Raju and Raman can complete a piece of work in 12 days and 16 days, respectively. If they work on alternate days, starting with Raju, in how many days will the work be completed?

Explanation

Formula used: Raju’s work rate = 1/12 Raman’s work rate = 1/16 Combined work rate over 2 days = (1/12) + (1/16) Solution: Combined work rate over 2 days = (1/12) + (1/16) = (4/48) + (3/48) = 7/48 Work completed in 2 days = 7/48 Total work = 1 (complete work) Calculating the number of 2-day cycles needed to finish the work: = 1 / (7/48) = 48/7 = 6 cycles (6 full 2-day cycles) Work completed in 6 full cycles (12 days): = 6 × (7/48) = 42/48 = 7/8 of the total work Remaining work = 1 - 7/8 = 1/8 On the next day, Raju works (13th day): Raju’s one-day work rate = 1/12 To finish the remaining 1/8 of the work: Let x be the fraction of the day Raju needs to work: = (1/12) × x = 1/8 x = (1/8) × 12 x = 12/8 x = 3/2 Since Raju only works for 1 full day, he completes his share in that time, leaving: = 1/8 - 1/12 = (3/24) - (2/24) = 1/24 Raman completes the remaining 1/24 of the work in: = (1/24) / (1/16) = 16/24 = 2/3 of a day Total time taken: = 12 days (for 6 cycles) + 1 day (Raju’s 13th day) + 2/3 day (Raman’s work) = 13 + 2/3 Final Answer: 132⁄3 days