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Rate/Work Problem 1

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Rate/Work Problem 1

by knight247 » Thu Mar 14, 2013 11:41 pm
It takes six days for three women and two men working together to complete a piece of work. Three men would do the same work five day sooner than nine women. By how many times does the output of a man exceed that of a woman?

(A) 3 times
(B) 4 times
(C) 5 times
(D) 6 times
(E) 7 times

OA - D

I would appreciate a direct solution to this problem using equations only, as I've already figured out the backdoor method to solving this problem.
Last edited by knight247 on Fri Mar 15, 2013 2:27 am, edited 2 times in total.
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by Anju@Gurome » Fri Mar 15, 2013 12:59 am
knight247 wrote:It takes six days for three women and two men working together to complete a piece of work. Three men would do the same work five day sooner than nine women. How many days does the output of a man exceed that of a woman?
That's a very bad wording. I'm not sure what does that mean.

Let us assume that it takes M and W days for a man and a woman to complete the job individually, respectively. Hence, in one day, a man does 1/M of the work and a woman does 1/W of the work.

Hence, 6*(2/M + 3/W) = 1
And (W/9 - M/3) = 5 ---> (W/3 - M) = 15 ---> W/3 = (15 + M) ---> 3/W = 1/(15 + M)

So, 6*(2/M + 1/(15 + M)) = 1
--> 6*[(30 + 2M + M)/(M*(15 + M)] = 1
--> 6*(30 + 3M) = M*(15 + M)
--> 180 + 18M = 15M + M²
--> M² - 3M - 180 = 0
--> M² - 15M + 12M - 12*15 = 0
--> (M - 15)(M + 12) = 0

As M cannot be negative, M = 15

Now, W = 3*(15 + M) = 3*(M + M) = 6M

This means a woman take 6 times as much days as a man take to finish the work, i.e. a woman take 5 times more days than what a man take to finish the work.
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by knight247 » Fri Mar 15, 2013 2:27 am
Thanks a ton, Anju. I just edited my post. There was the type-o. My exact question was: By how many TIMES does the output of a man exceed that of a woman?
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by GMATGuruNY » Fri Mar 15, 2013 7:00 am
knight247 wrote:It takes six days for three women and two men working together to complete a piece of work. Three men would do the same work five day sooner than nine women. By how many times does the output of a man exceed that of a woman?

(A) 3 times
(B) 4 times
(C) 5 times
(D) 6 times
(E) 7 times

OA - D
I think it would be easier to plug in the answer choices.

Answer choice C: rate for each man = 5 times the rate for each woman
Let the rate for each woman = 1 unit per day and the rate for each man = 5 units per day.
Thus, the combined rate for 3 women and 2 men = 3*1 + 2*5 = 13 units per day.
In 6 days, the amount of work produced = 6*13 = 78 units.
Since the rate for 9 women = 9*1 = 9 units per day, the time for 9 women to complete 78 units = 78/9 = 8+ days.
Since the rate for 3 men = 3*5 = 15 units per day, the time for 3 men to complete 78 units = 78/15 = 5+ days.
Here, the men finish about 3 days sooner than the women.
Since the men must finish 5 days sooner than the women, the men must work FASTER.

Answer choice D: rate for each man = 6 times the rate for each woman
Let the rate for each woman = 1 unit per day and the rate for each man = 6 units per day.
Thus, the combined rate for 3 women and 2 men = 3*1 + 2*6 = 15 units per day.
In 6 days, the amount of work produced = 6*15 = 90 units.
Since the rate for 9 women = 9*1 = 9 units per day, the time for 9 women to complete 90 units = 90/9 = 10 days.
Since the rate for 3 men = 3*6 = 18 units per day, the time for 3 men to complete 90 units = 90/18 = 5 days.
Here, the men finish 5 days sooner than the women.
Success!

The correct answer is D.
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