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Wow, this is an insane problem. I wonder if someone can come up with a quick solution but it took me a long time to figure it out. Too much calculation. I came up with 1452/175 days or about 8.3 days.

1) M/12 + W/16 = 5
2) M/13 + W/24 = 4

Multiply second line by -3/2 to eliminate variable W:
2) -3M/26 - W/16 = -6

Add up line 1 and 2:
M/12 - 3M/26 = 5 - 6
(-18 + 13)M/156 = -1
-5M = -156
M = 156/5

Now plug in second equation to get W:
156/65 + W/24 = 4
12/5 + W/24 = 20/5
W/24 = 8/5
W = 192/5

Plug in back to the question asked:
M/7 + W/10 = X
156/(5*7) + 192/(5*10) = X
156/35 + 96/25 = X
(672+780)/175 = X
X = 1452/175 = approx 8.3 days

These are some the toughest problems that I have. I do have the solutions, but I am throwing them out in the hope that somebody can come up with a simpler solution as well as to challenge people.

Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]
15 Jan 2014, 23:24

Here is how I solved it:

Create two equations based on the information given. Start from the basics. Each man can do 1/x work per day and each woman can to 1/y work per day. If we have 12 men and 16 women, then in total they can do 12/x work per day and 16/y work per day together. We also know that they can do the work together in 5 days. So in 1 day, how much of the job did they finish? 1/5 or 20%. Repeat this logic to create the second formula. 12/x + 16/y = 1/5 13/x + 24/y = 1/4

Solve for common denominators in each formula: 12y+16x=((xy)^2)/5 13y+24x=((xy)^2)/4

Multiply by denominator of fraction in each formula to get nice numbers. Since both would equal the same variable, make them equal to each other: 60y+80x=52y+96x 8y=16x y=2x

Plug y=2x into the first equation to get: 12/x+16/2x=1/5 12/x+8/x=1/5 20/x=1/5 x=100

If x=100 and y=2x, y=200

Now in the original equation, it asks us how long 7 men and 10 women can do the work. We can create the following formula, similar to what we did in the first step. This formula tells us that in one day 7 men and 10 women can do X% of the job. We can simply take the reciprocal of the fraction to get the number of days the job will be completed in: 7/x+10/y=1/z (we want to solve for z) 7/100+10/200=1/z 7/100+5/100=1/z 12/100=1/z z=100/12 or 8.33

Last edited by psal on 16 Jan 2014, 12:36, edited 1 time in total.

Using equations (I) and (II) we need to find the sum of 7M and 10W. We can get a multiple of 7M in various ways: (3M + 4W = 1/20) * 5 gives 15M + 20W = 1/4 Adding this to equation II, we get 28M + 44W = 1/2 7M + 11W = 1/8

7 men and 11 women complete the work in 8 days. 7 men and 10 women will take a little more than 8 days to complete the work. If you do get such a question in GMAT, you will easily be able to manipulate the equations to get the desired equation. Finding the values of M and W is way too painful to interest GMAT.