Rakesh started a business with some money . After four months Ram and Shyam joined his business with investments of Rs 31500 and Rs 27000 respectively . Total profit earned at the end of year was Rs 19200. If profit received by Rakesh was Rs 7500, then find Rakesh 's investment in the business ?

Options

A. A)27000
B. B)24000
C. C)26000
D. D)25000

Answer

To find Rakesh's investment in the business, we can follow these steps:

### Step 1: Define Variables
Let Rakesh's investment be \( K \).

### Step 2: Determine Time of Investment
- Rakesh invested for 12 months (1 year).
- Ram and Shyam joined after 4 months, so they invested for 8 months.

### Step 3: Calculate the Effective Investment
- Rakesh's effective investment = \( K \times 12 \)
- Ram's investment = Rs. 31,500 for 8 months = \( 31,500 \times 8 \)
- Shyam's investment = Rs. 27,000 for 8 months = \( 27,000 \times 8 \)

Calculating these:
- Rakesh's effective investment = \( 12K \)
- Ram's effective investment = \( 31,500 \times 8 = 252,000 \)
- Shyam's effective investment = \( 27,000 \times 8 = 216,000 \)

### Step 4: Calculate Total Effective Investment
Total effective investment = \( 12K + 252,000 + 216,000 \)

### Step 5: Calculate Profit Shares
The total profit earned at the end of the year is Rs. 19,200. Rakesh's share of the profit is Rs. 7,500.

The profit share ratio is proportional to the effective investments:
- Rakesh's share = \( \frac{12K}{12K + 252,000 + 216,000} \times 19,200 \)

### Step 6: Set Up the Equation
We know Rakesh's profit is Rs. 7,500, so we can set up the equation:
\[ \frac{12K}{12K + 468,000} \times 19,200 = 7,500 \]

### Step 7: Simplify the Equation
Cross-multiplying gives:
\[ 12K \times 19,200 = 7,500 \times (12K + 468,000) \]

Expanding this:
\[ 230,400K = 7,500 \times 12K + 7,500 \times 468,000 \]
\[ 230,400K = 90,000K + 3,510,000,000 \]

### Step 8: Rearranging the Equation
Rearranging gives:
\[ 230,400K - 90,000K = 3,510,000,000 \]
\[ 140,400K = 3,510,000,000 \]

### Step 9: Solve for K
Dividing both sides by 140,400:
\[ K = \frac{3,510,000,000}{140,400} = 25,000 \]

### Conclusion
Rakesh's investment in the business is Rs. 25,000.

---