A can do a piece of work in 12 days and B can do same work in 18 days. Both A and B start the work and after some days A left and B complete the remaining work in 8 days. After how many days will A left work?

Answer

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

### Step 1: Determine the work done by A and B in one day.
- A can complete the work in 12 days, so A's work per day = 1/12 of the work.
- B can complete the work in 18 days, so B's work per day = 1/18 of the work.

### Step 2: Find the total work in terms of units.
- To find a common unit of work, we can take the LCM of 12 and 18.
- LCM(12, 18) = 36 units of work.

### Step 3: Calculate the efficiency of A and B in terms of units.
- A's efficiency = 36 units / 12 days = 3 units per day.
- B's efficiency = 36 units / 18 days = 2 units per day.

### Step 4: Set up the equation for the total work done.
- Let X be the number of days A and B worked together before A left.
- In X days, the work done by A and B together = (A's efficiency + B's efficiency) * X = (3 + 2) * X = 5X units.
- After A left, B completed the remaining work in 8 days. The work done by B in 8 days = B's efficiency * 8 = 2 * 8 = 16 units.

### Step 5: Write the equation for total work.
- Total work = Work done by A and B together + Work done by B alone.
- Therefore, we have: 36 = 5X + 16.

### Step 6: Solve for X.
- Rearranging the equation: 5X = 36 - 16.
- This simplifies to: 5X = 20.
- Dividing both sides by 5 gives: X = 4.

### Conclusion:
- A left the work after 4 days.