The below question is from where my doubt had arisen.
1st Problem:
Question: A, B, C can do a job in 10, 20, and 40 days respectively. In how many days A can complete the job if he is assisted by B and C on every third day?
Answer:
A's one-day work = 1/10
B's one-day work = 1/20
C's one-day work =1/40
(A+B+C)'s one day work = 1/10 + 1/20 + 1/40 = 7/40
Work done in three days will be the sum of A's two-day work and (A+B+C)'s one day work.
A's two-day work = 1/10 * 2 = 1/5
Therefore, the work is done in three days = 1/5 + 7/40 = 3/8
3/8 part of the job is done in 3 days.
The entire job will be done by A in = 3 * 8/3 = 8 days.
I decided to follow the same approach from the above question to another same kind of my own problem.
2nd Problem:
My Question: A, B can do a job in 11, 11/3 days respectively. In how many days A can complete the job if he is assisted by B on every second day?
My Own approach to this Question [Correct me if I'm wrong]: A's one day of work = 1/11 B's one day of work = 3/11
So, On 1st day, 1/11 work is done
On 2nd day, 4/11 work is done [Since B's work is also added on 2nd day]
On 3rd day, 6/11 work is done totally [Since 1st day's work(1/11) + 2nd day's work(4/11) + 3rd day's work(1/11)]
On 4th day, 10/11 work is done totally [Since 1st day's work(1/11) + 2nd day's work(4/11) + 3rd day's work(1/11) + 4th day's work(4/11)]
On 5th day, 11/11 work is done totally [Since 1st day's work(1/11) + 2nd day's work(4/11) + 3rd day's work(1/11) + 4th day's work(4/11) + 5th day's work(1/11)]
So from this, we can know that it takes 5 days to complete the work by A. Simple right? But when I follow the previous method from the first problem to this problem, it gives me different answers. I'll show it below,
A's one-day work = 1/11
B's one-day work = 3/11
(A+B)'s one day work = 1/11 + 3/11 = 4/11
Work done in two days will be the sum of A's one-day work and (A+B)'s one day work.
A's one-day work = 1/11
Therefore, the work is done in two days = 1/11 + 4/11 = 5/11
So, 5/11 part of the job is done in 2 days.
The entire job will be done by A in = 2 * 11/5 = 22/5 days. When I follow my own method I get 5 days which I think is the correct answer to my problem which is 5 days. But when I follow the previously given method, I get a different answer which is 22/5 days. But both the 1st problem and the 2nd problem is of similar type except the numbers and some minor things are different.
My doubt is If my own answer which is 5 days from the 2nd problem is right means then, Is the 1st problem wrong or the approach to the first problem is wrong? But the 1st problem's solution is quite famous all over the internet.
As @JMoravitz suggests in the comments the provided approach for question $1$ is not usually valid. The method assumes that the work done during any given day is constant and is given by the average work done during the period after which the working pattern repeats. But this not true as you found using the valid method you suggest. The suggested method is only valid if it's the case that after an integer multiple of this working period the job is done. This is because the average work done over every day considered would be equal to the periodic average in this case.