A and B undertaken to do a piece of work for Rs 1200. A alone can do it in 8 days, while B can do it in 6 days. With the help of C, they complete it in 3 days. Find Cs share.

Options

A. `Rs.450`
B. Rs. 300
C. Rs. 150
D. Rs. 100

Answer

To solve the problem step by step, we will calculate the efficiencies of A, B, and C, and then determine C's share of the total payment of Rs. 1200.

### Step 1: Calculate the work done by A and B
- A can complete the work in 8 days. Therefore, A's work per day = 1/8 of the work.
- B can complete the work in 6 days. Therefore, B's work per day = 1/6 of the work.

### Step 2: Calculate the total work done by A and B in one day
- To find the total work done by A and B together in one day, we add their individual work rates:
\[ \text{Total work per day by A and B} = \frac{1}{8} + \frac{1}{6} \]
- To add these fractions, we need a common denominator. The least common multiple of 8 and 6 is 24.
\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{6} = \frac{4}{24} \]
- Therefore,
\[ \text{Total work per day by A and B} = \frac{3}{24} + \frac{4}{24} = \frac{7}{24} \]

### Step 3: Calculate the total work done by A, B, and C together in one day
- They complete the work together in 3 days, so their combined work rate is:
\[ \text{Total work per day by A, B, and C} = \frac{1}{3} \]

### Step 4: Calculate C's work rate
- We know that the total work done by A, B, and C together in one day is:
\[ \frac{1}{3} = \text{Total work per day by A and B} + \text{C's work per day} \]
- Substituting the value we found for A and B:
\[ \frac{1}{3} = \frac{7}{24} + \text{C's work per day} \]
- To isolate C's work rate, we convert \(\frac{1}{3}\) to a fraction with a denominator of 24:
\[ \frac{1}{3} = \frac{8}{24} \]
- Now, we can solve for C's work rate:
\[ \frac{8}{24} = \frac{7}{24} + \text{C's work per day} \]
\[ \text{C's work per day} = \frac{8}{24} - \frac{7}{24} = \frac{1}{24} \]

### Step 5: Calculate the total efficiency
- The total efficiency of A, B, and C is:
\[ \text{Total efficiency} = \text{A's efficiency} + \text{B's efficiency} + \text{C's efficiency} = 3 + 4 + 1 = 8 \]

### Step 6: Calculate C's share of the payment
- The total payment for the work is Rs. 1200. C's share is based on his efficiency relative to the total efficiency:
\[ \text{C's share} = \left(\frac{\text{C's efficiency}}{\text{Total efficiency}}\right) \times \text{Total payment} \]
\[ \text{C's share} = \left(\frac{1}{8}\right) \times 1200 = 150 \]

### Final Answer
C's share of the payment is Rs. 150.

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