A alone can do a piece of work in 6 days and B alone can do it in 8 days. A and B undertook to do it for Rs. 320 with the help of C, they finished it in 3 days. How much is paid to C ?

Options

A. Rs. 37.50
B. Rs. 40
C. Rs. 60
D. Rs. 80

Answer

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

### Step 1: Determine the work done by A and B
- A can complete the work in 6 days, so A's work rate is:
\[ \text{Work rate of A} = \frac{1}{6} \text{ (work per day)} \]

- B can complete the work in 8 days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1}{8} \text{ (work per day)} \]

### Step 2: Calculate the combined work rate of A and B
- The combined work rate of A and B is:
\[ \text{Combined work rate of A and B} = \frac{1}{6} + \frac{1}{8} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 6 and 8 is 24.
- Converting the fractions:
\[ \frac{1}{6} = \frac{4}{24}, \quad \frac{1}{8} = \frac{3}{24} \]
- Therefore:
\[ \text{Combined work rate of A and B} = \frac{4}{24} + \frac{3}{24} = \frac{7}{24} \]

### Step 3: Calculate the total work done by A, B, and C in 3 days
- A, B, and C together complete the work in 3 days, which means their combined work rate is:
\[ \text{Combined work rate of A, B, and C} = \frac{1}{3} \text{ (work per day)} \]

### Step 4: Determine the work rate of C
- We know that:
\[ \text{Work rate of A, B, and C} = \text{Work rate of A and B} + \text{Work rate of C} \]
- Thus:
\[ \frac{1}{3} = \frac{7}{24} + \text{Work rate of C} \]
- To find the work rate of C, we rearrange the equation:
\[ \text{Work rate of C} = \frac{1}{3} - \frac{7}{24} \]

### Step 5: Calculate the common denominator and solve for C's work rate
- The common denominator for 3 and 24 is 24:
\[ \frac{1}{3} = \frac{8}{24} \]
- Therefore:
\[ \text{Work rate of C} = \frac{8}{24} - \frac{7}{24} = \frac{1}{24} \]

### Step 6: Calculate the total work done by A, B, and C
- The total work done by A, B, and C in 3 days is:
\[ \text{Total work} = 3 \times \left(\frac{1}{3}\right) = 1 \text{ (complete work)} \]

### Step 7: Calculate the payment distribution
- The total payment for the work is Rs. 320.
- The total efficiency (work rates) of A, B, and C combined is:
\[ \text{Total efficiency} = \frac{1}{6} + \frac{1}{8} + \frac{1}{24} = \frac{4}{24} + \frac{3}{24} + \frac{1}{24} = \frac{8}{24} = \frac{1}{3} \]
- The share of C in the total efficiency is:
\[ \text{Share of C} = \frac{\text{Work rate of C}}{\text{Total efficiency}} = \frac{\frac{1}{24}}{\frac{1}{3}} = \frac{1}{24} \times \frac{3}{1} = \frac{3}{24} = \frac{1}{8} \]

### Step 8: Calculate C's payment
- C's payment is:
\[ \text{Payment to C} = \text{Total payment} \times \text{Share of C} = 320 \times \frac{1}{8} = 40 \]

### Final Answer
C is paid Rs. 40.

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