A alone can finish a work in 10 days and B alone can do it in 15 days. If they work together and finish it, then out of a total wages of Rs. 75, A will get….

Options

A. Rs. 30
B. Rs. 37.50
C. Rs. 45
D. Rs. 50

Answer

To solve the problem, we need to determine how much A will earn when he and B work together to complete a task, given their individual work rates and the total wages.

### Step-by-Step Solution:

1. **Determine the Work Rates of A and B:**
- A can complete the work in 10 days. Therefore, A's work rate is:
\[ \text{A's work rate} = \frac{1 \text{ work}}{10 \text{ days}} = \frac{1}{10} \text{ work/day} \]
- B can complete the work in 15 days. Therefore, B's work rate is:
\[ \text{B's work rate} = \frac{1 \text{ work}}{15 \text{ days}} = \frac{1}{15} \text{ work/day} \]

2. **Calculate the Total Work Rate When A and B Work Together:**
- To find their combined work rate, we add their individual work rates:
\[ \text{Combined work rate} = \text{A's work rate} + \text{B's work rate} = \frac{1}{10} + \frac{1}{15} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 10 and 15 is 30. Therefore:
\[ \frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30} \]
- Now, adding these gives:
\[ \text{Combined work rate} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \text{ work/day} \]

3. **Calculate the Total Work Done:**
- Since they can complete the work together at a rate of \(\frac{1}{6}\) work per day, they will finish the work in:
\[ \text{Total time to complete the work} = \frac{1 \text{ work}}{\frac{1}{6} \text{ work/day}} = 6 \text{ days} \]

4. **Determine the Share of Wages Based on Work Contribution:**
- In 6 days, A's total work contribution is:
\[ \text{A's total work} = 6 \text{ days} \times \frac{1}{10} \text{ work/day} = \frac{6}{10} = \frac{3}{5} \text{ of the work} \]
- In 6 days, B's total work contribution is:
\[ \text{B's total work} = 6 \text{ days} \times \frac{1}{15} \text{ work/day} = \frac{6}{15} = \frac{2}{5} \text{ of the work} \]

5. **Calculate the Total Efficiency:**
- A's efficiency is 3 (from the work done in 10 days), and B's efficiency is 2 (from the work done in 15 days). Therefore, the total efficiency is:
\[ \text{Total efficiency} = 3 + 2 = 5 \]

6. **Calculate A's Share of the Wages:**
- The total wages are Rs. 75. A's share of the wages is calculated as follows:
\[ \text{A's share} = \left(\frac{\text{A's efficiency}}{\text{Total efficiency}}\right) \times \text{Total wages} = \left(\frac{3}{5}\right) \times 75 \]
- Simplifying this gives:
\[ \text{A's share} = \frac{3 \times 75}{5} = \frac{225}{5} = 45 \]

### Final Answer:
A will get Rs. 45 out of the total wages of Rs. 75.

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