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Bunel-I got a quadratic equation while solving these two eqn. Is there a simple way of solving them?

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 \(m=15\).

Just a smal typo: in the discriminant, it should be \(3^2\) and not \(3^3.\)
_________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

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

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15 Sep 2012, 11:53

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

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.
_________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

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 \(w=90\). \(\frac{w}{m}=6\).

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.

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 \(w=90\). \(\frac{w}{m}=6\).

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 \(w=90.\)

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?

Completely agree... I did GMAT two years ago and by far the questions were not that complicated. Complicated, of course. But not that complicated. As you can see on my blog, studying GMAT is basically using the Official book and maybe one or two other books for reinforcement. Look for my advice. There is not "rocket science".

I also say GMAT preparation should not mean spending more than $100.

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

Show Tags

19 Feb 2013, 03:13

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

Maybe a method only suitable for GMAT :

We know that Work = rate*time.

Let m = rate of work of each man in one day and so on for w(each women)

As the work done is the same for 3 men/9 women;

\(3*m*t = 9*w*(t+5)\)

or \(\frac{m}{w} = 3\frac{(t+5)}{t}\) = \(3*(1+\frac{5}{t})\) = \(3+3*\frac{5}{t}\)

Now we go back to the options, and see that each of them is an integer. Thus, m/w, which is required can only be an integer and also a multiple of 3. The only multiple present can be 6, for t=5.

Bunel-I got a quadratic equation while solving these two eqn. Is there a simple way of solving them?

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 \(m=15\).

My friend, I'm ending up with 12w+18m=mw and 9m+135=3w

2/m+3/w=1/6

take lcm and you end up with 12w+18m=mw

same with m/3+5=w/9, I end up with 9m+135=3w

With that I get some very large numbers when trying to solve so I'm pretty sure I'm not doing it right.

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 \(w=90\). \(\frac{w}{m}=6\).

Answer: D.

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 --> \(\frac{m}{3}+5=\frac{w}{9}\). _________________

Like my post Send me a Kudos It is a Good manner. My Debrief: http://gmatclub.com/forum/how-to-score-750-and-750-i-moved-from-710-to-189016.html

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 \(w=90\). \(\frac{w}{m}=6\).

Answer: D.

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 --> \(\frac{m}{3}+5=\frac{w}{9}\).

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 \(w=90\). \(\frac{w}{m}=6\).

Answer: D.

I'm just not having a good day here haha.... if the rate for 3 men to do the same work 5 days sooner than 9 woman, shouldn't the equation be \(\frac{3}{m}+5=\frac{9}{w}\)? Since the rate of 1 woman is 1/w, and for one man is 1/m. If you have 3 men, then you have 3/m, and if you have 9 women you have 9/w...

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 \(w=90\). \(\frac{w}{m}=6\).

Answer: D.

I'm just not having a good day here haha.... if the rate for 3 men to do the same work 5 days sooner than 9 woman, shouldn't the equation be \(\frac{3}{m}+5=\frac{9}{w}\)? Since the rate of 1 woman is 1/w, and for one man is 1/m. If you have 3 men, then you have 3/m, and if you have 9 women you have 9/w...

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 \(w=90\). \(\frac{w}{m}=6\).

Answer: D.

Thanks for previous reply. I'm getting stuck at this point though...so the party I've highlighted in green, is that even necessary to solve? The part in red seems to be the key, and then did you just keep plugging numbers in until you got the solution for the blue? I see you said "solving:" before that line, but there's no computations, is it just fairly simple plugging in of numbers?

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 \(w=90\). \(\frac{w}{m}=6\).

Answer: D.

Thanks for previous reply. I'm getting stuck at this point though...so the party I've highlighted in green, is that even necessary to solve? The part in red seems to be the key, and then did you just keep plugging numbers in until you got the solution for the blue? I see you said "solving:" before that line, but there's no computations, is it just fairly simple plugging in of numbers?

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 \(w=90\). \(\frac{w}{m}=6\).

Answer: D.

I think the part I'm getting confused by is this:

Bunuel wrote:

So the rate of 1 woman is \(\frac{1}{w}\) job/day and the rate of 1 man is \(\frac{1}{m}\) job/day.

I get that part.

Bunuel wrote:

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

I get this part as well, because 3*\(\frac{1}{w}\)= \(\frac{3}{w}\), and the same for the men's rate. But here's where I get lost:

Bunuel wrote:

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

Using the logic from the second quote, 3 men doing the same work 5 days sooner than 9 women would be 3*\(\frac{1}{m}\)+5 = 9*\(\frac{1}{w}\). I don't understand how we can just switch whether the multiplier goes in the numerator or the denominator like that. Because their work rates are remaining the same; 1 man will do 1/m no matter what, so 3 men should always do 3/m, not m/3.

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 \(w=90\). \(\frac{w}{m}=6\).

Answer: D.

I think the part I'm getting confused by is this:

Bunuel wrote:

So the rate of 1 woman is \(\frac{1}{w}\) job/day and the rate of 1 man is \(\frac{1}{m}\) job/day.

I get that part.

Bunuel wrote:

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

I get this part as well, because 3*\(\frac{1}{w}\)= \(\frac{3}{w}\), and the same for the men's rate. But here's where I get lost:

Bunuel wrote:

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

Using the logic from the second quote, 3 men doing the same work 5 days sooner than 9 women would be 3*\(\frac{1}{m}\)+5 = 9*\(\frac{1}{w}\). I don't understand how we can just switch whether the multiplier goes in the numerator or the denominator like that. Because their work rates are remaining the same; 1 man will do 1/m no matter what, so 3 men should always do 3/m, not m/3.

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 \(w=90\). \(\frac{w}{m}=6\).

Answer: D.

I think the part I'm getting confused by is this:

Bunuel wrote:

So the rate of 1 woman is \(\frac{1}{w}\) job/day and the rate of 1 man is \(\frac{1}{m}\) job/day.

I get that part.

Bunuel wrote:

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

I get this part as well, because 3*\(\frac{1}{w}\)= \(\frac{3}{w}\), and the same for the men's rate. But here's where I get lost:

Bunuel wrote:

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

Using the logic from the second quote, 3 men doing the same work 5 days sooner than 9 women would be 3*\(\frac{1}{m}\)+5 = 9*\(\frac{1}{w}\). I don't understand how we can just switch whether the multiplier goes in the numerator or the denominator like that. Because their work rates are remaining the same; 1 man will do 1/m no matter what, so 3 men should always do 3/m, not m/3.

ooooh wait, I get it now, if you have 3 men working at rate m, then it will take 1/3 as long to complete their portion as a single man would do. Whereas in the other quote, you're just using the rate of 1/m, and then not adjusting the rate, just stating that 2 of them working at that rate will complete a job in 2/m, and in the green quote, we have the actual number of days it takes given, whereas in the last quote we only have each gender's rate for comparison. Yes?

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