A, B and C working alone can finish a work in 80 days, 50 days and 90 days respectively. A worked alone for 21 days, then B took over from A. B worked on it alone for 22 days and then C took over from B. In how many days will C finish the remaining work? (in days)

Options

A. A)`31(31)/(40)`
B. B)`26(31)/(40)`
C. C)`30(27)/(40)`
D. D)`28(31)/(40)`

Answer

To solve the problem step by step, we will first determine the work done by A, B, and C individually, then calculate the total work done after A and B have worked, and finally find out how long it will take C to finish the remaining work.

### Step 1: Determine the work efficiency of A, B, and C
- A can finish the work in 80 days, so A's work efficiency = 1/80 of the work per day.
- B can finish the work in 50 days, so B's work efficiency = 1/50 of the work per day.
- C can finish the work in 90 days, so C's work efficiency = 1/90 of the work per day.

### Step 2: Find the least common multiple (LCM) of the days
To find a common measure of work, we calculate the LCM of 80, 50, and 90.
- LCM(80, 50, 90) = 3600 units of work.

### Step 3: Calculate the work done by each worker per day
- A's efficiency = 3600 / 80 = 45 units/day
- B's efficiency = 3600 / 50 = 72 units/day
- C's efficiency = 3600 / 90 = 40 units/day

### Step 4: Calculate the work done by A in 21 days
- Work done by A in 21 days = 45 units/day * 21 days = 945 units.

### Step 5: Calculate the work done by B in 22 days
- Work done by B in 22 days = 72 units/day * 22 days = 1584 units.

### Step 6: Calculate the total work done by A and B
- Total work done by A and B = Work done by A + Work done by B = 945 + 1584 = 2529 units.

### Step 7: Calculate the remaining work
- Remaining work = Total work - Work done by A and B = 3600 - 2529 = 1071 units.

### Step 8: Calculate the time taken by C to finish the remaining work
- Time taken by C to finish the remaining work = Remaining work / C's efficiency = 1071 units / 40 units/day.

### Step 9: Simplify the time taken by C
- Time taken by C = 1071 / 40 = 26.775 days, which can be expressed as 26 whole days and 31/40 of a day.

### Final Answer
C will finish the remaining work in **26 whole days and 31/40 of a day**.

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