Hi,
Let 'n' be the number of days Victor takes to finish a job when supervised.
So, he will finish a job when unsupervised in (n+3) days.
He is supervised for 72 days and unsupervised for 72 days.
Number of jobs finished when supervised is 72/n
Similarly, Number of jobs finished when supervised is 72/(n+3)
Given that he finished 36 jobs in total.
So, 72/n + 72/(n+3) = 36.
Solving this, we get n=3.
He will finish a job in 3 days when supervised and 6 days when unsupervised.
So, he will take 6*10 = 60 days to finish 10 jobs without any supervision.

It is worth understanding this step:

So, 72/n + 72/(n+3) = 36.
Solving this, we get n=3.

72/n + 72/(n+3) = 36

Divide by 36 to work with smaller numbers
2/n + 2/(n+3) = 1

Multiply everything by n*(n+3) to eliminate denominators
2(n)(n+3)/n + 2(n)(n+3)/(n+3) = 1(n)(n+3)

Cancel on the left
2(n+3) + 2(n) = 1(n)(n+3)

Simplify
2n + 6 + 2n = n^2 + 3n

Bring terms to the left
0 = n^2 - n - 6

Factor:
0 = (n - 3)(n + 2)

n = 3 or n = -2

Only the positive answer makes sense since this about jobs completed.

chaitanya.mehrotra wrote:Victor's job requires him to complete a series of identical jobs. If Victor is supervised at
work, he finishes each job three days faster than if he is unsupervised. If Victor works for
144 days and is supervised for half the time, he will finish a total of 36 jobs. How long
would it take Victor to complete 10 jobs without any supervision?

We can reason our way to the correct answer.
The times in this problem are almost certainly restricted to positive integers.

36 jobs in 144 days is an average of = 144/36 = 4 days for each job.

Victor works at two different rates (supervised and unsupervised) for equal amounts of time (72 days each).
To yield an average of 4 days per job, at the unsupervised rate it must take Victor MORE than 4 days to complete a job, at the supervised rate it must take Victor LESS than 4 days to complete a job.
Since the number of days when unsupervised is 3 more than when supervised, the two times must be factors of 72 that are 3 apart.

Thus, only two times are possible: 6 days per job when unsupervised, 3 days per job when supervised.

To confirm:
Let unsupervised = 6 days per job, supervised = 3 days per a job.
In 72 unsupervised days, number of jobs completed = 72/6 = 12.
In 72 supervised days, number of jobs completed = 72/3 = 24.
Total number of jobs completed in 144 days = 12+24 = 36.
Success!

Time to complete 10 jobs at a rate of 6 days per job = 10*6 = 60.

chaitanya.mehrotra wrote:Victor's job requires him to complete a series of identical jobs. If Victor is supervised at
work, he finishes each job three days faster than if he is unsupervised. If Victor works for
144 days and is supervised for half the time, he will finish a total of 36 jobs. How long
would it take Victor to complete 10 jobs without any supervision?

Although this question can be solved by using - 72/n + 72/(n+3) = 36, it can also easily be solved by using answer options. Since the question asks for 10 jobs, it makes our job easier as calculation will be quick.