P and Q together can finish a task in 72 days while Q and R can do it in 48 days. If P worked for 18 days, Q worked for 30 days, and then R finished the remaining task in 54 days, in how many days can R complete the whole task working alone?

Let the total work = 288 units (LCM of 72 and 48) Work done by P and Q together in one day = 288/72 = 4 units Work done by Q and R together in one day = 288/48 = 6 units Work done by P and Q together in 18 days = 18 × 4 = 72 units ---- (I) Work done by Q and R together in 12 days = 12 × 6 = 72 units ---- (II) Q’s 18 days work and R’s 12 days work have been included in (I) and (II), respectively. Work done by R alone in 42 days = 288 – 72 – 72 = 144 units Work done by R in one day = 144/42 = 24/7 units Time taken by R alone to complete the full work = 288 ÷ (24/7) = 84 days