It takes 6 days for 3 women and 2 men working together to

Let women do \(w\) units of work per day and men do \(m\) units of work per day, then question asks, what is m/w

Total units of work to be done = \(6*(3*w+2*m)\)

Time taken by 9 women to do this work on their own = \(6*(3*w+2*m)\)/\((9*w)\) = \(2 + 4/3*m/w\)

Time taken by 3 men to do this work on their own = \(6*(3*w+2*m)/(3*m)\) = \(6/(m/w) + 4\)

Let m/w be x

then we know

\(2 + (4/3)*x -6/x -4 = 5\)

or

\((4/3)*x - 6/x= 7\)

Now substituting for x from choices will quickly give us x = 6 so D

I like this as it reduces quickly to the required form of m/w and it obviates the need for a quadratic equation and also lest w and m remain in numerator most of the time

I received a PM asking me to respond. First, this is definitely not a realistic GMAT question. For one thing, it's horribly written (the phrase 'to complete a work' is not English, the question should read '*By* how many times...', the question needs to make clear that each man works at the same rate, as does each woman, the word 'sooner' is non-idiomatic, the word 'output' is misused, etc). For another, it's terribly contrived, and altogether tedious if you take any normal approach; I don't see any direct way to solve that will allow you to avoid a quadratic equation. Real GMAT questions are never designed in such a way, so you can confidently move on to better material and ignore this question (incidentally, where is it from?).

While it isn't especially fast either, you can work backwards from the answers here relatively easily. This might at least be less confusing for some than a direct (algebraic) approach. Say we get 1 unit of work per woman per day. If you test, say, answer C, we'd then get 5 units of work per man per day. The job would then require 6(3 + 2*5) = 78 units of work. Notice that, to find how long it will take 9 women to do the job, we'll need to get an integer when we divide 78 by 9, so C cannot be right. If you move next to D, we have 6 units of work per man per day, and the job requires 6(3 + 2*6) = 90 units of work. Thus 9 women do the job in 10 days, and 3 men would do the job in 90/(6*3) = 5 days, so D is correct.

Clear D:

x: days 9 women doing the job --> 1 woman works W=1/9x per day
x-5: days 3 men doing the job --> 1 man works M=1/[3*(x-5)] per day

The answer to the question is: M/W

And, how much is x?

As per first data:

2*M+3*W=1/6
You can solve x, and obtain 2 values, 10 and 1 (1 is impossible because that would imply that the 3 men take -4 days), so x=10. So M/W=6 times

Guys, even if you know the solution right away, it takes several minutes (definitely more than 3) to just write it down to find the answer. Is it a real GMAT question? Can something like that be expected on the real test?

Hm, i got stuck cuz I got something a little different:
YOURS: 3 men would do the same work 5 days sooner than 9 women --> \(\frac{m}{3}+5=\frac{w}{9}\).

MINE: 3 men would do the same work 5 days sooner than 9 women --> \(\frac{3}{m}=\frac{9}{w}+5\)

In the above equation you also have for 2 men: \(\frac{2}{m}\) - so why do u suddenly use the reciprocal? And why don't we add the 5 to women, because they take longer, hence their side is smaller...

I also got quadratic equation (\(m^2-3m-180=0\)) and it wasn't too hard to solve (discriminant would be perfect square \(d=3^3+4*180=729=27^2\)) --> \(m=-12\) or \(m=15\).

First set : 3/w+2/m=1/6
Second set: 3/m = 1/x and 9/w=1/(x+5)
==> 1/m = 1/3x and 1/w = 1/9(x+5)
Enter in first equation and solve for x
1/(3x+15) + 2/3x = 1/6
simplify => x^2-x-20=0
solve for x : x=(1 +- (1+80)^,5)/2= (1+- 9)/2.
X can only be positive ==> x= 5
enter in 2nd set ==> 3/m=1/5 and 9/w=1/10 => 3/2m=9/w => m/w=1/6

The fastest and easiest way to solve this question was already proposed by IanStewart.

I am trying another algebraic approach.

