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## Problem solving

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# GMAT: Math 13

## Video transcript

We're on problem 70. If x plus 5 is greater than 2,
and x minus 3 is less than 7, the value of x must be between
which of the following pairs of numbers? Well for this top one, let's
just subtract 5 from both sides of this equation. If you subtract 5 from both
sides, you get x is greater than, what's 2 minus 5, x
is greater than minus 3. For this bottom one, let's
add four to both sides of the equation. So x minus 3-- sorry,
lets add 3 to both sides of this equation. So x minus 3 plus 3, that's x,
is less than, what's 7 plus 3? 10. So x is greater than minus 3--
I just wrote this the other way around-- and it's
less than 10. So that's A. Problem 71, a gym class can be
divided into 8 teams with an equal number of players on each
team-- 8, that means you get an equal number-- or into 12
teams with an equal number on each team. Fair enough. What is the lowest possible
number of students in the class? So essentially it's saying, that
the number of students in the classes is divisible
by both 8 and 12. So what they want to know is,
what's the smallest number that's divisible by
both 8 and 12. That's the least common
multiple of 8 and 12. That might ring a bell
from middle school. So what's the smallest number
that's divisible by 8 and 12? So let's think about
8's multiples. 8 is indivisible by 12. 16 is indivisible by 12. What about 24? Well yeah, sure, 24 is
divisible by 12. So the least common multiple
is 24, which is B. Clearly you can divide a gym
class into 3 groups of 8, or 2 groups of 12. 72. At least 2/3 of the 40 members
of a committee must vote in favor of a resolution
for it to pass. What is the greatest number of
members who could vote against the resolution and still
have it to pass? So essentially they're saying,
what is the least number that we need to have it pass, and
then 40 minus that is the greatest number that could
vote against it. So we have to have at
least 2/3 of 40. So what's 2/3 times 40? Well that's equal to 80/3. That's 3 goes into 80,
let's figure it out. 3 goes into 8 2 times. 2 times 3 is 6, 20. 3 goes into 20 6 times. 6 times 3 is 18. You bring down a
2 and then a 0. So it's going to just
keep repeating. So you have to have at least
26.6 people vote for it. But clearly you can't have
a 0.6 person vote for it. 26 people isn't going
to be good enough. So you need at least
27 people. So if 27 are going to
vote for, how many going to vote against? Well that means 40 minus
27, that means 13 against, which is E. That is the greatest number. Because if a larger number voted
against it, you would have less people
voting for it. Problem 73. In the Johnson's monthly budget,
the dollar amounts allocated to household expenses,
food-- so I'll say h for household expenses, f for
food-- and miscellaneous items, are in the ratio of
5:2:1-- fair enough-- respectively. If the total amount allocated
to these three categories is $1,800, so $1,800 is the total,
what is the amount allocated to food? So we could view it this way. So food is going to be twice
times miscellaneous, and household items is going to
be equal to 5 times the miscellaneous. So we could say that household
plus food plus miscellaneous is going to be equal to $1,800
This ratio tells us that food is 2 times miscellaneous,
household is 5 times miscellaneous. So let's just substitute
that in. 5 times miscellaneous plus 2
times miscellaneous plus miscellaneous is equal to $1,800
Let's see, 5 plus 2 is 7 plus 1 is 8. 8 times miscellaneous is equal
to $1,800 So m is equal to $1,800 over 8. Let's see, what does
that turn into? That is equal to 900 over 4,
which is equal to 450 over 2. That's how my brain works,
I have to go in steps. What's 450 divided by 2? That's 225. That's miscellaneous. They want to know how much
is allocated to food. Well, food is 2 times
miscellaneous. So 2 times 225 is equal
to, we're back at 450. And that is choice D. Problem 74. There are 4 more women
than men on Centerville's Board of Education. So 4 more women, so women
is equal to men plus 4. If there are 10 members on the
board, how many are women? So we know that men plus
women is equal to 10. We want to solve for women. So let me solve for men here,
then substitute there. So you get men is equal
to 10 minus the women. We could substitute
that back there. We get women is equal to m, or
the men, which we just figured out is 10 minus the
women plus 4. My brain was jumping ahead. So let's add w to both sides. So you get 2w is equal to 10
plus 4, which is equal to 14. w is equal to 7. There are seven women on
the board, so that's D. Problem 75. After this video, I think I'm
going to call it for a night and go watch a movie
or something. 75. Leona bought a 1 year $10,000
certificate of deposit that pays interest at an
annual rate of 8%. So 1 year, $10,000, and
it paid a rate of 8%. Compounded semi-annually, what
was the total amount of interest paid on the certificate
at maturity? So when you say compound
semi-annually, that means you collect half of the interest
every half year. That's something I don't know
if the GMAT, they should be expecting you to know that, it's
something you probably would learn in business
school. But anyway, so after 6 months
you're going to get 4% on the original amount. So you're going to have
$10,000 dollars. And you're going to 4%
percent of this. So they're going to owe you--
compounded semi-annually, what was the total amount of
interest paid on the certificate at maturity? So they're going to pay you
$10,000 times half of it, because only six months
have gone by. 8% is for the full year. So for half the year, you're
going to get times 4%. So let's see, $10,000 times
4%, what is that? $10,000 times 0.04. Let's look at it this way, 4
times $10,000 is $40,000. We have two numbers behind the
decimal place, so it's $400. So they're going to
pay you $400. So then you can view
it essentially, they owe you $10,400. That's essentially how
much they borrowed. Because they're not going to
give it to you just then. You're going to get all your
money back at the end. So at the end, they're going to
have to pay you another 4%. But this time, it's not going
to be off of the $10,000. It's going to be off
of the $10,400. So $10,400 times 0.04. 4 times 0 is 0. 4 times 0 is 0. 4 times 4 is 16. 4 times 0 is 0 plus 1 is 1. 4 times 1 is 4. Then we have two numbers behind
the decimal point. So now they're going to pay you
$416, the second payment. What was the total amount
of interest? So it's going to be $416 plus
$400, which is $816, which is choice C. I'm a little surprised that they
expect you to know what compounding semi-annually
means. But since it's there, I guess
you should know it. When it compound semi-annually,
it means that each semi-annual point in time,
you divide the annual interest rate by 2, and
you get that much. But then the next interest you
get is off of, not just the initial amount, but the initial
amount plus the previous interest, because they
haven't paid you yet. But anyway, I hope you don't
find that too confusing. You might want to look it
up on the internet. You could also compound
monthly, or daily, or continuously. I actually have a couple of
videos on the Khan Academy. But anyway, I'm all out of time,
so I'll see you in the next video.