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Math Expert V
Joined: 02 Sep 2009
Posts: 59075
Re: Time n Work Problem  [#permalink]

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mrwells2 wrote:
Bunuel wrote:
nonameee wrote:
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?

Below is another solution which is a little bit faster.

It takes 6 days for 3 women and 2 men working together to complete a work.3 men would do the same work 5 days sooner than 9 women.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

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.

It takes 6 days for 3 women and 2 men working together to complete a work --> sum the rates: $$\frac{3}{w}+\frac{2}{m}=\frac{1}{6}$$.

3 men would do the same work 5 days sooner than 9 women --> $$\frac{m}{3}+5=\frac{w}{9}$$.

Solving: $$m=15$$ and $$w=90$$. $$\frac{w}{m}=6$$.

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.

No, that's not correct.

m and w in both equations represent the same thing: time.

w is the number of days (time) one woman needs complete the job.
m is the number of days (time) one man needs complete the job.

The following posts might help:
it-takes-6-days-for-3-women-and-2-men-working-together-to-82718.html#p751436
it-takes-6-days-for-3-women-and-2-men-working-together-to-82718-20.html#p1272526
it-takes-6-days-for-3-women-and-2-men-working-together-to-82718-40.html#p1295389
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An alternate approach for a problem in a locked thread:

Quote:
It takes 6 days for 3 women and 2 men working together to complete a work. 3 men would do the same work 5 days sooner than 9 women. How many times does the output of a man exceed that of a woman ?

a. 3
b. 4
c. 5
d. 6
e. 7

We can PLUG IN THE ANSWERS, which represent how many times a man's output exceeds that of a woman.
When the correct answer is plugged in, 3 men will take 5 fewer days to complete the job than 9 women.

B: 4
Let the rate for each woman = 1 unit per day and the rate for each man = 4 units per day.
Since it takes 6 days for 3 women and 2 men to complete the job, the job = (rate for 3 women and 2 men)(6 days) = (3*1 + 2*4)(6) = 66 units.
Time for 9 women to complete the job $$= \frac{work}{rate-for-9-women} = \frac{66}{(9*1)} = \frac{22}{3}$$ ≈ 7 days.
Time for 3 men to complete the job $$= \frac{work}{rate-for-3-men} = \frac{66}{(3*4)} = \frac{66}{12} = \frac{11}{2}$$ ≈ 5 days.
Doesn't work:
Here, 3 men take only about 2 fewer days than 9 women.
Eliminate B.
For 3 men to take 5 fewer days, the rate for each man must INCREASE.
Eliminate A.

D: 6
Let the rate for each woman = 1 unit per day and the rate for each man = 6 units per day.
Since it takes 6 days for 3 women and 2 men to complete the job, the job = (rate for 3 women and 2 men)(6 days) = (3*1 + 2*6)(6) = 90 units.
Time for 9 women to complete the job $$= \frac{work}{rate-for-9-women} = \frac{90}{(9*1)} = \frac{90}{9} = 10$$ days.
Time for 3 men to complete the job $$= \frac{work}{rate-for-3-men} = \frac{90}{(3*6)} = \frac{90}{18} = 5$$ days.
Success!
Here, 3 men take 5 fewer days than 9 women.

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Re: It takes 6 days for 3 women and 2 men working together to  [#permalink]

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2
virtualanimosity wrote:
It takes 6 days for 3 women and 2 men working together to complete a work. 3 men would do the same work 5 days sooner than 9 women. 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

We can PLUG IN THE ANSWERS, which represent how many times a man's output exceeds that of a woman.
When the correct answer is plugged in, 3 men will take 5 fewer days to complete the job than 9 women.

B: 4
Let the rate for each woman = 1 unit per day and the rate for each man = 4 units per day.
Since it takes 6 days for 3 women and 2 men to complete the job, the job = (rate for 3 women and 2 men)(6 days) = (3*1 + 2*4)(6) = 66 units.
Time for 9 women to complete the job = $$\frac{work}{rate-for-9-women} = \frac{66}{(9*1)} = \frac{22}{3} ≈ 7$$ days.
Time for 3 men to complete the job = $$\frac{work}{rate-for-3-men} = \frac{66}{(3*4)} = \frac{66}{12} = \frac{11}{2} ≈ 5$$ days.
Doesn't work:
Here, 3 men take only about 2 fewer days than 9 women.
Eliminate B.
For 3 men to take 5 fewer days, the rate for each man must INCREASE.
Eliminate A.

D: 6
Let the rate for each woman = 1 unit per day and the rate for each man = 6 units per day.
Since it takes 6 days for 3 women and 2 men to complete the job, the job = (rate for 3 women and 2 men)(6 days) = (3*1 + 2*6)(6) = 90 units.
Time for 9 women to complete the job = $$\frac{work}{rate-for-9-women} = \frac{90}{(9*1)} = \frac{90}{9} = 10$$ days.
Time for 3 men to complete the job = $$\frac{work}{rate-for-3-men} = \frac{90}{(3*6)} = \frac{90}{18} = 5$$ days.
Success!
Here, 3 men take 5 fewer days than 9 women.

