40 employees take 30 days,working at 8 hrs per day,to complete a task.

Hi All,

We're told that 40 employees take 30 days (working at 8 hrs per day) to complete a task. 40 employees start the work - BUT after 10 days, 20 workers leave and are replaced by employees who are 1/2 as productive. We're asked for the number hours per day that the new team must work to complete the job in the scheduled timeline. With these types of rate questions, it helps to first figure out the total amount of 'work' needed to complete the job, then use that information with whatever other information you've been given.

The original team would need (40)(30)(8) = 9600 worker-hours to complete the job

For the first 10 days, all 40 employees work as planned, so (40)(10)(8) = 3200 worker-hours are completed, leaving 9600 - 3200 = 6400 worker-hours to go

20 of the 40 employees are replaced with workers who are HALF as productive, meaning that each of those employees completes 1/2 a worker-hour of work per 1 hour. With those 20 replacement workers, the 'new' team of 40 employees will complete LESS work per hour than the original team did.

New team:
20 original workers complete 20 worker-hours per hour
20 new workers complete 10 worker-hours per hour
20 + 10 = 30 worker-hours completed per hour

With 20 days remaining and 6400 worker-hours to go...
6400/20 = 320 worker-hours must be completed each day.

At the new rate, that would require...
320/30 = 10 2/3 hours of work per worker each day

Final Answer:

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

Hi...

the TIMER shows all have gone wrong, so here is something which may help ..

two ways..
1) LOGICAL...
for the remaining days 20 are working for 8 hrs and other 20 will require to work for 16hrs..
so all 40 work for entire 8 hrs and then share the remaining work of 16-8=8 hrs..
these 8 hrs are of the speed of 1/2 productive and if we add full productive person, it is SAME as adding 2 half-productive persons..

so these 8 hrs is being shared by 1+2=3 person, MEANING each person requires to do \(\frac{8}{3}=2.66\) hrs
so the hours required is \(8+2.66=10.66\)
C

PROPER method..
40 require \(30*8=240\) hrs, so 1 full productive will require \(240*40\)h.....1hr work = \(\frac{1}{240*40}\), so
1 hr work of half productive = \(\frac{1}{2}*\frac{1}{240*40}= \frac{1}{480*40}\)..
combined work of these two - set of productive+half productive = \(\frac{1}{240*40}+\frac{1}{480*40} = \frac{3}{480*40}\)
now there are 20 sets so 1 hr work of all 40 = \(20*\frac{3}{480*40}=\frac{1}{320}\)
so they require to work for 320 h for complete work but remaining work is \(\frac{30-10}{30}=\frac{2}{3}\), this will be completed in \(\frac{320*2}{3}\)
and these hours are equally spread across 20 days, so per day = \(\frac{320*2}{3*20}=\frac{32}{3} = 10.66\)

C

The rate for the 40 workers is 1/(30 x 8) = 1/240 task/hour

So after 10 days, the amount of work completed is 1/240 x 10 x 8 = 80/240 = 1/3 of the job and thus 2/3 is left to be completed.

Since 20 workers leave, the rate of the remaining 20 workers is 1/2 x 1/240 = 1/480 task/hour and the 20 new workers who join in have a rate that is half of 1/480, or 1/960, task/hour. Thus the new rate of the 40 workers (20 original and 20 new workers) is 1/480 + 1/960 = 3/960 = 1/320 task/hour.

They still have to finish the task in 20 more days. If we let n = the number of hours they work per day, then it must be true that:

1/320 x 20 x n = 2/3

1/16 x n = 2/3

n = 2/3 x 16

n = 32/3 = 10 ⅔ ≈ 10.6

Alternate Solution:

The number of worker-hours required for the entire job is 40 workers x 30 days x 8 hours/day = 9600 worker-hours. In the first 10 days, the workers have accomplished 40 workers x 10 days x 8 hours/day = 3200 worker-hours, leaving 6400 worker-hours to be accomplished.

