A and B can do a piece of work in 45 and 40 days repectively. They began the work together but A left after some days and B finished the remaining work in 23 days. A left after

Options

A. 6 days
B. 9 days
C. 12 days
D. 5 days

Answer

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

### Step 1: Determine the total work
To find the total work, we need to calculate the least common multiple (LCM) of the number of days A and B take to complete the work individually.

- A can complete the work in 45 days.
- B can complete the work in 40 days.

**LCM of 45 and 40:**
- Prime factorization of 45: \(3^2 \times 5^1\)
- Prime factorization of 40: \(2^3 \times 5^1\)

The LCM is found by taking the highest power of each prime:
- LCM = \(2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 360\)

**Total work = 360 units.**

### Step 2: Calculate the work done by A and B in one day
Now, we calculate how much work A and B can do in one day.

- Work done by A in one day = \(\frac{360}{45} = 8\) units
- Work done by B in one day = \(\frac{360}{40} = 9\) units

### Step 3: Calculate the total work done by B in 23 days
Since B worked alone for 23 days after A left, we calculate the work done by B in that time.

- Work done by B in 23 days = \(9 \times 23 = 207\) units

### Step 4: Calculate the remaining work after B's contribution
Now, we find out how much work was completed before B worked alone.

- Total work = 360 units
- Work done by B = 207 units
- Remaining work = \(360 - 207 = 153\) units

### Step 5: Calculate the combined work done by A and B together
Next, we calculate how much work A and B can do together in one day.

- Combined work done by A and B in one day = \(8 + 9 = 17\) units

### Step 6: Determine how many days A and B worked together
Now, we find out how many days A and B worked together to complete the remaining work of 153 units.

- Days worked together = \(\frac{153}{17} = 9\) days

### Conclusion
A and B worked together for **9 days** before A left the work.

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