Twenty-four men can complete a work in sixteen days.Thirty

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pmFollowed by: 23 members

by BTGmoderatorDC » Sat Sep 08, 2018 10:59 pm

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Twenty-four men can complete a work in sixteen days.Thirty-two women can complete the same work in twenty-four days.Sixteen men and sixteen women started working for twelve days.How many more men are to be added to complete the work remaining work in 2 days?

A .16
B. 24
C. 36
D. 48
E. 54

OA B

Source: Veritas Prep

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Source: Beat The GMAT — Problem Solving | Sat Sep 08, 2018 10:59 pm

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sun Sep 09, 2018 3:11 am

BTGmoderatorDC wrote:Twenty-four men can complete a work in sixteen days.Thirty-two women can complete the same work in twenty-four days.Sixteen men and sixteen women started working for twelve days.How many more men are to be added to complete the work remaining work in 2 days?

A .16
B. 24
C. 36
D. 48
E. 54

Let the rate for each man = 1 widget per day, implying that the rate for 24 men = 24 widgets per day.
Since 24 men complete the job in 16 days, the total amount of work = rt = 24*16 widgets.

Since 32 women require 24 days, the rate for 32 women = w/t = (24*16)/24 = 16 widgets per day.
Since 32 women produce 16 widgets per day, the rate for each woman = (daily output)/(number of women) = 16/32 = 1/2 widget per day.
Thus, the combined rate for 16 men and 16 women = (16*1) + (16 * 1/2) = 24 widgets per day.

In 12 days, the amount of work produced by 16 men and 16 women = rt = 24*12 widgets.
Remaining work = (total work) - (work produced in the first 12 days) = (24*16) - (24*12) = 24(16-12) = 24*4 = 96 widgets.

To complete the remaining work in 2 days, the required rate = (remaining work)/(number of days) = 96/2 = 48 widgets per day.
Since the rate must increase from the value in red to the value in blue -- an increase of 24 widgets per day -- 24 more men are required.

The correct answer is B.

It seems sexist to make the rate for each man twice that for each woman.
Official problems strive to avoid this sort of bias.

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fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

Twenty-four men can complete a work in sixteen days.Thirty

by fskilnik@GMATH » Sun Sep 09, 2018 8:11 am

BTGmoderatorDC wrote:Twenty-four men can complete a work in sixteen days.Thirty-two women can complete the same work in twenty-four days.Sixteen men and sixteen women started working for twelve days.How many more men are to be added to complete the work remaining work in 2 days?

A .16
B. 24
C. 36
D. 48
E. 54

Nice problem!

LetÂ´s suppose each man does 1 task/day , and each woman does k tasks/day (where k>0, not necessarily an integer).

By the question stem, we know that:

> 24 men do 1 work in 16 days, hence 1 work = 24*16 tasks (1)
> 32 women do 1 work in 24 days, hence 1 work = 32*24*k tasks implies, by (1) above, that k = 1/2

> 16 men and 16 women will do in 12 days exactly 16*12*1 + 16*12*1/2 = 24*12 tasks, therefore by (1) we have still 24*(16-12) = 24*4 tasks to be done.

> If x is the number of men to be added (our FOCUS), we have a group of (16+x) men and 16 women performing (16+x)*1 + 16*1/2 = (24+x) tasks/day , and the final touch is done
with UNITS CONTROL, one of our methodÂ´s most powerful tools:

\[24 \cdot 4\,\,\,{\text{tasks}}\,\,\, = \,\,2\,\,days\,\,\left( {\frac{{24 + x\,\,\,{\text{tasks}}}}{{1\,\,\,{\text{day}}}}\,\,\,\begin{array}{*{20}{c}}
\nearrow \\
\nearrow
\end{array}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,24 + x = 24 \cdot 2\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = x = 24\]
Obs.: arrows indicate licit converters.

(If you realize this solution is absolutely clear and safe, you are the perfect candidate for our method... try our free and super-complete test drive!)

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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Scott@TargetTestPrep: GMAT Instructor; Posts: 8199; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Mon Sep 17, 2018 5:32 pm

BTGmoderatorDC wrote:Twenty-four men can complete a work in sixteen days.Thirty-two women can complete the same work in twenty-four days.Sixteen men and sixteen women started working for twelve days.How many more men are to be added to complete the work remaining work in 2 days?

A .16
B. 24
C. 36
D. 48
E. 54

Since 24 men can complete a work in 16 days, the rate of 24 men is 1/16, and hence the rate of 1 man is (1/16)/24 = 1/384.

Similarly, 32 women can complete the same work in 24 days, so the rate of 32 women is 1/24, and hence the rate of 1 woman is (1/24)/32 = 1/768.

Thus, the rate for 16 men and 16 women is 16 x (1/384) + 16 x (1/768) = 1/24 + 1/48 = 2/48 + 1/48 = 3/48 = 1/16.

Since these 16 men and 16 women worked for 12 days, they completed (1/16) x 12 = 12/16 = 3/4 of the job.

Thus, 1/4 of the job is left to be done. To find the number of additional men needed to complete the remaining work in 2 days, we can let n = the additional men needed and create the following equation (keep in mind that the rate of each man is 1/384 and the rate of the existing 16 men and 16 women is 1/16):

n(1/384)(2) + (1/16)(2) = 1/4

n/192 + 1/8 = 1/4

n/192 = 1/8

8n = 192

n = 24

Alternate Solution:

If 24 men can do the job in 16 days, then 16 men, which is 2/3 of 24 men, can do the job in 16/(2/3) = 24 days.

Similarly, since 32 women can do the same job in 24 days, 16 women, which is half of 32 women, can do the same job in 24/(1/2) = 48 days.

Thus, in one day, 1/24 of the job is done by men and 1/48 of the job is done by women; thus 1/24 + 1/48 = 3/48 = 1/16 of the job is done in total. Therefore, with 16 men and 16 women working, the job will be finished in 16 days. Since 16 men and 16 women have already worked for 12 days, the remaining job would have been completed by 16 men and 16 women in 16 - 12 = 4 days.

Since 16 men complete the job in exactly half the time compared to 16 women, the men work twice as fast as women; i.e. 1 man can do the same job done by 2 women. Thus, the work done by 16 men and 16 women is equivalent to the work done by 16 + 16/2 = 24 men.

Now, if we want the remaining job to be completed in 2 days instead of 4; in other words, if we want the remaining job to be completed in half the time, we should double the number of men. Since 16 men and 16 women is equivalent to 24 men, we should add 24 more men to get the remaining job done in 2 days.

Answer: B

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