The labourers A, B, C were given a contract of Rs.750 for doing a certain piece of work. All the three together can finish the work in 8 days. A and C together can do it in 12 day, while A and B together can do it in `13(1)/(3)` days. The money will be divided in the ratio

Options

A. `4:5:6 `
B. `4:7:5 `
C. `5:7:4 `
D. `5:6:8`

Answer

To solve the problem step by step, we will determine the efficiencies of laborers A, B, and C based on the given information and then find the ratio in which the money will be divided.

### Step 1: Determine the Total Work
The total work can be calculated using the least common multiple (LCM) of the days taken by different combinations of workers.

- A, B, and C together can finish the work in 8 days.
- A and C together can finish the work in 12 days.
- A and B together can finish the work in \(13 \frac{1}{3}\) days, which is equivalent to \(\frac{40}{3}\) days.

To find the LCM of 8, 12, and \(\frac{40}{3}\):
- Convert \(\frac{40}{3}\) to a whole number by multiplying by 3: \(40\).
- The LCM of 8, 12, and 40 is 120.

Thus, the total work is **120 units**.

### Step 2: Calculate the Efficiency of A, B, and C Together
The efficiency of A, B, and C together can be calculated as follows:
\[ \text{Efficiency of A + B + C} = \frac{\text{Total Work}}{\text{Time}} = \frac{120}{8} = 15 \text{ units per day} \]

### Step 3: Calculate the Efficiency of A and C Together
The efficiency of A and C together can be calculated as follows:
\[ \text{Efficiency of A + C} = \frac{\text{Total Work}}{\text{Time}} = \frac{120}{12} = 10 \text{ units per day} \]

### Step 4: Calculate the Efficiency of A and B Together
The efficiency of A and B together can be calculated as follows:
\[ \text{Efficiency of A + B} = \frac{\text{Total Work}}{\text{Time}} = \frac{120}{\frac{40}{3}} = 120 \times \frac{3}{40} = 9 \text{ units per day} \]

### Step 5: Calculate the Individual Efficiencies
Now we can find the individual efficiencies of A, B, and C.

1. **Efficiency of B**:
\[ \text{Efficiency of B} = \text{Efficiency of A + B + C} - \text{Efficiency of A + C} = 15 - 10 = 5 \text{ units per day} \]

2. **Efficiency of C**:
\[ \text{Efficiency of C} = \text{Efficiency of A + B + C} - \text{Efficiency of A + B} = 15 - 9 = 6 \text{ units per day} \]

3. **Efficiency of A**:
\[ \text{Efficiency of A} = \text{Efficiency of A + B + C} - \text{Efficiency of B} - \text{Efficiency of C} = 15 - 5 - 6 = 4 \text{ units per day} \]

### Step 6: Determine the Ratio of Efficiencies
The efficiencies of A, B, and C are:
- Efficiency of A = 4 units per day
- Efficiency of B = 5 units per day
- Efficiency of C = 6 units per day

Thus, the ratio of their efficiencies is:
\[ \text{Ratio of A : B : C} = 4 : 5 : 6 \]

### Step 7: Conclusion
The total amount of Rs. 750 will be divided among A, B, and C in the ratio of their efficiencies, which is **4 : 5 : 6**.

---