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Re: It takes 6 days for 3 women and 2 men working together to  [#permalink]

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virtualanimosity wrote:
It takes 6 days for 3 women and 2 men working together to complete a work. 3 men would do the same work 5 days sooner than 9 women. 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

We can let the number of days for 1 man to finish the work = m and that for 1 woman = w. Thus, the rate of 1 man = 1/m and the rate of 1 woman = 1/w.

Since it takes 6 days for 3 women and 2 men working together to complete the work, we have:

6(3/w + 2/m) = 1

Since 3 men would do the same work 5 days sooner than 9 women, we have:

1/(3/m) + 5 = 1/(9/w)

m/3 + 5 = w/9

Multiplying the above by 9, we have:

3m + 45 = w

Substituting this into 6(3/w + 2/m) = 1, we have:

6(3/(3m + 45) + 2/m) = 1

6(1/(m + 15) + 2/m) = 1

Multiplying the above by m(m + 15), we have:

6m + 12(m + 15) = m(m + 15)

6m + 12m + 180 = m^2 + 15m

m^2 - 3m - 180 = 0

(m - 15)(m + 12) = 0

m = 15 or m = -12

Since m can’t be negative, m = 15. So w = 3(15) + 45 = 90. We see that it takes a man 15 days to complete the work but 90 days for a woman to complete the same work. So a man’s output is 6 times the output of a woman.

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Re: It takes 6 days for 3 women and 2 men working together to  [#permalink]

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1
1
virtualanimosity wrote:
It takes 6 days for 3 women and 2 men working together to complete a work. 3 men would do the same work 5 days sooner than 9 women. 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

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Re: It takes 6 days for 3 women and 2 men working together to  [#permalink]

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virtualanimosity wrote:
It takes 6 days for 3 women and 2 men working together to complete a work. 3 men would do the same work 5 days sooner than 9 women. 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

3w+2m= 1/6 (in one day )

NEXT: 3 men would do the same work 5 days sooner than 9 women , SO let X be the job done by woman and let x-5 be the work done by men

HENCE: $$\frac{1}{x}- \frac{1}{(x-5)} = \frac{x-5-x}{x(x-5)}$$ i.e $$\frac{-5}{x(x-5)}$$ (i know i got negative in numerator but still continued solving further )

$$\frac{-5}{x(x-5)}$$ = $$\frac{1}{6}$$ cross multiply

$$x(x-5)=-30$$

$$x$$ = $$\frac{-30}{x(x-5)}$$ now if i divide -30 by -5 i get 6. this was my solution to correct answer Bunuel, pushpitkc generis do you think my solution is correct ? Intern  B
Joined: 13 Aug 2018
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Re: It takes 6 days for 3 women and 2 men working together to  [#permalink]

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virtualanimosity wrote:
It takes 6 days for 3 women and 2 men working together to complete a work. 3 men would do the same work 5 days sooner than 9 women. 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

Hello all,

I couldn't solve this question within time. But i did attempt it the following manner and got the answer right. Can someone tell me if this approach is correct/incorrect?

Let the work be W.

Now, based on the data provided
- one man can do 1/3 of W
- one woman can do 1/9 of W

Here i am now assuming a number for W - let's say the work needed to be done is eating 27 pies. i.e. W=27

So one man can eat 9 pies and one woman can eat 3 pies.

Because the question asks How many times does the output of a man exceed that of a woman?, I have understood this as Output of a man (9 pies) is x more than output of a woman (3 pies).
so 9 = 3+x therefore, X=6.

Let me know if this makes sense or I was just lucky to get the answer right (won't work all the time)
Intern  B
Joined: 25 Jan 2018
Posts: 8
Re: It takes 6 days for 3 women and 2 men working together to  [#permalink]

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Hi Bunuel,

I have no trouble setting up the two equations. However, I don’t know how to solve them quickly and efficiently since w and m are found in the denominators of the 1st equation and in the numerators of the 2nd equation. Please help! Thank you so much

Bunuel wrote:
nonameee wrote:
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?

Below is another solution which is a little bit faster.

It takes 6 days for 3 women and 2 men working together to complete a work.3 men would do the same work 5 days sooner than 9 women.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

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.

It takes 6 days for 3 women and 2 men working together to complete a work --> sum the rates: $$\frac{3}{w}+\frac{2}{m}=\frac{1}{6}$$.

3 men would do the same work 5 days sooner than 9 women --> $$\frac{m}{3}+5=\frac{w}{9}$$.

Solving: $$m=15$$ and $$w=90$$. $$\frac{w}{m}=6$$.

Posted from my mobile device
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Re: It takes 6 days for 3 women and 2 men working together to  [#permalink]

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Rurugu wrote:
Hi Bunuel,

I have no trouble setting up the two equations. However, I don’t know how to solve them quickly and efficiently since w and m are found in the denominators of the 1st equation and in the numerators of the 2nd equation. Please help! Thank you so much

Bunuel wrote:
nonameee wrote:
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?

