P completes 2/3 of a certain job in 6 days. Q can complete 1/3 of the same job in 8 days and R can complete 3/4 of the work in 12 days. All of them work together for 4 days and then P and R quit. How long will it take for Q to complete the remaining work alone?

P's rate: (2/3)/6 = 1/9 per day. Q's rate: (1/3)/8 = 1/24 per day. R's rate: (3/4)/12 = 1/16 per day. Combined work in 4 days: 4(1/9 + 1/24 + 1/16) = 4(1/9 + 7/48) = 4(55/144) = 55/36. Remaining work: 1 - 55/36 = -19/36 (this is negative, indicating an error in my calculation). Let me recalculate: 1/9 = 16/144, 1/24 = 6/144, 1/16 = 9/144. Combined = 31/144 per day. In 4 days: 124/144 = 31/36 done. Remaining: 5/36. Q's time for 5/36 = (5/36)/(1/24) = 5/36 × 24 = 10/3 = 3.33 days.