which is equal to 1/15, and what is that as
a percentage? 15 goes into 100 6 times? It's not 7. Yeah, 6 times. 6 times 5 is 30. 6 times 1 is 6 plus 90. And then I have a
10, so it keeps going on, 0.0666 repeating. So it's 6 and 2/3 percentages,
or 6 point-- well, let's see, they have choice-- yeah,
choice D looks right. 6.67% if you round this. It just keeps repeating
with 6. So that is choice D. Problem 157. If x plus 5y is equal to 16 and
x is equal to minus 3y, then what does y equal? So let's just substitute
this over here. So we get minus 3y plus
5y is equal to 16. Add those together, you get
2y is equal to 16, or y is equal to 8. And that is choice E. Problem 158. A committee is composed of w
women and m men, so w and m. If 3 women and 2 men are added
to the committee and if one person is selected at random
from the enlarged committee, then the probability that a
woman is selected can be represented as? OK, the probability of a woman
being selected is essentially the fraction of the population
that are women. So how many women are there? There's going to be
w plus 3 women. And what's the total
population? There's w plus 3 women, and
then there's m plus 2 men. So this is the probability we
select a woman, and then we can just simplify this a bit. That equals w plus 3 over
w plus m plus 5. And that is choice E. Problem 159. If the product of the integers
w, x, y, and z is 70. So let me just write that, w,
x, y, z is equal to 770. And if 1 is less than w, which
is less than x, which is less than y, which is less than z,
what is the value of w plus z? Ok, this is interesting. So they're telling us that
these are integers. We have four integers, and
they're all greater than 1. Let's see if we can do a prime
factorization of 770, because I have a feeling that that is
what it's going to come down to, if we just look at
the factors of 770. So let's see, that's
equal to 7. 7 times 11 is 77. 7 times 11 times 10, right? 7 times 11 times 10 is 770. 110. And then 10 is-- so that's equal
to 7 times 11 times-- let's see, what's 10? 10 we can rewrite
it as 2 times 5. So that's its prime
factorization. And lo and behold, we have
exactly four numbers, and let's put them in order. So that equals 2 times
5 times 7 times 11. And so this is what w, x, y,
and z have to be because they're different numbers. They're all greater than 1,
and they're all integers. So let's see, they want to know
w plus z, so w plus z is 2 plus 11 is 13, and
that's choice B. And looks like we have time
even for another one. Let's do this one. The figure above shows
a circular flowerbed. Let me see if I can draw it. So that's the outside and then
they drew the inside. So if I do it like-- oh,
I didn't do a good job. Let me see if I can
draw the inside, maybe if I put it there. It's not showing up
the way I want to. Let me move it around
a little bit. Let me just erase it and try
to draw it one more time. Well, I think you
get the idea. And they tell us that the
distance from the center to here is 8 feet. The figure above shows a
circular flowerbed with it's center at 0 and surrounded
by a circular path that is 5 feet wide. So this is 5. That is 3 feet wide. I'm having trouble reading. 3 feet wide. What is the area of the
path in square feet? So it's essentially the area of
this outside circle minus the area of this
inside circle. That would be the area
of this path. So what's the area of
the outside circle? Well, the area of the outside
circle, you go 8 to get to this edge, then you
have 3 more. So its radius is 8 plus 3 is 11,
so the area of the outside is equal to pi times
11 squared. And what's the area
of the inside? The area of the inside is equal
to pi times the inside radius times 64 squared. So the area of our path is
going to be area of the outside minus the area of
the inside, which is equal to 121 pi. That's 11 squared times pi
minus-- oh, sorry, it's 8 squared, which is 64. I should have written
8 squared there, so that's minus 64 pi. And that is equal to--
what's 121 minus 64? 121 minus 64 is-- let's see,
we have 21 plus 36, that's equal to 57 pi, and
that is choice D. See you in the next video.