hi i'm erika one of the senior curricular managers here at magoosh and a gmat expert
if you've watched my video on the six hardest gmat math problems you know that work rate is
one of the word problems concepts that a lot of test takers really struggle with
in this video we're going to walk through the two equations you need to know to solve work
rate problems and then we're going to work through a particularly nasty example problem
let's jump in now if you haven't seen kallee's video on distance problems on the gmat i really
recommend checking that out because work rate builds on distance rate questions but if you
have then you know the equation distance equals rate times time or d equals r t sometimes known as
dirt the first work rate equation is actually very very similar to that it's going to be work equals
rate times time or w equals r t or wrt so with this equation if we have any two of the variables
we can solve for the third so if we have rate and time we can solve for work if we have work and
rate we can solve for time and if we have work and time we can solve for rate so as an example
if we have a printer that prints one page every two seconds and we want to find its rate the work
is one page the time is two seconds rearranging that we get that rate is one over two or one page
per two seconds relatively simple application as a second example if we have another printer that
prints one page every three seconds what is its rate here work is one page time is three seconds
plugging that in we get that the rate is one over three or one page per three seconds now the second
equation we need to know for work rate is the combined rate equation and that's basically the
idea that if we are looking for the rate of two or three or four or five things working together
all we need to do is add up the rate of each of those things working on their own so as an example
if we have our two printers the one that prints one page every two seconds and the one that prints
one page every three seconds and we want to find their rate working together we'll add up those two
rates that we already solved for so the rate of the first printer is one page every two seconds
or one over two the rate of the second printer is one page every three seconds or one over three
so we have one half plus one third equals our combined rate of the two printers so finding
a common denominator we get three over six plus two over six equals five over six so we know that
those printers working together are going to print five pages every six seconds so let's put this all
together in a gmat style work rate problem so in this problem we have our two printers we know
that they start printing at the same time and we want to know how long it will take them to
print a 2 000 page manuscript okay so the first thing we would do is use our work rate equation
to solve for the individual rates of each printer so 1 over 2 and 1 over 3. then we're going to use
our combined rate equation to solve for their rate working together so that's one-half plus one-third
equals five-sixths and then we're gonna go back to our work rate equation in order to solve for the
time given the work using our combined rate so our work is two thousand pages our combined rate is
five over six plugging that in we get two thousand equals five over six times t rearranging we're
able to solve for the fact that time is going to be 2400 seconds which reduces to 40 minutes
and that's gmat work rate problems in a nutshell now if you know that you struggle with word
problems in general i really recommend subscribing to our channel and coming back next week because
we got a video on overlapping sets word problems coming out soon so that would be an example of a
relatively easy work rate problem let's go ahead and try a particularly hard one from magoosh
okay so in this problem we have a really classic work rate scenario we've got two pumps that are
emptying a pool now even though this is a pretty classic scenario there are a couple of things that
make this problem complicated let's work through those so starting with the first sentence pump
a can empty a pool in a minutes and pump b can empty the same pool in b minutes so notice that
they're giving us variables instead of numbers and we can see that in our answer choices as well
this is a classic move for pretty much any type of word problem to make it more complicated and
to force us to do algebra instead of arithmetic um so that is one thing we're gonna deal with that in
a second. second sentence pump a begins emptying the pool for one minute before pump b joins okay
so this is something that's specific to rate problems and basically what's happening here is
we're splitting up our process of emptying the pool into two phases with two different rates
so in this first phase we've got a working alone that's going to have the rate of just a and then
in the second phase we've got a and b working together so that's going to be one of those
combined rates okay interesting finally last sentence beginning from the time pump a starts
how many minutes will it take to empty the pool so we are looking for the total time
okay so the first thing that i'm gonna do is deal with the variables and the way i'm gonna handle
that is the way that we often handle it with word problems which is picking my own numbers so
