Hi I'm Erika one of the senior curriculum managers here at Magoosh and a GMAT expert
and in today's video we're continuing our series on GMAT word problems with a video
on overlapping sets, let's jump in. Now if you haven't seen the previous video in the series
our video on work rate I recommend checking that one out but this one we're going to approach
a little bit differently so instead of first telling you the equations that you need to
know to succeed we're instead of going to teach this one by going through our example
problem so strap in for a hard Magoosh overlapping sets problem. Okay starting from the top Paracelsus
university has 2 kinds of professors academic professors and professional professors okay
so first thing I'm gonna notice here is that academic and professional or what's called
complementary so that means that professors can either be academic or professional but
they cannot be both academic and professional and they cannot be neither academic nor professional
so they really have to be one or the other this is a really common set up in overlapping
sets problems and it's going to really inform the way that we approach our math so we're
just gonna know that now it's going to be important later okay at Paracelsus university
60 percent of the professors are academic professors cool and 70 percent of the professors
are tenured now this is a new idea right so we know that professors can be either academic
or professional but we know that on top of that they can either be tenured or not tenured
okay so we have this implied compliment to tenured of not tenured all right and we're
starting to see that there could be some overlaps here so we could be academic and tenured we
could be professional and tenured or we could be academic and not tenured or professional
and not tenured okay so we're starting to see the potential for overlapping sets here
all right moving up if 90 percent of the professors at Paracelsus university our academic professors
or tenured or both if we weren't sure that this was an overlapping sets problem yet that
word both should clue us in and both as an overlap so words like both work only are going
to be kind of clue words that we have an overlapping sets problem all right then what percent of
the professional professors there are tenured that's the thing that we are solving for okay
awesome okay so at this point we know we're dealing with overlapping sets we know we're
dealing with academic versus professional and tenured versus not tenured but how do
we organize this information and deal with all of the numbers in this question. The best
way in most cases when we're dealing with 2 overlapping sets when we have complementary events
or complementary features whatever is going to be setting up a double matrix I'm gonna
show you how to do that so what I'm gonna do is I'm gonna draw 3 vertical lines. And
then 3 horizontal lines going through them doesn't have to be pretty. And what I'm gonna
do is I'm going to put my compliments across the top and down the side so let's start with
academic and professional so I'm gonna skip this one. And I'm gonna put academic right
here. And professional. Right here now because those are complementary we know that all of
the academic professors and all of the professional professor should add up to our total which
I'm gonna label with a capital T. okay now going down the side so we know that our other
set of compliments are tenured and not tenured okay and once again those are compliments
so they add up to the total now what this does is it makes beautiful little cells that
contain all of the different types of professors we could have so this cell right here is going
to be academic and tenure this cell right here is going to be professional and tenured
this cell right here is going to be our total tenured professors okay here we have academic and
not tenured here we have professional and not tenured and here we have our total not
tenured down here we have our total academic here we have our total professional and here
we have our total total for all of our professors okay now at this point if I had a bunch of
values I would just start throwing the values in the table however I don't they made this
problem a little bit more complex by only giving me percent 60 percent 70 percent to
90 percent and I'm selling for 8 percent okay awesome. So this is kind of like what they
did to us in our work rate problem from last weeks video where they gave us doubles right
and instead of deciding to do it algebraically with those variables we decided instead we're
going to plug in our own numbers for those variables and then solve in terms of those
numbers we're gonna do the same thing here just to get rid of that element of complexity
that unknown so since we have a percent here and we do not know what the total is we can
make up that total and usually when we have persons the easiest total to work with is
100 so I'm just gonna decide that there are 100 professors in total at Paracelsus university
then I'm going to solve using those percents being very very careful that I'm taking my
percents for the right thing okay jumping in. First piece of data is that 60 percent
of the professors are academic okay so I know that I have 100 professors and I know that
60 percent of 100 is 60 so I have 60 total academic professors now look at the magic
of the stable because I know that I have 2 values in this bottom row I can solve for
the third of value in the bottom row so I have 60 academic professors I want 100 total
professors because academic and professional are complementary I know that there must be
40 professional professors all right let's do that again 70 percent of the professors
are tenured okay so I have 100 professors 70 percent of 100 is 70 so I have 70 tenured
professors which means that I must have 30 not tenured professors very cool all right
but we have a little bit more information and this is where this gets a little bit nasty
if 90 percent or 90 percent of 100 90 of the professors at Paracelsus university are academic
or tenured or both okay let's work through that so academic and tenured that both situation
is right here but we know that it's cool if they're just academic right that's what I
say academic or tenure or both so it's cool they're just academic so academic and not
tenured it's also cool if they're just tenured so tenured and professional instead of academic
so we know that any of those circled cells are okay here and that makes up that 90 percent
or that 90 out of 100 so what does that mean about the remaining cell we know that it's
all supposed to add up to 100 so those 3 cells make up 90 the remaining cell makes up. 10.
Now that we have that we can fill in our other cells so something plus 10 equals 30 that's
got to be 20. Something plus 10 equals 40 that's got to be 30 and then confirming with
that last sell something plus 20 equals 60 that's going to be 40 and 40 plus 30 does equal 70
and then just to confirm against that 90 percent thing if I add up my circled ones though should
add up to 90 40 plus 20 and 60 plus 30 is 90 okay great now we're not out of the woods
yet because we have to make sure that we saw for the right thing so what percent of the
professional professors there are tenured now mistake that a lot of people are going
to make is they're going to go cool and looking for professional and tenured and that is okay
it's 30 so taking a look and there's no 30. Which is actually a blessing in disguise if
there had been a 30 a lot of people would have picked it and even if they did all of
this work right up until now we got the question wrong because that's not what they're asking
for they're asking for of the professional professors how many are tenured so we're looking
for out of the total professional professors which is 40 out of that how many of them are
tenured well that's that 30 number so we're looking for 30 over 40 which is 3 fourths
which is 75 percent and the answer is E. So overlapping sets word problems feature a lot
of different pieces of information however with careful organization using a tool
like a double matrix with really careful reading of what we have and what we're solving for
and using the word problems tricks like picking our own numbers we're gonna be able to solve
these problems effectively and in a reasonable time frame now you may be wondering why I
didn't use a venn diagram for that problem and the reason is because I find that the
vast majority of my students find a double matrix easier to label and they find a venn
diagram I see that a lot of students end up getting confused about what to label their
circles whether to put numbers inside the circle or outside the circle whereas with
a double matrix or matrix box there is a pretty clear cell for every piece of information
so I really recommend using a double matrix whenever you can so when you have 2 overlapping
sets where each of those sites have compliments now that said there is a place for a venn
diagram on the GMAT and that place is when you have 3 overlapping sets because a double
matrix cannot handle that let me know in the comments if you'd like a video specifically
on 3 overlapping sets it comes up pretty rarely on the GMAT but when it does come up the way
to handle is going to be a venn diagram and that's another one of the most common word
problems on the team that now if you're looking for other hard math topics on the GMAT I recommend
checking out our video on the 6 hardest GMAT math questions we talk about the 6 topics
that most test takers find confusing no matter what level the questions are at let us know
if this video help you by liking and subscribe for more videos from me in the other experts
in addition and if you have any questions or comments or suggestions for future videos
leave those in the comments below thanks for watching.