The remaining work will be accomplished by 20 original workers plus 20 new workers who work at half-speed. The total amount of work accomplished, then, is equivalent to the work of 30 original workers.

With 6400 worker-hours needing to be done by (the equivalent of) 30 workers, we see that each worker will have to work for 6400/30 ≈ 213.33 hours. This needs to be done in 20 days, so each worker will have to work 213.33/20 ≈ 10.6 hours per day.

Answer: C

Hi...

the TIMER shows all have gone wrong, so here is something which may help ..

two ways..
1) LOGICAL...
for the remaining days 20 are working for 8 hrs and other 20 will require to work for 16hrs..
so all 40 work for entire 8 hrs and then share the remaining work of 16-8=8 hrs..
these 8 hrs are of the speed of 1/2 productive and if we add full productive person, it is SAME as adding 2 half-productive persons..

so these 8 hrs is being shared by 1+2=3 person, MEANING each person requires to do \(\frac{8}{3}=2.66\) hrs
so the hours required is \(8+2.66=10.66\)
C

generis please help with this question using your method.

Buttercup3 - Sure. The approach, though, works when all workers are equally productive.
When all the workers are not equally productive, we have to finesse the formula a bit.*
These employees are all going to be men. Time is in hours.

Rewrite the standard work formula RT=W by adding (# of workers) to LHS:
(# of workers) * (Rate) * (Time) = Work

Scenario 1, FIRST 10 DAYS - equally productive workers
1) Find the rate of each man
# * R * T = W
# = 40
R = ??
T = 240 (30 days * 8 hrs per day)
W = 1
40 * R * 240 = 1
R = \(\frac{1}{(40)(240)}=\frac{1}{9600}\) = rate of individual man

2) How much work is finished?**
W = # * R * T
# \(= 40\)
\(R = \frac{1}{9600}\)
\(T = 80\) (10 days * 8 hrs per day)
\(W = 40 * \frac{1}{9600} * 80\)
\(W = \frac{3200}{9600}=\frac{1}{3}\)
Work is \(\frac{1}{3}\) finished, \(\frac{2}{3}W\) remains

Scenario 2: NEXT 20 DAYS, workers not equally productive
20 men leave. Replaced by 20 men who are half as productive
Adding individual rates (fast and slow) and simply multiplying by number of men (40) will not work because the workers are not equally productive.

chetan2u shows one way to handle the different levels of productivity.
Pair one fast man with one slow man, add their rates, then multiply by the number of pairs (20 pairs).

You can also find each group's rate (or each "set's" rate), then add them.
Group rate in this context is simply # (of men) * \(R\) (individual man) -- i.e. two of the three variables on LHS

Find each group's / set's rate, then add

• FAST set's rate: (# \(* R) = (20 * \frac{1}{9600}) = \frac{20}{9600}\)
• SLOW set's rate: (# \(* R * \frac{1}{2}) =\)
\((20*\frac{1}{9600}*\frac{1}{2})= \frac{20}{(9600)(2)} = \frac{20}{19200}\)

FAST set + SLOW set = new rate = \(R_2\)

\(R_{2}= (\frac{20}{9600}+\frac{20}{19200})=(\frac{40}{19200} + \frac{20}{19200})=\)
\(\frac{60}{19200}=\frac{6}{1920}= \frac{1}{320} = R_2\)

With \(R_2\) , we have calculated (#\(* R)\) on LHS.
We can substitute \(R_{2}\) for those two variables, thus \(R_2 * T = W\)

Number of hours per day? Total hours divided by number of days

Total hours, \(R_2 * T = W\), so \(T = \frac{W}{R_{2}}\)
\(W = \frac{2}{3}\)
\(R_{2} = \frac{1}{320}\)
\(T = \frac{\frac{2}{3}}{\frac{1}{320}}= \frac{2}{3} * 320 = \frac{640}{3}\) total hours

Hours per day: \(\frac{TotalHours}{NumberOfDays}\)
\(\frac{\frac{640}{3}}{20}= \frac{640}{60} = \frac{64}{6} = \frac{32}{3} \approx{10.66}\approx{10.6}\) hours per day

Answer C

Hope that helps.

