A can do a piece of work in 120 days and B can do it in 150 days. They work together for 20 days. Then B leaves and A alone continues the work. 12 days after that C joins A and the work is completed in 48 days more. In how many days can C do it if he works alone?

Options

A. 230 days
B. 225 days
C. 240 days
D. 220 days

Answer

To solve the problem step by step, we will first calculate the work done by A and B together, then find out how much work is left for A and C to complete.

### Step 1: Calculate the work done by A and B together.

1. **Find the work done by A in one day:**
- A can complete the work in 120 days.
- Work done by A in one day = \( \frac{1}{120} \) of the work.

2. **Find the work done by B in one day:**
- B can complete the work in 150 days.
- Work done by B in one day = \( \frac{1}{150} \) of the work.

3. **Find the combined work done by A and B in one day:**
- Combined work per day = \( \frac{1}{120} + \frac{1}{150} \)
- To add these fractions, we need a common denominator. The LCM of 120 and 150 is 600.
- Convert the fractions:
- \( \frac{1}{120} = \frac{5}{600} \)
- \( \frac{1}{150} = \frac{4}{600} \)
- Combined work per day = \( \frac{5}{600} + \frac{4}{600} = \frac{9}{600} = \frac{3}{200} \)

4. **Calculate the work done in 20 days:**
- Work done in 20 days = \( 20 \times \frac{3}{200} = \frac{60}{200} = \frac{3}{10} \)

### Step 2: Calculate the remaining work after A and B work together for 20 days.

1. **Total work = 1 (whole work)**
2. **Remaining work = Total work - Work done in 20 days**
- Remaining work = \( 1 - \frac{3}{10} = \frac{7}{10} \)

### Step 3: Calculate the work done by A alone for 12 days.

1. **Work done by A in 12 days:**
- Work done by A in 12 days = \( 12 \times \frac{1}{120} = \frac{12}{120} = \frac{1}{10} \)

2. **Remaining work after A works alone for 12 days:**
- Remaining work = \( \frac{7}{10} - \frac{1}{10} = \frac{6}{10} = \frac{3}{5} \)

### Step 4: Calculate the work done by A and C together for 48 days.

1. **Let C's work rate be \( \frac{1}{C} \) (C can do the work in C days).**
2. **Combined work rate of A and C:**
- Work done by A and C together = \( \frac{1}{120} + \frac{1}{C} \)

3. **Work done in 48 days:**
- Work done in 48 days = \( 48 \left( \frac{1}{120} + \frac{1}{C} \right) = \frac{3}{5} \)

### Step 5: Set up the equation and solve for C.

1. **Set up the equation:**
- \( 48 \left( \frac{1}{120} + \frac{1}{C} \right) = \frac{3}{5} \)

2. **Multiply both sides by 120C to eliminate the fractions:**
- \( 48C + 48 \times 120 = \frac{3 \times 120C}{5} \)
- \( 48C + 5760 = 72C \)

3. **Rearranging gives:**
- \( 5760 = 72C - 48C \)
- \( 5760 = 24C \)
- \( C = \frac{5760}{24} = 240 \)

### Conclusion:
C can complete the work alone in **240 days**.