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### Course: GMAT > Unit 1

Lesson 1: Problem solving- GMAT: Math 1
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# GMAT: Math 32

163-166, pg. 174. Created by Sal Khan.

## Want to join the conversation?

- if the length of the longest rod that can be placed in a cubical room is 4*2^0.5 m

,then the area of the four walls of the room is(1 vote)- Well, the dimensions of a rectangular room are length, width and height (l, w, h) so

area will be the sum of the areas of each wall.

For a cubical room, if all we want is the area of the 4 walls, this simplifies to the sum of 4 squares of side**s**

which is 4s²

So A₄ = 4s²

For the rectangular room, the distance of a diagonal is √(l² + w² + h²)

For a cubical room, we can use s = w = l = h

so the length of the diagonal is √(s² + s² + s²) = √(3 s²) which simplifies to s√3

Now you were given a length of 4√2, so we set that equal to s√3 to solve for s

s√3 = 4√2

We COULD divide both sides by √3 and then rationalize the denominator, but,

since we need s², in order to solve the final part, I am going to square both sides

(s√3)² = (4√2)²

s²∙3 = 16 ∙ 2

s² = 16 ∙ 2/3

s²=32/3

4s²=4 ∙ 32/3

=128/3 m² or 42 2/3 m²(1 vote)