Denote by \(W\) the rate of a woman, by \(M\) that of a men, and by \(T\) the time it takes 9 women to complete the work.
We have the following equations:
\(6(3W + 2M) = 9WT = 3M(T-5)\), or, after reducing by 3, \(2(3W + 2M) = 3WT = M(T - 5).\)
We are looking for the ratio \(M/W\) which we can denote by \(n.\) Substituting in the above equations \(M = nW,\) we can write:
\(2(3W + 2nW) = 3WT = nW(T - 5).\)

Divide through by \(W,\) so \(6 + 4n = 3T = nT - 5n.\) Solving for \(T\) (equality between the last two expressions) we obtain \(T=\frac{5n}{n-3}.\)
Taking the equality of the first two expressions, we get \(6+4n=\frac{3\cdot{5}n}{n-3}.\)
From the possible answer choices we can deduce that \(n\) must be a positive integer.
We need \(\frac{15n}{n-3}\) to be a positive integer. We can see that \(n\) cannot be odd and it must be greater than 3.
We have to choose between B and D.
Only \(n = 6\) works.

Answer D.

How did you solve for m and w in the very last part? I do the algebra and can't get the right answer. You have one equation with 2 unknown variables.

You have 2 equations with two unknowns:
First equation \(\frac{3}{w}+\frac{2}{m}=\frac{1}{6}.\)
Second equation \(\frac{m}{3}+5=\frac{w}{9}\).
After getting rid of the denominators (multiply first equation by \(6wm\) and the second by 9), for example express \(w\)
from the second equation and substitute it into the first. You obtain a quadratic equation for \(m\):

\(m^2-3m-180=0\)

This equation has one positive and one negative root. The sum of the two roots must be 3 and their product -180.
Using factorization for 180, you can find -12 and 15.
So \(m=15\) and \(w=90.\)

For another algebraic approach see:
it-takes-6-days-for-3-women-and-2-men-working-together-to-82718.html#p1121807

Can you please help me to understand on what logic did you make this explanation- 3 men would do the same work 5 days sooner than 9 women --> \(\frac{m}{3}+5=\frac{w}{9}\).

One man completes the job in \(m\) days --> 3 men in m/3 days.
One woman completes the job in \(w\) days --> 9 women in w/9 days.

We are told that m/3 is 5 less than w/9 --> \(\frac{m}{3}+5=\frac{w}{9}\).

Hope it's clear.

My question is this, on the second equation how did you KNOW to put m/3, whereas in the first it was 2/m? In both cases aren't you figuring out the rate? In the first equation, you know that a man does 1/m of the job, and that 2 would do 2/m. In the second equation the rate is still 1/m, but you have 3 men, so should it not be 3/m+5=9/m?

Please read the solution carefully:
First equation gives the rate: the rate of 1 man is \(\frac{1}{m}\) job/day, then the rate of 2 men will be \(\frac{2}{m}\) job/day.

Second equation gives time: 1 man needs \(m\) days to do the job 3 men will need \(\frac{m}{3}\) days to do the job.

Hope it's clear.

bunuel, pls help

if i equate time i am not getting it pls tell me where i am going wrong

lets take 1 woman can complete the work in 'w' days and 1 man can complete in 'm' days
so, it becomes w/3+m/2=6
and m/3+5=w/9

but i am getting the answer wrong.

thanks in advance

That's because your equations are wrong. If one woman complete the job in \(w\) days and one man in \(m\) days, then w/3 is the time one woman needs to complete 1/3 of the work and m/2 is the time one man needs to complete 1/2 of the work. Adding them makes no sense. We can add rates but not times.

Check here: it-takes-6-days-for-3-women-and-2-men-working-together-to-82718.html#p751436

Hope this helps.

I stumbled on this answer and think it's worth clarifying:

In the second equation: 3 men would do the same work 5 days sooner than 9 women --> \(\frac{m}{3}+5=\frac{w}{9}\).

m and w are representing TOTAL work done by men and women.

Whereas in the first equation: Let one woman complete the job in \(w\) days and one man in \(m\) days. So the rate of 1 woman is \(\frac{1}{w}\) job/day and the rate of 1 man is \(\frac{1}{m}\) job/day.

m and w are representing the RATE of work done by men and women.

I hope this is correct (Bunuel can you confirm?) and has helped some grasp the concept.