Three workers A, B, C working individually can complete a task

then i think the question framing is wrong :
when the question says every second and third day:
It means second (alternate day)
and third means (every third day)
so basically on 6th day they three will be working

Apologies but the question language is little misleading

Yes, you're right. Look at the solution, on Day 6 they all work together (Day 6 - A+B+C - \(\frac{1}{30}\) + \(\frac{1}{15}\) + \(\frac{1}{10}\) = \(\frac{1}{5}\)
)

\(A\) works every day so let us start with a rate of 1/30. Next, B has a rate of 1/15 if he were to work every day. Instead, he works every two days. We may average out his work by saying he does 1/30 every day instead. Similarly, for C we average out his work so that he does 1/30 every day.

Thus the rate is 1/30*3 = 1/10. Let us use this rate for only 6 days however as we need to decide which exact day the task will be completed.

There is 40% of the task left, we may subtract from 60% instead to find how many days are required. The last day of each 6th day, all of ABC are working together. Their combined rate is 1/30 + 1/15 + 1/10 = 6/30 = 1/5 = 20% of the task. Thus if we take away the 6th day, they complete only 40% for the first 5 days of the 6. Then they need 6 + 5 = 11 days to complete 100%.

Ans: E

Hi ravigupta2912 ,

You were assuming constant rates throughout the course of the job. That is ok every 6 days, but for the last part we need to add the rates (or subtract from 60%) one by one since they get the job done unevenly during the last 40% of the job (the 6th day as much more weight compared to the average daily rate).

Thanks TestPrepUnlimited. Understood, I needed to multiply that rate with 6 days (since 1/10 is avg for 6 day period) to derive the work done (6/10) in 6 days and then compute time needed for the balance 4/10 or 40% of work.

apologies, I forgot to check the solution.
Ye sit is then correct . we have to check till 6 days how much is completed and then the rest

CONCEPT: Time and Work/Rate of completing a work
Efficiency is inversely proportionate to time.

SOLUTION: Let the total work = LCM(30,15,10)=30 units
A's efficiency =30/30=1unit/day; B's efficiency=30/15=2unit/day 'C's efficiency=30/10=3units/day.
Work done in three consecutive days= [1+(1+2)+(1+3)]=8 units.
Hence, work completed in 9 days= 8*3=24 units.
Work completed by A on 10th day=1 unit.
Work completed by A & B on 11th day =3 units.
Total work completed as of now =24+1+3=28 units.
On 12th day, the remaining 2 units of work is completed by utilizing 2/4=1/2 of the day.
(As 4 units is the efficiency of every 3rd day by A and C)
Hence, total days needed to complete the task is 11 1/2 days. (E)

Hope this helps. Keep studying
Devmitra Sen

On rates questions, rather than saying one of the workers completes 1/30 or 1/15 or 1/10 of the job, I strongly recommend just making up what the job is in such a way that you can avoid messing around with the fractions!!

Let's say the task is to solve 30 GMAT questions.
A can complete the task in 30 days, so A solves one GMAT question per day.
B can complete the task in 15 days, so B solves two GMAT questions per day.
C can complete the task in 10 days, so C solves three GMAT questions per day.

On day 1, only A works, so 1 question gets solved.
On day 2, A and B work, so 1+2=3 questions get solved. We have now completed 4 questions.
On day 3, A and C work, so 1+3=4 questions get solved. We have now completed 8 questions.
On day 4, A and B work, so 1+2=3 questions get solved. We have now completed 11 questions.
On day 5, only A works, so 1 question gets solved. We have now completed 12 questions.
On day 6, all three work, so 1+2+3=6 questions get solved. We have now completed 18 questions.
On day 7, only A works, so 1 question gets solved. We have now completed 19 questions.
On day 8, A and B work, so 1+2=3 questions get solved. We have now completed 22 questions.
On day 9, A and C work, so 1+3=4 questions get solved. We have now completed 26 questions.
On day 10, A and B work, so 1+2=3 questions get solved. We have now completed 29 questions.
On day 11, only A works, so 1 question gets solved. We have now completed all 30 questions.

11 days is not listed as an answer choice.

Answer choice E.


FWIW, I wrote all that to make it clear what I was doing, but this is what I wrote on scratch paper:
1 A 1 1
2 AB 3 4
3 AC 4 8
4 AB 3 11
...
...
11 A 1 30

Shouldn't A be working alone on Day 4? Then joined by B on Day 5? Then again by C on day 6? Then alone again on day 7? Isn't that what the wording of the question said? I'm sorry but I'm having trouble understanding the way you drew the pattern. Excellent explanation though. Kudos awarded

I think this might boil down to a misunderstand of how the word “respectively” in a situation like this? “B and C helps A on every second and third day respectively” means that B helps on every second day and C helps on every third day. So, B helps on days 2, 4, 6, 8,…and C helps on 3, 6, 9,… Does that help?

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A works everyday - 1, 2, 3, 4, 5, 6, 7, 8...
B works alternately - 2, 4, 6, 8, 10, 12 ...
C works on thirds - 3, 6, 9, 12...

We have a cycle of 6 days. (the LCM of 1, 2, 3).

In 6 days, A works every day and completes 6/30 = 1/5 of work.
B works on 3 days and completes 3/15 = 1/5 of work.
C works on 2 days and completes 2/10 = 1/5 of work.
So they all complete 3/5th work in first 6 days. So for 2/5th of work ( remaining work), they will take less than 6 days. Hence answer would be (A) or (E).
On the 6th day, they complete 1/30 + 1/15 + 1/10 = 6/30 = 1/5th of work. So in 5 days, they do 2/5th of the work (because they do 3/5th in 6 days).

Hence total number of days taken = 6 + 5 = 11

Answer (E)

A's daily rate is 1/30, so after 10 days 1/3 of the job is completed

Since B works every other day, he worked 10/2= 5 days multiplied by his 1/15 rate =
1/3 of the job

In 10 days C worked only the 3rd, 6th and 9th day = 3 days
multiplied by his rate of 1/10 = 3/10 of the job.

Total completed in 10 days:

1/3 + 1/3 + 3/10 = 29/30

So, 1/30th of the job remaining.

On the 11th day only A works and because his daily rate is 1/30, he works the whole day to complete the remaining 1/30 of the job

11 days

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