A can do a piece of work in 10 days, while B alone can do it in 15 days. They work together for 5 days and the rest of the work is done by C in 2 days. If they get Rs. 450 for the whole work, how should they divide the money ?

Options

A. Rs. 225, Rs. 150, Rs. 75
B. Rs. 250, Rs. 100, Rs. 100
C. Rs. 200, Rs. 150, Rs. 100
D. Rs. 175, Rs. 175, Rs. 100

Answer

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

### Step 1: Determine the work done by A and B
- A can complete the work in 10 days, so A's work rate is \( \frac{1}{10} \) of the work per day.
- B can complete the work in 15 days, so B's work rate is \( \frac{1}{15} \) of the work per day.

### Step 2: Calculate the combined work rate of A and B
- The combined work rate of A and B is:
\[ \text{Combined rate} = \frac{1}{10} + \frac{1}{15} \]
- To add these fractions, we find the least common multiple (LCM) of 10 and 15, which is 30:
\[ \frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30} \]
- Thus, the combined rate is:
\[ \text{Combined rate} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \]
- This means A and B together can complete \( \frac{1}{6} \) of the work in one day.

### Step 3: Calculate the work done by A and B in 5 days
- In 5 days, A and B together will complete:
\[ \text{Work done in 5 days} = 5 \times \frac{1}{6} = \frac{5}{6} \]

### Step 4: Determine the remaining work
- The total work is considered as 1 (or 30 units). Therefore, the remaining work after A and B have worked for 5 days is:
\[ \text{Remaining work} = 1 - \frac{5}{6} = \frac{1}{6} \]

### Step 5: Determine the work done by C
- C completes the remaining \( \frac{1}{6} \) of the work in 2 days. Therefore, C's work rate is:
\[ \text{C's rate} = \frac{\frac{1}{6}}{2} = \frac{1}{12} \]

### Step 6: Calculate the total payment per unit of work
- The total payment for the work is Rs. 450. Since the total work is 1 unit, the payment per unit of work is:
\[ \text{Payment per unit} = \frac{450}{1} = 450 \text{ Rs.} \]

### Step 7: Calculate the payment for A, B, and C based on the work done
- **Work done by A**:
- A's work in 5 days = \( 5 \times \frac{1}{10} = \frac{1}{2} \)
- Payment for A = \( \frac{1}{2} \times 450 = 225 \text{ Rs.} \)

- **Work done by B**:
- B's work in 5 days = \( 5 \times \frac{1}{15} = \frac{1}{3} \)
- Payment for B = \( \frac{1}{3} \times 450 = 150 \text{ Rs.} \)

- **Work done by C**:
- C's work = \( \frac{1}{6} \)
- Payment for C = \( \frac{1}{6} \times 450 = 75 \text{ Rs.} \)

### Step 8: Final payment distribution
- A receives Rs. 225, B receives Rs. 150, and C receives Rs. 75.

### Summary of Payments
- A: Rs. 225
- B: Rs. 150
- C: Rs. 75