I believe the intention of the problem is as follows:
Working together, printers A and B can do a certain printing job in 12 days. Working together, printers B and C can do the job in 16 days. Over the past 13 days, printer A worked for 5 days, printer B worked for 7 days, and printer C worked for 13 days, with the result that the job was completed at the end of the 13 days. How many days would it take printer C to do the job on its own?
Let the job = 48 pages.
Since A and B working together can do the job in 12 days, the combined rate for A and B = w/t = 48/12 = 4 pages per day.
Since B and C working together can do the job in 16 days, the combined rate for B and C = w/t = 48/16 = 3 pages per day.
Printer A worked for 5 days and printer B worked for 7 days:
Since A and B each worked for at least 5 days, the amount of work produced by A and B together over these 5 days = r*t = 4*5 = 20 pages.
Since B worked for 2 additional days, the amount of work produced by B and C together over these 2 days = r*t = 3*2 = 6 pages.
Remaining work = 48 - 20 - 6 = 22 pages.
Printer C worked for 13 days:
The work produced by C over 2 of these 13 days has already been counted, leaving 11 days for C to work alone.
Since C prints the remaining 22 pages over these 11 days, C's rate alone = w/t = 22/11 = 2 pages per day.
Thus:
Time for C to do the entire job on its own = w/r = 48/2 = 24 days.
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