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If 12 men and 16 women can do a piece of work in 5 days and

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If 12 men and 16 women can do a piece of work in 5 days and

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Source: GMAT Prep

If 12 men and 16 women can do a piece of work in 5 days and 13 men and 24 women can do it in 4 days, how long will 7 men and 10 women take to do it?

A. 4.2 days
B. 6.8 days
C. 8.3 days
D. 9.8 days
E. 10.2 days

The OA is C.

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BTGmoderatorLU wrote:
Source: GMAT Prep

If 12 men and 16 women can do a piece of work in 5 days and 13 men and 24 women can do it in 4 days, how long will 7 men and 10 women take to do it?

A. 4.2 days
B. 6.8 days
C. 8.3 days
D. 9.8 days
E. 10.2 days

The OA is C.
Let M = the rate for each man and W = the rate for each woman.

The TIME RATIO for 12 men and 16 women to 13 men and 24 women is 5 days to 4 days.
Since time and rate are RECIPROCALS, the RATE RATIO for 12M+16W and 13M+24W is equal to the reciprocal of the time ratio:
(12M+16W)/(13M+24W) = 4/5
60M + 80W = 52M + 96W
8M = 16W
M = 2W.

Let W = 1 widget per day, implying that M = 2 widgets per day.

Work produced each day by 12 men and 16 women = (12*2) + (16*1) = 40 widgets per day.
Thus, the total work produced by 12 men and 16 women over 5 days = 40*5 = 200 widgets.

Work produced each day by 7 men and 10 women = (7*2) + (10*1) = 24 widgets per day.
Time for 7 men and 10 women to produce 200 widgets = 200/24 = 25/3 ⩳ 8.3 days.

The correct answer is C.

I doubt that this problem is contained in GMATPrep.
While the question stem asks for an exact time, the OA is an approximation.
The GMAT does not use the phrase "piece of work."
Also, it seems sexist for the men's rate to be twice the women's rate.
The GMAT strives to avoid this sort of bias.

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BTGmoderatorLU wrote:
Source: GMAT Prep

If 12 men and 16 women can do a piece of work in 5 days and 13 men and 24 women can do it in 4 days, how long will 7 men and 10 women take to do it?

A. 4.2 days
B. 6.8 days
C. 8.3 days
D. 9.8 days
E. 10.2 days
We can let the time it takes 1 man to finish the work = m, and thus the rate of 1 man = 1/m. Likewise, we can let the time it takes 1 woman to finish the work = w, and thus the rate of 1 woman = 1/w.

Thus, the combined rate of 12 men and 16 women is 12/m + 16/w. Since they can finish the work in 5 days, their combined rate is also equal to 1/5. Thus, we have:

12/m + 16/w = 1/5

Multiplying both sides of the equation by 5mw, we have:

60w + 80m = mw

Similarly, the combined rate of 13 men and 24 women is 13/m + 24/w. Since they can finish the work in 4 days, their combined rate is equal to 1/4. Thus, we have:

13/m + 24/w = 1/4

Multiplying both sides of the equation by 4mw, we have:

52w + 96m = mw

So, we have 60w + 80m = 52w + 96m (since they both equal mw).

60w + 80m = 52w + 96m

8w = 16m

w = 2m

We can now substitute w = 2m into the first equation, 12/m + 16/w = 1/5, to solve for m:

12/m + 16/(2m) = 1/5

12/m + 8/m = 1/5

20/m = 1/5

m = 100

Since m = 100 days, w = 200 days. The rate of 1 man is 1/100 and the rate of 1 woman is 1/200. Thus, the rate of 7 men and 10 women is 7/100 + 10/200 = 7/100 + 5/100 = 12/100, and the time for them to finish the same work is 1/(12/100) = 100/12 = 8.3 days.

Alternate Solution:

Alternatively, we can interpret the equation w = 2m as follows:

The time required for 1 woman to finish the job is twice that of a man; or in other words, the job done by 2 women is equivalent to the job done by 1 man. We know 12 men and 16 women finish the job in 5 days and as per the above discussion, this is equivalent to the job done by 12 + 16/2 = 20 men.

The question is asking for the time required to finish the job with 7 men and 10 women working, which is equivalent to 7 + 10/2 = 12 men working. We can set up an inverse proportion to find the required time: If 20 men finish a job in 5 days, then 12 men finish the job in how many days? Letting this unknown quantity be x, we have

20*5 = 12*x

x = 100/12 = 8.3 days

Answer: C

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BTGmoderatorLU wrote:
Source: GMAT Prep

If 12 men and 16 women can do a piece of work in 5 days and 13 men and 24 women can do it in 4 days, how long will 7 men and 10 women take to do it?

A. 4.2 days
B. 6.8 days
C. 8.3 days
D. 9.8 days
E. 10.2 days
\[7\,\,{\text{men}}\,\, \cup \,\,\,{\text{10}}\,\,{\text{women}}\,\,\, - \,\,\,1\,\,{\text{work}}\,\,\,\, - \,\,\,?\,\,{\text{days}}\]

Is there a systematic way of dealing with this kind of problem, to be able to do it in a few minutes "naturally"?

