Rushi and Arun are two business partners. They invest their capital in the ratio 7 : 9. After 6 months, a third person Ram joins them with a capital of two third of that of Arun. After a year, profit share of Rushi was Rs. 5600, then find total profit?

Options

A. A) 26,250
B. B) 19,200
C. C) 18,650,
D. D) 15,200

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the Capital Contributions
Let Rushi's capital be \( 7x \) and Arun's capital be \( 9x \).

### Step 2: Calculate Ram's Capital
Ram joins after 6 months with a capital of two-thirds of Arun's capital. Therefore, Ram's capital is:
\[ \text{Ram's capital} = \frac{2}{3} \times 9x = 6x \]

### Step 3: Determine the Time Period of Investment
- Rushi's investment duration: 12 months
- Arun's investment duration: 12 months
- Ram's investment duration: 6 months

### Step 4: Calculate the Effective Capital Contribution
To find the effective capital contribution, we multiply the capital by the time period for which it was invested:
- Rushi's effective capital = \( 7x \times 12 = 84x \)
- Arun's effective capital = \( 9x \times 12 = 108x \)
- Ram's effective capital = \( 6x \times 6 = 36x \)

### Step 5: Calculate the Total Effective Capital
Now, we sum up all the effective capital contributions:
\[ \text{Total effective capital} = 84x + 108x + 36x = 228x \]

### Step 6: Determine the Profit Sharing Ratio
The profit sharing ratio is based on their effective capital contributions:
- Rushi's share = \( 84x \)
- Arun's share = \( 108x \)
- Ram's share = \( 36x \)

Thus, the ratio of their shares is:
\[ \text{Rushi : Arun : Ram} = 84 : 108 : 36 \]

### Step 7: Simplify the Ratio
To simplify the ratio, we can divide each term by 12:
\[ \text{Rushi : Arun : Ram} = 7 : 9 : 3 \]

### Step 8: Calculate Total Profit Using Rushi's Share
We know that Rushi's profit share is Rs. 5600. Let the total profit be \( P \). The share of Rushi can be expressed as:
\[ \frac{7}{19} P = 5600 \]
(where \( 19 \) is the sum of the ratio parts: \( 7 + 9 + 3 = 19 \))

### Step 9: Solve for Total Profit \( P \)
To find \( P \), we rearrange the equation:
\[ P = 5600 \times \frac{19}{7} \]
Calculating this gives:
\[ P = 5600 \times 2.7143 \approx 15200 \]

### Final Answer
The total profit is Rs. 15200.

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