A contractor estimated that his 10-man crew could complete

Check out the work rate video here first: https://youtu.be/88NFTttkJmA

You can also look at this problem in terms of units of work.

10 people can finish the work in 110 days so there are 1100 units of work (1 unit done by 1 man in 1 day)
10 men worked for 55 days and did 10*55 = 550 units (since it rained for 5 days so day 56th to day 60th were all washed out). Now 550 units are left. [Mind you, it is an assumption that there was no rain in the first 55 days. Ideally, the question should have explicitly stated it.)

Now there are 16 people to do 550 units of work. Work will be done in 550/16 = 34.4 days i.e. 35 days are needed.
After 60th day, we have total 40 days. Out of these, 35 are needed so it must have rained on 5 days.

Official Solution:

A contractor estimated that his 10-person crew could complete a construction project in 110 days if there were no rain delays. (Assume the crew does not work on rainy days and that rain is the only factor preventing the crew from working). However, on day 61, after having experienced a total of 5 days of rain in the previous days, the contractor hired 6 additional workers and managed to finish the project early. If the project was completed in 100 days, how many rainy days occurred after day 60?

A. 4
B. 5
C. 6
D. 7
E. 8


Given: A 10-person crew requires 110 days to complete the construction project.

"On day 61, after experiencing 5 days of rain...": Since there were 5 days of rain out of the first 60 days, the crew must have been working for 55 days, thus completing half of the job, with the other half remaining (55 days of work for the 10-person crew).

Subsequently, the contractor hired "6 additional workers": This increased the construction speed by 1.6 times. Therefore, the new 16-person crew would require \(\frac{55}{1.6} \approx 34.4\) days to complete the construction. However, after the additional workers were hired, the project was completed in \(100-60=40\) days. As a result, there must have been 5 rainy days during this period. (The crew needed MORE than 34 days to finish the job, so if there had been 6 rainy days, they wouldn't have been able to complete the project within 100 (40) days.)


Answer: B
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Given: 10-man crew needs 110 days to complete the construction.

"On the 61-st day, after 5 days of rain ..." --> as it was raining for 5 days then they must have bee working for 55 days thus completed 1/2 of the job, 1/2 is left (55 days of work for 10 men).

Then contractor "hired 6 more people" --> speed of construction increased 1.6 times, so the new 16-man crew needed 55/1.6=~34.4 days to complete the construction, but after they were hired job was done in 100-60=40 days --> so 5 days rained. (They needed MORE than 34 days to finish the job, so if it rained for 6 days they wouldn't be able to finish the job in 100(40) days.)

Answer: B.
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You can visualize it in another way:
10 man crew needs 110 days means the total work = 10*110 = 1100 man-days
(man-days means the amount of work 1 man does in 1 day)

In 60 days, it rained for 5 days and the crew worked for 55 days and completed 10*55 = 550 man-days work and left over work is 1100 - 550 = 550 man-days

From 61st day on, there are 16 people working and will need 550/16 = 34.4 days to complete the remaining work. So out of 40 days (from 61st to 100th day), 35 days were used (though the 100th day wasn't completely utilized) and it must have rained on 5 days.

your method makes sense, but i got kind of lost at the end.
so if it takes 34.4 days, wouldn't it be 61+34=95 days, so it rained for 5 days?
thanks.

Yes it rained on 5 days but the job took 34.4 days, so more than 34 (full 34 days and the part of 35th day). Next, you should count from 61st day (61st day is the first day when 16 men crew started to work) so 60+35=95 (not 61+...) --> 100-95=5.
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As bunuel pointed out above, it rained for 5days in the first 60 days. Thereafter, in the remaining 100 - 60 = 40 days, they worked for 34.4 days ie 35 days (the 35th day, they worked partly). The rest of the 5 days, it rained. Snce the work was not over in 34 days, they needed the 35th day. You cannot round it off to 34.

Bunnel - Could you please explain the step
"Then contractor "hired 6 more people" --> speed of construction increased 1.6 times, so the new 16-man crew needed 55/1.6=~34.4 days"

10 men need 55 days to complete a job. How many days does 20 men need to complete the same job? In 55/2 days (the rate of 20 men group is twice the rate of 10 men group).