* If all 40 workers were replaced by workers half as productive, you would stay with an unmodified (# * R * T) = W
You would: 1) Find "normal" rate 2) Find work finished after 10 days; 3) Find the slower rate (1/2 * R); 4) use the formula straightforwardly. No modifications
**We know this already, from direct proportionality between time and work when rate is constant.

Total work \(= 40 * 30 * 8 = 9600\) units

Work done by \(40\) employees in \(10\) days \(= 40*10*8 = 3200\) units.

Remaining work \(= 9600-3200 = 6400\) units.

In order to complete the work on scheduled time \(40\) employees must work for \(\frac{6400}{(40*20)} = 8\) hours a day, but since \(20\) of the current employees are only \(\frac{1}{2}\) as efficient, \(20\) employees with \(100\%\) efficiency would equal to \(10\) with \(50\%\) efficiency, hence the total number of employees can be treated as \(30\).

So, the no. of hours to be worked = \(\frac{6400}{(30*20)} = \frac{6400}{600} = 10.6\). Ans - C.

Here is my way of solving this problem:

In the first 10 days, the original set of employees did 1/3 of the work, so the remain is 2/3 of the work. After 10 days, 20 of the employees will be replaced with new employees that is half as productive. So:

Original: 40 employees --- 8 h/day --- 30 days --- 1 work
Now : 40 employees --- 8 h/day --- 30 days --- 1/2 + 1/4 = 3/4 work (because new employees are half as productive)

We know from the problem that the new crew needed to increase hours per day to complete the remaining work in 20 days, hence:

40 employees --- 8 h/day --- 30 days --- 3/4 work (the work rate of new crew)
40 employees --- x h/day --- 20 days --- 2/3 work (the remaining work needed to be done in 20 days)

We got the following equation: x/8 * 20/30 = (2/3) / (3/4), equal to x = (12 * 8)/9 = 10.66

Answers C
There was a link to an article to easily solve this kind of problem by Karishma but i can't post in since my post count is way too low

hoang221 , welcome!

Please send me a PM with the link. I will be happy to post it for you here, with credit to you and VeritasPrepKarishma.

The excellent article to which hoang221 refers is Work-Rate Questions Made Easy!

I tried it by unitary method :

if total work expected per day is 16(8+8)

but because of 1/2 productivity from 20 people

12(8+4) units of work is done in 8 hours
1 unit in 8/12 hours
16 units in (8/12)*16 = 10.66

Let the rate for each of the 40 employees = 2 widgets per hour, implying the following:
Hourly rate for 40 employees = 2*40 = 80 widgets
Work produced over each 8-hour day = 8*80 = 640 widgets
Total amount of work to be produced for the 30-day job = 30*640 = 19200 widgets

Since the daily rate for 40 workers = 640 widgets, the work produced over the first 10 days = 10*640 = 6400 widgets.
Remaining work = 19200 - 6400 = 12800 widgets

Since the 20 replacement workers are half as productive as the 20 original employees, the resulting rate for the new team = 20*1 + 20*2 = 60 widgets per hour.
To complete 12800 widgets in the remaining 20 days, the daily work required \(= \frac{12800}{20} = 640\) widgets.
For 640 widgets to be produced each day at a rate of 60 widgets per hour, the number of hours required \(= \frac{640}{60}\) ≈ 10.6

Yes, this method has its roots in variation. The link is shared by generis above and here is the link to the posts:
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2015/1 ... made-easy/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2013/0 ... variation/

You can break down the information given to make two whole scenarios.

40 men......30 days ...... 8 hrs ...... 1 work
30 men ..... 20 days ....... ?? hrs..... 2/3 work

(Note that after 10 days, 1/3rd the work will be done so 2/3rd work will be left. Also, if 20 men leave and are replaced by 20 men half as productive, it is like getting back only 10 men)

No of hrs = 8 * (40/30) * (30/20) * (2/3) = 10.66 hrs

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