A and B alone can complete a piece of work in 9 days and 12 days, respectively. In how many days will the work be completed if they work on alternate days starting with A?

Explanation

Given: A can complete the work alone in 9 days. B can complete the work alone in 12 days. Step 1: Calculate the work done by A and B per day The amount of work A can do in one day = 1/9 The amount of work B can do in one day = 1/12 Step 2: Calculate the work done in two days (one cycle of A and B working on alternate days) If A works on the first day and B works on the second day, then: Work done in two days = (1/9) + (1/12) Finding a common denominator (36) to add these fractions: Work done in two days = (4/36) + (3/36) = 7/36 Step 3: Calculate how many such cycles are needed to complete the work Since the total work is 1, let’s divide 1 by the work done in each two-day cycle: Number of cycles required = 1 / (7/36) = 36/7 ≈ 5.14 Step 4: Calculate the exact number of days This means 5 complete two-day cycles (10 days) will be nearly enough, but not quite complete the work. After 5 cycles, the work done = 5 × (7/36) = 35/36 Remaining work = 1 - 35/36 = 1/36 Step 5: Complete the remaining work On the next day (10 + 1/3 days), A will work and complete exactly 1/9 of the work. Since 1/9 is enough to cover the remaining 1/36 of the work, the work is completed within that day. Answer: The work will be completed in 10 1/3 days.