A and B can complete a piece of work in 45 and 40 days respectively. Both started to work together, but after some days A left and B alone completed the rest work in 23 days. For how many days did A work?

Options

A. 12 days
B. 10 days
C. 8 days
D. 9 days

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the total work
A can complete the work in 45 days, and B can complete it in 40 days. We can find the total work in terms of work units. The least common multiple (LCM) of 45 and 40 will give us the total work.

- LCM of 45 and 40 = 360 units of work.

### Step 2: Calculate the work efficiency of A and B
Next, we calculate the work efficiency of A and B.

- A's efficiency = Total work / Time taken by A = 360 / 45 = 8 units/day.
- B's efficiency = Total work / Time taken by B = 360 / 40 = 9 units/day.

### Step 3: Calculate the work done by B alone
B worked alone for 23 days. We can calculate how much work B completed during this time.

- Work done by B in 23 days = B's efficiency * Number of days = 9 units/day * 23 days = 207 units.

### Step 4: Calculate the remaining work
Now, we need to find out how much work is left after B has completed his part.

- Remaining work = Total work - Work done by B = 360 - 207 = 153 units.

### Step 5: Calculate the combined efficiency of A and B
When A and B work together, their combined efficiency is:

- Combined efficiency = A's efficiency + B's efficiency = 8 units/day + 9 units/day = 17 units/day.

### Step 6: Calculate the time taken by A and B together to finish the remaining work
Now we can find out how many days A and B worked together to complete the remaining work.

- Time taken to complete remaining work = Remaining work / Combined efficiency = 153 units / 17 units/day = 9 days.

### Step 7: Conclusion
Since A and B worked together for 9 days, we conclude that A worked for 9 days before leaving the work to B.

### Final Answer:
A worked for **9 days**.

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