## Video transcript

I was interrupted in the last
video by a phone call. I was actually expecting a phone
call from someone whose call I wanted to take, but that
ended up being one of those robocallers. So, anyway, I was
on problem 163. Let me just start it over so
that I don't interrupt your train of thought. A fruit salad mixture consists
of apples, peaches, and grapes in the ratio of 6:5:2. So the ratio of apples to
peaches to grapes is equal to 6:5:2 by weight. If 39 pounds of the mixture is
prepared, the mixture include how many more pounds of
apples than grapes? So they essentially want us to
figure out what the apples minus the grapes are equal to. So we have 39 pounds, so the
pounds of apples plus the peaches plus the grapes,
that equals 39 pounds. And to do a ratio problem like
this, the easiest way is to say let's just define the number
of pounds of apples being 6 times some number x. If the apples are 6 times
some number x, what are the peaches? We know that the ratio of the
apples to the peaches are 6:5, so the peaches are going
to have to be 5 times some number x. By the same logic, the grapes
are going to have to be 2 times some number x. And if you try out the ratio the
apples to the grapes is 6x over 2x, which is 6:2. So it all works out. So let's use this information
to substitute back there and see what x is equal to. So we get 6x plus 5x plus
2x is equal to 39. 6 plus 5 is 11 plus 2 is 13. 13x is equal to 39. x is equal to 3. And so how many apples
are there? 6x, there are 18 apples. We don't have to figure out the
peaches, but we can do it fast. There are 15 peaches, and
then there are 6 grapes. And if you want to know how
many more apples there are than grapes, 18 minus 6 is equal
to 12-- actually these are all in pounds. There are 12 pounds more
of apples than grapes. And that's choice B. Problem 164. Louise has x more dollars
than Jim has. So L is equal to Jim plus x, and
together they have a total of y dollars. So L plus J is equal to y. Which of the following
represents the number of dollars that Jim has? So they want Jim equals,
and they want it in terms of x and y. So let's take this L and
substitute it here so that we have an equation that this has
J's, x's, and y's in it. So if L is equal to J plus
x, we can substitute. So we get J plus x, instead of
an L, plus J is equal to y. That is that, just that. And you get 2J plus
x is equal to y. Subtract x from both sides. 2J is equal to y minus x. And we get J is equal
to y minus x over 2. And that is choice A. Problem 165. Let's switch colors. During a certain season, a team
won 80% of its first 100 games, and 50% of
the remaining. If the team won 70% of its games
for the entire season, so they won 70% of the total,
what was the total number of games that the team played? So how do we figure out the
total number of games that they played? It'll be 100 plus remainder. So let's think of it this way. If we wanted to figure out the
total percentage that they won, we would want the
number that they won. So how many games
did they win? They won 80% of their first 100
games, so that's 80 games, plus 50% of the remainder
games. So plus 0.5 times
the remainder. That's how many games
they've won. Now, how many games
did they play? The total number of games is
going to be 100 plus whatever this remainder number
of games are. That's their total
number of games. And the problem tells us that
this ratio, the total percentage-- this is of all the
games they won divided by all the games they played--
that that is equal to 0.7 or 70%. If we multiply both sides of
this equation by 100 plus r, we get 80 plus 0.5r is equal
to-- 0.7 times 100 is equal to 70 plus 0.7r. Now let's subtract 70
from both sides. You get 10 plus 0.5r
is equal to 0.7r. And then if we subtract 0.5r
from both sides, we get 10 is equal to 0.2r, or r is
equal to 10 over 0.2. And 10 over 0.2, 0.2 is
the same thing as 1/5. So this is 10 times 5, so
that's equal to 50. So the remainder games were 50,
so they want to know the total number of games
the team plays. So it's going to be the
first 100 games plus the remainder games. So 100 plus 50. They played 150 games,
which is choice D. We're on problem 166. Of 30 applicants for a job,
14 had at least 4 years of experience. So experience greater
than 4, we have 14. I just make up notation
as we go. Of 30 applicants for a job,
14 had at least 4 years of experience, 18 had degrees--
so we say degrees, we have 18-- and 3 had less than 4 years
of experience and did not have a degree. So experience was less than
4 years' experience and no degree, and that is 3 people. How many of the applicants had
at least 4 years of experience and a degree? So let's see how we can
think about it. There's 30 applicants
for a job. So let me draw our universe. If I did all of the
applicants, let me draw it like this. So that's the whole applicant
universe, and that's 30. And we know that 14 had more
than 4 years of experience. Let me do that in this color. So this attribute, I'll make it
look like that, so that's the 14 that have experience
greater than 4 years. And that's 14 of them. And then 18 have a degree. And I'm assuming that there's
some overlap between the degrees in the 14. We don't know that for sure, but
let's say that this point right here I'll draw
another circle. So let's say that circle looks
something like that. So let's call this the
degreed people. And there's 18 of them
in this whole circle. And then they say that there
were 3 people with experience less than 4 and no degree. So they're outside of both
of these circles. So this right here is 3. Now, what are they asking? I've already forgotten that. How many of the applicants had
at least 4 years experience and a degree? So they're asking for this
intersection right here. So there's one question
we can answer. What's the sum of this
population, the population that has more than 4 years of
experience or a degree? So let me circle. What's this sum right here? This and this. It's going to be whatever's left
after you take out the 3 people who have nothing. So that's going to
be 27 people. So 27 people have at
least 4 years of experience or a degree. Now, how can relate that
to the 14 and 18? So what happens if we just
add the 14 and the 18? When we have the 14, we're
going to count these people once. And if we were add that
to 18, we would count these people twice. So if we wanted this number--
if we wanted the number of people who are in this circle or
this circle-- what we would want to do is add the 14 to the
18, but then subtract out once the people who are
this intersection. Let's call them B. That's a B. I know you can't read it. Let's say there B people
in that intersection. You'll subtract that out once. I want that to really
make sense for you. If we want to know the total
number of people in this circle and this circle combined,
when we say 18 people, that's including
some people here. And if we were add that to 14
people, where once again some of those 14 are here again,
are these people. So when we just add 18 to 14,
we're counting this twice. So if we want to know the actual
number in the circle or the circle, since we add 18 to
14 we count it twice, we wants to subtract these
people out once. Hopefully that makes sense. But anyway, now we're
ready to solve. We get 27 is equal to--
14 plus 18 is equal to 32 minus B. 27 from both sides, you get
0 is equal 5 minus B. Add B the both sides. You get B is equal to 5. And that is choice E. See you in the next video.