Similarly, if 10 men need 55 days to complete a job, 16 men (10*1.6) will need 55/1.6 days to complete the job (the rate of 16 men group is 1.6 times greater than the rate of 10 men group).

Or: there was 10*55=550 man-days of work left which can be done by 16 men in 550/16=~34.4 days.
Refer to two-consultants-can-type-up-a-report-126155.html for theory on work problems and practice questions.

Hope it helps.
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I thought of the problem in units of work. The total project needed 1100 units of work. 550 of these units were completed by the 55th day (10 x 55). It rained on the 56th, 57th, 58th, 59th, and 60th day. On the 61st day, the contractor hired 6 more laborers thus bringing the total laborer population to 16. With 550 units of work left, these 16 laborers should have completed the remaining work in 34.375 days.

Here's where I went wrong. I added 34.375 to 61 to bring total days needed (assuming no rain after the 60th) to 95.375 (essentially 96 days of work). However, this would imply only 4 days of rain since the project actually took 100 days to complete. I believe I went wrong in adding the 34.375 to 61 instead of 60 since by doing the former I didn't credit them for one day of work. Could someone confirm this?

Thank you.

You are right. You simply took away one work day. The 61st day of the project was a work day, in fact the first day when the extended team of 16 took over the job.
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Think of it this way: 60 days are accounted for. From the 61st day to the 100th day, there are 40 days. You need 35 of these to complete the work so you are left with 5 days.
When you add 35 to 61, you are saying that 61 days are accounted for but that is not the case.

Hi,

please tell me one thing:

Estimated time = 110 days
actual completed time = 100 days
no work days = 110-100 = 10 days
we are given only rain is a factor because of which no work will be done and it has already rained for 5 days before the 60th. thus 10-5 = 5 more days it will rain after the 60th

why do we have get into work and rates for this?

It took 100 days to complete the work instead of the estimated 110 because he employed 6 extra people, not because they did not work on those days. The workers actually worked for only 90 days. Out of the 100 days, it rained for 10 days and they worked for the rest of the 90 days. It is just a co-incidence that the extra workforce reduced the working days by 10, the same number of days for which it rained.

Given 10 men took 110 days.
Total work = 1100 Man Days

Given on the 61-st day, after 5 days of rain, which means the 10 men worked for 55 days and completed:
55 X 10 = 550 Man Days of work.

Remaining work = 1100-550 = 550 Man Days of work.

Problem states that the work was completed early in 100 days by the 16 men.
Assume it rained on x days, so the work was completed in 40-x days
Hence , 16 (40-x)=550
or x ~ 5 days

Hi,
I wanted to understand if the above mentioned step in red is right.

Yes, but ensure that you understand how to deal with fractions.

16 (40-x)=550
40 - x = 34.375
x = 5.625
Now will you say it rained for 5 days or 6 days? Note that you will say that it rained for 5 days because the workers needed the extra .375 day to complete the work. If it were to rain on the 6th day too, the workers would not have been able to work on that day and hence the work would not be complete.
The fraction means that on the 100th day, the workers worked i.e. there was no rain but they worked for only a fraction of the day because the work got over.

110 days * 10 workers= 1100 units of work in total..

55 days of work done with help of 10 workers..therefore, 55 days * 10 workers = 550 units of work done..it means 550 units of work remaining..

It means, 550 units /16 workers = 34.3 days required to finish the work..

Since the work done in 100 days,we have 40 days remaining, but the days required are 34.3 or we can say 35 since,until 0.3 units of work is not completed,we cannot claim that work is completed..

Hence 5 days of work remain...


Hope it helps..
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Following your approach, I did this, but the answer came out wrong.

after 60 days have passed, the 16 member crew is able to do 550 mandays of work in 40 days.

What would've happened if it worked for the 40 days? they would have done 640 mandys of work.
But they did only 550 mandays of work. so they did not work for the number of days it would have taken them to do 640-550 = 90 mandays of work

90/16 = 5.625 rounding off gives 6!

The answer is the same - you need to use logic here to figure out whether you will round off to 5 or 6. Note that in my explanation above, I rounded off 34.4 days to 35 days. Why? Because they worked for a part of the 35th day too. The work got completed in 0.4 day but the 35th day was used. Hence it rained for 5 days only.
You will use the same logic.
5.6 days were not used. It must have rained on 5 days and a part of the 6th day must have been used so it must not have rained on it.