A Band Cinvest their capital into a partnership business in the fol lowing manner: A invests one-half of the capital for three- fourth of the time, B invests one-third of the capital for one-half of the time and Cinvests the remaining capl tal for the whole time. If the profit earned is 510, how should A get?

Options

A. 260
B. 250
C. 270
D. 280

Answer

To solve the problem, we need to determine how much A should receive from the total profit of 510 based on their respective investments and the time for which they invested.

### Step-by-Step Solution:

1. **Identify the Investments:**
- Let the total capital be denoted as \( C \).
- A invests \( \frac{1}{2}C \) for \( \frac{3}{4} \) of the time.
- B invests \( \frac{1}{3}C \) for \( \frac{1}{2} \) of the time.
- C invests the remaining capital, which is \( C - \left(\frac{1}{2}C + \frac{1}{3}C\right) \).

2. **Calculate C's Investment:**
- First, find a common denominator for the fractions:
- The common denominator of 2 and 3 is 6.
- A's investment: \( \frac{1}{2}C = \frac{3}{6}C \)
- B's investment: \( \frac{1}{3}C = \frac{2}{6}C \)
- Total investment by A and B: \( \frac{3}{6}C + \frac{2}{6}C = \frac{5}{6}C \)
- Therefore, C's investment is:
\[ C - \frac{5}{6}C = \frac{1}{6}C \]

3. **Calculate the Effective Investment:**
- A's effective investment (capital × time):
\[ A's \, investment = \frac{1}{2}C \times \frac{3}{4} = \frac{3}{8}C \]
- B's effective investment:
\[ B's \, investment = \frac{1}{3}C \times \frac{1}{2} = \frac{1}{6}C \]
- C's effective investment:
\[ C's \, investment = \frac{1}{6}C \times 1 = \frac{1}{6}C \]

4. **Convert to a Common Denominator:**
- The common denominator for \( \frac{3}{8}, \frac{1}{6}, \) and \( \frac{1}{6} \) is 24.
- Convert each investment:
- A: \( \frac{3}{8}C = \frac{9}{24}C \)
- B: \( \frac{1}{6}C = \frac{4}{24}C \)
- C: \( \frac{1}{6}C = \frac{4}{24}C \)

5. **Calculate the Total Effective Investment:**
- Total effective investment:
\[ \frac{9}{24}C + \frac{4}{24}C + \frac{4}{24}C = \frac{17}{24}C \]

6. **Determine the Profit Share:**
- The total profit is 510. To find the value of one part:
\[ \text{Total parts} = 9 + 4 + 4 = 17 \]
- Value of one part:
\[ \text{Value of one part} = \frac{510}{17} = 30 \]

7. **Calculate A's Share:**
- A's share of the profit:
\[ A's \, share = 9 \times 30 = 270 \]

### Final Answer:
A should receive **270** from the total profit of 510.