A and B undertake to complete a piece of work for Rupees
1200. A can do it in 8 days, B can do it in 12 days and with
the help of C they complete the work in 4 days. Find the
share of C ?

Options

A. 100
B. 200
C. 300
D. 400

Explanation

To solve the problem, we need to determine the share of C in the total payment of Rupees 1200 for the work done by A, B, and C together. Here’s a step-by-step breakdown of the solution:

### Step 1: Determine the work done by A and B individually.
- A can complete the work in 8 days, so A's work in one day is:
\[ \text{Work done by A in 1 day} = \frac{1}{8} \]
- B can complete the work in 12 days, so B's work in one day is:
\[ \text{Work done by B in 1 day} = \frac{1}{12} \]

### Step 2: Determine the combined work done by A and B in one day.
- The combined work done by A and B in one day is:
\[ \text{Combined work of A and B} = \frac{1}{8} + \frac{1}{12} \]
- To add these fractions, we need a common denominator. The least common multiple of 8 and 12 is 24.
\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24} \]
\[ \text{Combined work of A and B} = \frac{3}{24} + \frac{2}{24} = \frac{5}{24} \]

### Step 3: Determine the total work done by A, B, and C together in one day.
- Together, A, B, and C complete the work in 4 days, so their combined work in one day is:
\[ \text{Combined work of A, B, and C} = \frac{1}{4} \]

### Step 4: Calculate the work done by C in one day.
- We know the combined work of A, B, and C is:
\[ \frac{1}{4} = \text{Work done by A and B} + \text{Work done by C} \]
- Substituting the work done by A and B:
\[ \frac{1}{4} = \frac{5}{24} + \text{Work done by C} \]
- To find the work done by C, we rearrange the equation:
\[ \text{Work done by C} = \frac{1}{4} - \frac{5}{24} \]
- Converting \(\frac{1}{4}\) to a fraction with a denominator of 24:
\[ \frac{1}{4} = \frac{6}{24} \]
- Now, substituting back:
\[ \text{Work done by C} = \frac{6}{24} - \frac{5}{24} = \frac{1}{24} \]

### Step 5: Determine the time taken by C to complete the work alone.
- Since C does \(\frac{1}{24}\) of the work in one day, C can complete the entire work in:
\[ \text{Time taken by C} = 24 \text{ days} \]

### Step 6: Calculate the share of C.
- The efficiencies of A, B, and C are:
- A: \(\frac{1}{8}\)
- B: \(\frac{1}{12}\)
- C: \(\frac{1}{24}\)

- To find the ratio of their work, we convert these to a common denominator (which is 24):
\[ \text{Efficiency of A} = \frac{3}{24}, \quad \text{Efficiency of B} = \frac{2}{24}, \quad \text{Efficiency of C} = \frac{1}{24} \]
- Thus, the ratio of their efficiencies is:
\[ A : B : C = 3 : 2 : 1 \]

### Step 7: Calculate the total parts and C's share.
- Total parts = \(3 + 2 + 1 = 6\)
- C's share of the total payment of Rupees 1200 is:
\[ \text{C's share} = \frac{1}{6} \times 1200 = 200 \]

### Final Answer:
C's share of the payment is **Rupees 200**.

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