P and Q started a business by investing `₹ 15,000` and `₹18,000` respectively. After four months R joined the business with a capital of `₹10,000`. After two more months Q left the business with his capital. At the end of the year P got a share of `₹4,500` in the profit. What is the total profit earned ?

Options

A. `₹ 6800`
B. `₹7600`
C. `₹8600`
D. `₹9200`

Explanation

To solve the problem step by step, we will calculate the share of profit for each partner based on their investments and the duration for which they invested.

### Step 1: Calculate the investment duration for each partner.
- **P's investment**: ₹15,000 for 12 months.
- **Q's investment**: ₹18,000 for 6 months (since he left after 6 months).
- **R's investment**: ₹10,000 for 8 months (joined after 4 months).

### Step 2: Calculate the capital contribution of each partner in terms of "capital months."
- **P's contribution**:
\[ 15,000 \times 12 = 180,000 \text{ capital months} \]
- **Q's contribution**:
\[ 18,000 \times 6 = 108,000 \text{ capital months} \]
- **R's contribution**:
\[ 10,000 \times 8 = 80,000 \text{ capital months} \]

### Step 3: Calculate the total capital months.
\[ \text{Total capital months} = 180,000 + 108,000 + 80,000 = 368,000 \]

### Step 4: Determine the ratio of their contributions.
- The ratio of contributions is:
\[ P : Q : R = 180,000 : 108,000 : 80,000 \]
Simplifying this ratio:
\[ P : Q : R = 180 : 108 : 80 \]
Dividing each term by 36:
\[ P : Q : R = 5 : 3 : \frac{80}{36} \approx 5 : 3 : 2.22 \text{ (approximately)} \]
To make it whole numbers, we can multiply by 9:
\[ P : Q : R = 45 : 27 : 20 \]

### Step 5: Calculate the total profit based on P's share.
Given that P's share of the profit is ₹4,500, we can set up the following proportion:
\[ \frac{P's \text{ share}}{Total \text{ profit}} = \frac{45}{Total \text{ parts}} \]
Where total parts = \(45 + 27 + 20 = 92\).

### Step 6: Calculate the total profit.
Let \(X\) be the total profit.
\[ \frac{4500}{X} = \frac{45}{92} \]
Cross-multiplying gives:
\[ 4500 \times 92 = 45X \]
\[ 414000 = 45X \]
\[ X = \frac{414000}{45} = 9200 \]

### Final Answer:
The total profit earned is **₹9,200**.

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