Certainly! Let´s do it:

Let "task" be the fraction of this (piece of) work that one man can do in 1 day, hence:

\[1\,\,{\text{man}}\,\,\, - \,\,\,1\,\,{\text{day}}\,\,\,\, - \,\,\,1\,\,{\text{task}}\]

Let k (k>0) be the fraction of the "task" defined above that one woman can do in 1 day (where k may be between 0 and 1, or equal to 1, or greater), hence:

\[1\,\,{\text{woman}}\,\,\, - \,\,\,1\,\,{\text{day}}\,\,\, - k\,\,{\text{tasks}}\,\]

Now the long-lasting benefit of this "structure": everything else becomes easy and "automatic":

\[\left. \begin{gathered}
{\text{12}}\,\,{\text{men}} - \,\,\,5\,\,{\text{days}}\,\,\, - \,\,\,\,12 \cdot \,5 \cdot 1\,\,\,{\text{tasks}}\, \hfill \\
{\text{16}}\,\,{\text{women}} - \,\,\,5\,\,{\text{days}}\,\,\, - \,\,\,\,16 \cdot \,5 \cdot k\,\,\,{\text{tasks}}\,\,\, \hfill \\
\end{gathered} \right\}\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{question}}\,\,{\text{stem}}} \,\,\,\,5\left( {12 + 16k} \right)\,\,{\text{tasks}}\,\,\,\, = \,\,\,\,1\,\,{\text{work}}\,\,\,\,\left( * \right)\]
\[\left. \begin{gathered}
{\text{13}}\,\,{\text{men}} - \,\,\,4\,\,{\text{days}}\,\,\, - \,\,\,\,13 \cdot \,4 \cdot 1\,\,\,{\text{tasks}}\, \hfill \\
{\text{24}}\,\,{\text{women}} - \,\,\,4\,\,{\text{days}}\,\,\, - \,\,\,\,24 \cdot \,4 \cdot k\,\,\,{\text{tasks}}\,\,\, \hfill \\
\end{gathered} \right\}\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{question}}\,\,{\text{stem}}} \,\,\,\,4\left( {13 + 24k} \right)\,\,{\text{tasks}}\,\,\,\, = \,\,\,\,1\,\,{\text{work}}\,\,\,\,\left( {**} \right)\]

\[\left( * \right) = \left( {**} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,5\left( {12 + 16k} \right) = 4\left( {13 + 24k} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,k = \frac{1}{2}\]
\[?\,\,\,\,:\,\,\,\,\,\left. \begin{gathered}
\boxed{{\text{7}}\,\,{\text{men}}} - \,\,\,1\,\,{\text{day}}\,\,\, - \,\,\,\,7\,\,\,{\text{tasks}}\, \hfill \\
\boxed{{\text{10}}\,\,{\text{women}}} - \,\,\,1\,\,{\text{day}}\,\,\, - \,\,\,\,10 \cdot k = 5\,\,\,{\text{tasks}}\,\,\, \hfill \\
\end{gathered} \right\}\,\,\,\,\,\mathop \Rightarrow \limits^{\boxed{{\text{FOCUSED - GROUP}}}} \,\,\,\,\frac{{12\,\,{\text{tasks}}}}{{1\,\,\,{\text{day}}}}\,\,\,\,\left( {***} \right)\]

\[\left( * \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,5\left( {12 + 16 \cdot \frac{1}{2}} \right) = 100\,\,{\text{tasks}}\,\,{\text{ = }}\,\,{\text{1}}\,\,{\text{work}}\,\]

And we finish in "high style", using UNITS CONTROL, one of the most powerful tools of our method!

\[\left( {***} \right)\,\,\,\,?\,\,\, = \,\,\,100\,\,{\text{tasks}}\,\,\,\,\left( {\frac{{1\,\,\,\,{\text{day}}}}{{12\,\,{\text{tasks}}}}\begin{array}{*{20}{c}}
\nearrow \\
\nearrow
\end{array}} \right)\,\,\,\, = \,\,\,\,\frac{{100}}{{12}} = \frac{{25}}{3} = \frac{{24 + 1}}{3} = 8\frac{1}{3}\,\,\,\,\left[ {{\text{days}}} \right]\]
Obs.: arrows indicate licit converter.


This solution follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.

_________________
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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Hi All,

We're told that 12 men and 16 women can do a piece of work in 5 days and 13 men and 24 women can do it in 4 days. We're asked how long it will take 7 men and 10 women to complete that same task. This question can be approached in a couple of different ways (some of which involve a lot of calculations). The answer choices are sufficiently 'spread out' that you can use a bit of 'ratio math' and a little logic to get to the correct answer.

We're going to focus on just the first piece of information: it takes 12 men and 16 women a total of 5 days to complete a task. If you were to DOUBLE the number of workers, then you would HALVE the amount of time (re: 24 men and 32 women would take 2.5 days to complete the task). If you were to HALVE the number of workers, then you would DOUBLE the amount of time that it takes to complete the task:

6 men and 8 women would take a total of 10 days to complete a task.

If we HALVE the number of workers again, we would again DOUBLE the amount of time needed to complete the task:

3 men and 4 women would take a total of 20 days to complete a task.

We're asked how long it would take 7 men and 10 women to complete that task. If we multiply the above 'work information' by 2.5, we get...

(2.5)(3) men and (2.5)(4 women) would take a total of 20/2.5 days to complete a task...
7.5 men and 10 women would take a total of 8 days to complete a task.
Notice how this is almost the exact question we were asked to solve for. The difference is that we're including an extra "1/2 of a man" in this calculation. With just 7 men (instead of 7.5 men), we would need slightly more than 8 days to complete the task. There's only one answer that matches....

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

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Contact Rich at Rich.C@empowergmat.com

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