Two worker's A and B working together completed a job in (\frac{15}{2}) days. If A worked twice as efficiently as he actually did and B worked thrice as efficiently as he actually did, the work would have been completed in (\frac{10}{3}) days. how many days B require[e to complete the job?

Let A and B's actual work rates be 'a' and 'b' jobs per day respectively. Working together: a + b = 1/(15/2) = 2/15. With increased efficiency: 2a + 3b = 1/(10/3) = 3/10. From the first equation: 2a + 2b = 4/15. Subtracting from the second: (2a + 3b) - (2a + 2b) = 3/10 - 4/15, giving b = 9/30 - 8/30 = 1/30. So B alone would take 1/(1/30) = 30 days. Option D is correct. Options A, B, and C don't satisfy the given conditions.