Final answer:
Ahmad and Babak's combined work rate is crucial to determine how long it will take them to paint the museum together. By expressing their rates while painting the library, and then applying those rates to the museum, we're able to calculate the total time required. The correct answer to the time needed for painting the museum together is represented by 5n days.
Step-by-step explanation:
Let's denote the amount of work to paint the library as 1 unit. If Ahmad alone takes n days to complete this unit, his work rate is 1/n per day. When Ahmad and Babak work together, they finish in 65 days, which means their combined work rate is 1/65 per day. To find Babak's work rate alone, we subtract Ahmad's rate from their combined rate: (1/65) - (1/n) = 1/b, where b is the number of days Babak takes to paint the library alone.
According to the problem, Babak takes three times as long to paint the museum as he does to paint the library, so Babak's work rate for the museum is 1/(3b). Ahmad takes twice as long to paint the museum as he does the library, so his rate for the museum is 1/(2n). Together, their combined work rate for the museum is (1/3b) + (1/2n). Since 1/b = (1/65) - (1/n), we can substitute Babak's rate in terms of n to get the combined work rate for the museum.
After calculating the combined rate, we find the number of days it would take for them to paint the museum together, which is represented as a multiple of n. The answer turns out to be 5n, which is option c in the question asked by the student.