A can do a work in 5 days while B can do the same work in 8 days. They worked together to complete the work and earned Rs. 6760. Find A's share ?

Options

A. Rs. 5070
B. Rs. 3600
C. Rs. 4160
D. Rs. 4056

Answer

To solve the problem, we need to determine A's share of the total earnings based on the amount of work done by A and B together.

### Step-by-Step Solution:

1. **Determine Work Rates**:
- A can complete the work in 5 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1 \text{ work}}{5 \text{ days}} = \frac{1}{5} \text{ work per day} \]
- B can complete the work in 8 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1 \text{ work}}{8 \text{ days}} = \frac{1}{8} \text{ work per day} \]

2. **Calculate Combined Work Rate**:
- When A and B work together, their combined work rate is:
\[ \text{Combined work rate} = \frac{1}{5} + \frac{1}{8} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 5 and 8 is 40.
- Convert the fractions:
\[ \frac{1}{5} = \frac{8}{40}, \quad \frac{1}{8} = \frac{5}{40} \]
- Now add them:
\[ \text{Combined work rate} = \frac{8}{40} + \frac{5}{40} = \frac{13}{40} \text{ work per day} \]

3. **Calculate Total Work**:
- Since A and B together can complete the work at a rate of \(\frac{13}{40}\) work per day, the total work can be considered as 1 unit of work.
- The time taken to complete the work together is:
\[ \text{Time} = \frac{1 \text{ work}}{\frac{13}{40} \text{ work per day}} = \frac{40}{13} \text{ days} \]

4. **Calculate Individual Contributions**:
- In \(\frac{40}{13}\) days, the amount of work done by A is:
\[ \text{Work done by A} = \text{Work rate of A} \times \text{Time} = \frac{1}{5} \times \frac{40}{13} = \frac{8}{13} \text{ of the work} \]
- The amount of work done by B is:
\[ \text{Work done by B} = \text{Work rate of B} \times \text{Time} = \frac{1}{8} \times \frac{40}{13} = \frac{5}{13} \text{ of the work} \]

5. **Calculate A's Share of Earnings**:
- The total earnings are Rs. 6760. A's share of the earnings is proportional to the work done by A:
\[ \text{A's share} = \text{Total Earnings} \times \text{Fraction of work done by A} \]
\[ \text{A's share} = 6760 \times \frac{8}{13} = 4160 \]

### Final Answer:
A's share of the earnings is Rs. 4160.