A and B can do a piece of work in 6 days and 8 days, respectively. With the help of C, they completed the work in 3 days and earned 1,848. What was the share of C ?

Options

A. 231
B. 924
C. 462
D. 693

Answer

To solve the problem step by step, we will determine the work done by A, B, and C, and then calculate C's share of the earnings.

### Step 1: Determine the work done by A and B in one day
- A can complete the work in 6 days. Therefore, the work done by A in one day is:
\[ \text{Work done by A in one day} = \frac{1}{6} \text{ of the work} \]

- B can complete the work in 8 days. Therefore, the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{1}{8} \text{ of the work} \]

### 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:
\[ \text{Total work done by A and B in one day} = \frac{1}{6} + \frac{1}{8} \]

- To add these fractions, we find a common denominator, which is 24:
\[ \frac{1}{6} = \frac{4}{24}, \quad \frac{1}{8} = \frac{3}{24} \]
\[ \text{Total work done by A and B in one day} = \frac{4}{24} + \frac{3}{24} = \frac{7}{24} \]

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

### Step 4: Calculate the work done by C in one day
- Let the work done by C in one day be \( C \). We can set up the equation:
\[ \frac{7}{24} + C = \frac{1}{3} \]

- To solve for \( C \), we first convert \( \frac{1}{3} \) to have a denominator of 24:
\[ \frac{1}{3} = \frac{8}{24} \]
\[ \frac{7}{24} + C = \frac{8}{24} \]
\[ C = \frac{8}{24} - \frac{7}{24} = \frac{1}{24} \]

### Step 5: Calculate the ratio of work done by A, B, and C
- The work done by A, B, and C in one day is:
- A: \( \frac{4}{24} \)
- B: \( \frac{3}{24} \)
- C: \( \frac{1}{24} \)

- The ratio of their work is:
\[ A : B : C = 4 : 3 : 1 \]

### Step 6: Calculate C's share of the earnings
- The total earnings are 1,848. The total parts of the ratio are:
\[ 4 + 3 + 1 = 8 \]

- C's share of the earnings is calculated as follows:
\[ \text{C's share} = \frac{1}{8} \times 1848 = \frac{1848}{8} = 231 \]

### Final Answer
C's share of the earnings is **231 rupees**.

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