Below is another solution which is a little bit faster.

It takes 6 days for 3 women and 2 men working together to complete a work.3 men would do the same work 5 days sooner than 9 women.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

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.

It takes 6 days for 3 women and 2 men working together to complete a work --> sum the rates: $$\frac{3}{w}+\frac{2}{m}=\frac{1}{6}$$.

3 men would do the same work 5 days sooner than 9 women --> $$\frac{m}{3}+5=\frac{w}{9}$$.

Solving: $$m=15$$ and $$w=90$$. $$\frac{w}{m}=6$$.

Posted from my mobile device

Rurugu You can refer to my method as well if it looks any easier to you.

https://gmatclub.com/forum/it-takes-6-d ... l#p2167832
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It takes 6 days for 3 women and 2 men working together to  [#permalink]

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Solving quadratic equations, especially when they involve large numbers and need a lot of calculations to form the equation in the first place, is quite time-consuming and the likelihood of calculation errors is also quite high. I doubt whether problems like this can be solved within the GMAT time-frame in the conventional manner. However, there is a much quicker way of solving this problem with a bit of logic and by taking a few hints from the answer choices:

Let the time taken by 3 men to do the work be 'x' days.
Then, the time taken by 9 women to do the work is (x+5) days which means time taken for 3 women will be 3(x+5) days. If M and W are the speeds or rates of men and women respectively, then:
M/W = 3(x+5)/x. In other words, a man is 3(x+5)/x times faster than a woman.
3(x+5)/x = 3 + 15/x. Since all the answer options are whole numbers 'x' must be a factor of 15, i.e. either 3, 5 or 15. It can't be 3 because that would make a man 8(3+5) times faster (7 is the highest option).
x=15 would mean that 3 men complete the work in 15 days or 5 men complete in (3*15)/5=9 days. But we know that 3 women and 2 men complete in 6 days. Since men are faster than women so, if we substitute the 3 women by 3 men and we have 5 men working on the job it can be finished in less than 6 days. So 'x' cannot be 15 which leaves only 5 as the only possible value for 'x'. So a man is (3 + 15/5) = 3 +3 = 6 times faster than a woman.

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Re: It takes 6 days for 3 women and 2 men working together to  [#permalink]

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virtualanimosity wrote:
It takes 6 days for 3 women and 2 men working together to complete a work. 3 men would do the same work 5 days sooner than 9 women. 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

given total work
5x/6=1/6
or say x=1/5
so for 3 men = 5+ 9 women
3*1/5*x = 5+9/5
3x=18
i.e 6 times
IMO D
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Re: It takes 6 days for 3 women and 2 men working together to  [#permalink]

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Archit3110

Could you please explain the assumptions underlying your answer? For example, what does 'x' stand for? I am a bit confused because first you got
x=1/5 and then you got x=6. A detailed explanation would help us understand the process through which you arrived at your answer. Thanks.
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Re: It takes 6 days for 3 women and 2 men working together to  [#permalink]

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virtualanimosity wrote:
It takes 6 days for 3 women and 2 men working together to complete a work. 3 men would do the same work 5 days sooner than 9 women. 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

We can let m = the number of days a man can complete the work by himself and w = the number of days a woman can complete the work by herself. Thus, 1/m = the rate of a man, and 1/w = the rate of a woman.

We can create the equations

6(3 x 1/w + 2 x 1/m) = 1

and

1/(3 x 1/m) = 1/(9 x 1/w) - 5

Simplifying the first equation, we have:

3/w + 2/m = 1/6

Multiplying the equation by 6wm, we have:

18m + 12w = mw

Simplifying the second equation, we have:

m/3 = w/9 - 5

Multiplying the equation by 9, we have:

3m = w - 45

3m + 45 = w

Substituting this in 18m + 12w = mw, we have:

18m + 12(3m + 45) = m(3m + 45)

18m + 36m + 540 = 3m^2 + 45m

3m^2 - 9m - 540 = 0

m^2 - 3m - 180 = 0

(m - 15)(m + 12) = 0

m = 15 or m = -12

Since m can’t be negative, m = 15. Hence, w = 3(15) + 45 = 90. We see that a man’s rate is 6 times that of a woman’s rate since the number of days a man can complete the job is only 1/6 of the number of days a woman can complete the job.

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Re: It takes 6 days for 3 women and 2 men working together to  [#permalink]

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maelstroem wrote:
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

I did it in a similar manner. I assumed that 3 men can complete in 'x' days which means that 9 women can complete in (x+5) days and, working forward from there, got the same equation:
x^2 - x - 20 = 0. What I want to point out here is that in this type of quadratic equation where factorization is possible it is quicker to factorize:

x^2 -x -20 = 0...> x^2 - 5x + 4x - 20 = 0...> x(x-5) + 4(x-5) = 0...> (x-5)(x+4) = 0. x=5
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