Anuj and Bishnu can complete a task together in 24 days, Bishnu and Raj can complete the same task together in 48 days, and Anuj and Raj can finish it in 36 days. If Anuj works alone for 16 days and Bishnu works alone for 12 days, how long will it take Raj to finish the remaining task by himself?

Let the total work be = 144 units {LCM (24, 36 and 48) } Let the efficiency of 'Anuj', 'Bishnu' and 'Raj' be 'x' units/day, 'y' units/day and 'z' units/day, respectively. ATQ; (x + y) = 144 ÷ 24 Or, x + y = 6 ........ (I) (y + z) = 144 ÷ 48 Or, y + z = 3 ......... (II) And (z + x) = 144 ÷ 36 Or, x + z = 4 ............ (III) On adding equation (I) , (II) and (III) , we have; (x + y + y + z + x + z) = 6 + 3 + 4 Or, 2 × (x + y + z) = 13 So, x + y + z = 6.5 .......... (IV) On subtracting equation (I) from equation (IV) , we have; (x + y + z) - (x + y) = 6.5 - 6 Or, z = 0.5 Similarly, x = 3.5 and y = 2.5 So, work done by 'Anuj' in 8 days = (3.5 X 16) = 56 units And work done by 'Bishnu' in 12 days = (2.5 × 12) = 30 units So, work remaining = 144 - (56 + 30) = 58 units So, time taken by 'Raj' to finish the remaining work alone = 58 ÷ 0.5 = 116 days