P,Q and R enter into partnership. P advances one-fourth of the capital for one-fourth of the time. Qcontributes one-fifth of the capital for half of the time. R contributes the ramaining capital for the whole time, How should they divide a profit of `₹ 2280` ?

Options

A. `350,400,1560`
B. `200, 320, 1760`
C. `400,500,1600`
D. `250,300,1730`

Explanation

To solve the problem, we need to determine how P, Q, and R should divide a profit of ₹2280 based on their contributions in terms of capital and time.

### Step-by-Step Solution:

1. **Identify Contributions**:
- Let the total capital be \( C \).
- P contributes \( \frac{1}{4}C \) for \( \frac{1}{4} \) of the time.
- Q contributes \( \frac{1}{5}C \) for \( \frac{1}{2} \) of the time.
- R contributes the remaining capital for the entire time.

2. **Calculate R's Contribution**:
- The total capital contributed by P and Q is:
\[ P's \, contribution + Q's \, contribution = \frac{1}{4}C + \frac{1}{5}C \]
- To add these fractions, find a common denominator (which is 20):
\[ \frac{1}{4}C = \frac{5}{20}C, \quad \frac{1}{5}C = \frac{4}{20}C \]
- Thus,
\[ P's \, and \, Q's \, total = \frac{5}{20}C + \frac{4}{20}C = \frac{9}{20}C \]
- Therefore, R's contribution is:
\[ R's \, contribution = C - \frac{9}{20}C = \frac{11}{20}C \]

3. **Calculate Effective Capital Contribution**:
- Now we need to calculate the effective capital contribution of each partner considering the time for which they contributed:
- P's effective contribution:
\[ P's \, effective \, contribution = \left(\frac{1}{4}C\right) \times \left(\frac{1}{4}\right) = \frac{1}{16}C \]
- Q's effective contribution:
\[ Q's \, effective \, contribution = \left(\frac{1}{5}C\right) \times \left(\frac{1}{2}\right) = \frac{1}{10}C \]
- R's effective contribution:
\[ R's \, effective \, contribution = \left(\frac{11}{20}C\right) \times 1 = \frac{11}{20}C \]

4. **Convert to a Common Denominator**:
- To compare the effective contributions, convert them to a common denominator (which is 80):
\[ P's \, effective = \frac{1}{16}C = \frac{5}{80}C \]
\[ Q's \, effective = \frac{1}{10}C = \frac{8}{80}C \]
\[ R's \, effective = \frac{11}{20}C = \frac{44}{80}C \]

5. **Calculate the Ratio of Contributions**:
- The ratio of their effective contributions is:
\[ P : Q : R = 5 : 8 : 44 \]

6. **Calculate Total Parts**:
- Total parts = \( 5 + 8 + 44 = 57 \)

7. **Calculate Each Partner's Share of Profit**:
- Total profit = ₹2280
- P's share:
\[ P's \, share = \frac{5}{57} \times 2280 = \frac{11400}{57} = ₹200 \]
- Q's share:
\[ Q's \, share = \frac{8}{57} \times 2280 = \frac{18240}{57} = ₹320 \]
- R's share:
\[ R's \, share = \frac{44}{57} \times 2280 = \frac{100320}{57} = ₹1760 \]

### Final Shares:
- P's share: ₹200
- Q's share: ₹320
- R's share: ₹1760