Hi I'm Erika one of the senior curriculum managers here at Magoosh and GMAT expert
and in today's video we're going to talk about another GMAT word problem. Now if you've watched
my video on the six hardest GMAT math problems you know that statistics is a concept that a
lot of GMAT test takers struggle with and that's mostly because there are so many fun variations
on basic statistical concepts that make them not so basic anymore. So that might be working
with consecutive integers in a set that might be working with tricky weighted average problems or
that might be working with things like min/max now if you want to check out a really cool video
on a strategy for weighted average problems I recommend checking out Chris's video on the
subject but in today's video we're going to talk about min/maxing specifically in the context of
word problems. So min/max problems are always going to be centered around our basic statistical
definition. So that's going to be mean or average it's going to be median that's going to be range
that's going to be mode so hopefully our definitions for those are rock solid.
So the trickiness of min/max problems is that they take one of those statistical definitions
and then they build a scenario or a set around it and then they pick one or several of the
numbers in there and they say they want to either minimize it or maximize it. So they
either want to make it as small as possible or as large as possible and then it's up to us
to figure out how to manipulate the other numbers within that statistical definition within that set
in order to make that happen so if we want to minimize one number do we want to maximize the
other numbers or do we want to minimize them as well if we want to maximize one number
do we want to do what to these other numbers. So these problems require kind of a logical approach
to this specific scenario to this specific statistical definition to the specific question
that they're asking. So with that in mind let's go ahead and try a GMAT min/max problem from Magoosh.
All right let's start by addressing the elephant on the slide which is that this is not really a
word problem. Yeah, I mean sure it has words but it doesn't really have a scenario that we'd come
to expect from a classic GMAT word problem. Now I picked this problem because we showed it in
our six hardest GMAT math problems video um but also because this problem could really easily be
turned into a word problem. So that's something we're going to explore after we work through
the problem itself. So starting from the top the average okay so we know that we're working with
an average here, so we know we're working with the average formula. Which as a reminder is sum
over number equals average. So the sum of the terms over the number of the terms equals the
average of the terms, okay. So we have the average of four different integers is 75. So working with
this average equation I'm gonna go ahead and say my four different integers are A,
B, C, and D just to represent them for now. Okay so the sum of the four different integers would
be A plus B plus C plus D right. The number we know from the question stem is 4 A, B, C,
and D and the average is 75 . Okay great I can work with that. Now I could keep going with the
question stem at this point but what I'm going to do first is I'm actually going to manipulate this
a little bit by multiplying by 4 or the number on both sides and that's just going to get rid
of my fraction and make this a little bit easier to look at to conceptualize and to work with. So
multiplying by 4 on both sides gives me A plus B plus C plus D equals 300.
Continuing, if the largest integer is 90 okay so we've got some max language with "largest" so that's
going to be important to note I'm going to go ahead and decide that my largest integer is D and
I'm just going to put in 90 for it okay so because this is the largest integer I know that A, B,
and C are not going to be larger than 90. Great, that's important information. Moving on, what is
the least possible value of the smallest integer okay so now I'm seeing some min language here. So
I am looking to minimize the smallest integer so I'm going to go ahead and decide that that's A.
Now what does that mean for B and C if I want to minimize A? Well thinking
about this logically I know that the sum is 300 so that means if a is going to go down
then B and C are going to have to come up an equal amount. So if I want to bring a down as far
as it'll possibly go make it the least minimize it that means I'm going to have to bring B and C
up as far as they will go so I want them to be the most that they can be the largest they can be
the maximum they can be all right. So what does that make B and C? Well if we were missing one
of the words in this question we would go okay 90 is the largest possible integer in this set
of numbers so that means B and C should also be 90 which would lead to one of our wrong answers
but there's actually a word in here that prevents that from being the case and that is this word
"different" and this is a really important word in GMAT math problems especially when we're dealing
with min/max problems like this because it means that B and C can't be 90 because we already have
a 90. So, what does that mean? Well if we want C to be as big as possible but not 90 and not
larger than 90, that makes it 89. Okay well then what does that make B? Well B again has to be a
different number as large as possible but not larger than 90. So it can't be 90, it can't be
89. So that makes it 88. Now we can manipulate and solve all right adding these numbers up. The trick
that I'm going to use is to add together three 90s and then subtract 3 and that's going to give me a
plus 267. All right equals 300 and then I'm going to subtract 267 from both sides
and that gives me A equals 33 or E. Now notice here if we had put in 90s for B and C we would
have gotten answer choice D and then these other problems are going to have to do with more of a
misunderstanding of the problem. So in order to get this problem right we had to have an understanding
of basic statistical concepts we had to be able to think logically about minimizing and maximizing.
So if we are minimizing one value we are likely maximizing other values and then we had to read
really really really carefully to make sure we didn't miss any critical information
in the question stem now like I mentioned earlier this question could really easily
be rewritten as a scenario driven word problem while retaining the exact same numbers and the
exact same method to solve. So looking at this one my immediate kind of go-to is test scores so
I would rewrite this problem by saying that there is a student who is in a college class
in which there are four equally weighted tests. So four equally weighted tests on which they
want to receive an average test score of 75. Maybe that's like a requirement for a scholarship or a
requirement for passing the class or for achieving their major whatever they want an average of 75.
Now maybe we know that they scored a 90 on their highest exam and we know that they scored
different but lower scores on their other exams and we want to know what is the worst
they can do on their fourth and final exam so what is the least possible score that they can receive
on their last exam. So really easily translated into this word problem.
Now of course test scores aren't the only possible approach here these could be scores in
a game or prices of an item or weights of boxes or really a ton of different things
but what we are going to see is we are going to see that statistical term like average we are
going to see that min/max language like most or highest or at least or worst or smallest that sort
of extreme language and we're going to see those numbers that we can fill in our equation with so
min max problems are going to have the exact same approach all we need to do is be able to recognize
it whether in a more standard numerical form like this problem or in a scenario driven word problem.
Now if you haven't watched our other videos on GMAT word problems there are two of them that
we put out recently on overlapping sets and work rate. So I highly recommend you check those out,
especially if you struggle with highly specific word problem types. If this video helped you let
us know by liking and subscribe for more videos from me and the other experts here at Magoosh
and if you have requests for future videos or for specific word problem types that you want
to work through leave those in the comments below thanks for watching and I'll talk to you later.