instead of working with a and b i'm gonna pick numbers that i think will be easy to work with
that i can do the problem with solve for an answer in terms of those numbers
and then put the answer choices in terms of those numbers as well and see which one
matches so i'm going to pick numbers here i'm going to pick a equals 3 and b
equals 4. you can pick other numbers those are just the ones that i chose okay so that eliminates
an element of complexity i'm going to work with those numbers as i solve the next thing i'm going
to do is i'm going to come up with my various rates so i know work equals rate times time
that can be rearranged to rate equals work
over time okay using that information i'm going to solve for each of my rates so the rate of a i know
that pump a can empty one pool in a minutes so my work is one pool and my time will be a
i have chosen that a is three so the rate of a is one over three great the rate of b i'm
going to solve for in a similar way b empties one pool in b minutes b is four one over four
okay let's deal with combined rate now i know that combined rate the rate of a and b
is going to be the rate of a plus the rate of b
okay rate of a is 1 3 rate of b is one-fourth adding that up i would find a common denominator
but i'm just going to tell you that it ends up being 7 over 12. awesome okay using that
information i'm going to start working through my phases now something that i really like to do
i'm just going to circle this is to make a little work rate table and the way that i do that
is setting up my little work equals rate times time equation and using that to make
columns and then i make a row for each of the phases of my work rate problem
so my first phase is a working alone then my second phase is a and b working together
okay great so what do i know well i know the rates for each of those phases so i know that the rate
for just a working alone is one third and i know that the rate for a and b working together is 7
over 12. awesome i also know from the problem that pump a in its phase works alone for one
minute so that's my time for this row okay now at this point i'm starting to see that i have
something i can solve for which is the work for a alone so i know work equals rate times time so
because i have write in time i can solve for work so i'm going to come up with the fact that my work
is one-third because one-third equals one-third times one great okay now i can
extrapolate a little bit more information so i go okay i know that a alone does one third of
the work well i know that we are going to empty the pool right that is what i'm solving for how
much time will it take to empty the pool so that means that i'm going to finish the whole pool's
work worth of work over these two phases so if i've got one third done already that must mean
that a and b together do the other two thirds of the work and once again i see that i have this w
equals r t equation in the second row now where i have two of the pieces so now i can solve for the
third so i know two-thirds equals seven over 12 times something now at this point what i would do
is i would write this out as a separate equation with a variable in for t and then isolate for t
in order to solve but in the interest of time and saving space on the whiteboard when you do that
you end up with 8 over 7 as the time for a and b and if we just want to check that right now we
can see that if we multiplied it right in time here the 7s would cancel we would get 8 over 12
which reduces to 2 thirds okay now that's not our answer because we know that here we're solving for
our total time so that means i'm going to be adding up my times for each phase so i've got
1 plus 8 over 7 or 7 over 7 plus 8 over 7 which gives me 15 over 7. okay so that means when i take
my values for a and b 3 and 4 and plug them into these answer choices the correct answer should
give me 15 over 7. all right so taking a look at my answer choices right away i'm gonna go ahead
and say i'm pretty sure it's not a because that's over 2. how am i going to get that to be 15 over
7 unlikely now taking a look at the other answer choices i see that the denominator is a plus b
or 3 plus 4 which will be 7. so all of those are looking equally likely i'm going to go ahead and
start with my easiest one which is c so here i've got 3 plus 4 in the denominator so that's 7 and a
times b or 3 times 4 which is 12 in the numerator okay so that is too small and that's a really
useful piece of information because if i take a look at d and e i see that those are subtracting
something so those are going to be even smaller than 12 over 7 definitely not going to be 15 over
7 which leaves me with b as my only answer i'm just going to go ahead and check and confirm that
so again my denominator is going to be 7 or 3 plus 4. my numerator is going to be 3
times 5 or 4 plus 1 which is going to be 15 nice so my correct answer here
is b so this problem represents a particularly complex application of our work rate equations
however it really just is those two equations and if we're able to stay organized and take advantage
of some tricks and some very neat organization we can get through it in a reasonable amount of time
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and the other experts at magoosh and as always if you have any comments or questions or suggestions
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