A, B and C invested Rs 400 , Rs 900 and rs . 500 respectively at the beginning of a year in a business venture .
Then at end of the second quarter A,B and C invested additional amount in the ratio of `5:4:10`.
Again at the end of the third quarter they invested additional amount in the ratio of `1:1:2`
they invested the whole amount for one year and the profit earned in the business is proportional to the investment and the period of investment .
If they had invested additional amount at the end of each quarter in the same ratio as they had invested at the end of the first quarter then what will be profit of B at the end of the year if the total profit at the end of the year is Rs. 17500.

Options

A. Rs .7500
B. Rs .8750
C. Rs. 12500
D. Rs 10000

Answer

To solve the problem step by step, we need to calculate the total investment of A, B, and C over the year, considering their initial investments and the additional investments made at the end of each quarter.

### Step 1: Initial Investments
- A's initial investment = Rs 400
- B's initial investment = Rs 900
- C's initial investment = Rs 500

### Step 2: Additional Investments at the End of the Second Quarter
The additional investments made by A, B, and C at the end of the second quarter are in the ratio of 5:4:10.

Let the additional amounts be:
- A's additional investment = 5x
- B's additional investment = 4x
- C's additional investment = 10x

### Step 3: Additional Investments at the End of the Third Quarter
The additional investments made at the end of the third quarter are in the ratio of 1:1:2.

Let the additional amounts be:
- A's additional investment = 1y
- B's additional investment = 1y
- C's additional investment = 2y

### Step 4: Total Investment Calculation
Now, we need to calculate the total investment for each person at the end of the year.

**For A:**
- Initial investment = 400 (for 12 months)
- Additional investment after 6 months = 5x (for 6 months)
- Additional investment after 9 months = 1y (for 3 months)

Total investment for A:
\[ \text{Total A} = 400 \times 12 + 5x \times 6 + 1y \times 3 \]

**For B:**
- Initial investment = 900 (for 12 months)
- Additional investment after 6 months = 4x (for 6 months)
- Additional investment after 9 months = 1y (for 3 months)

Total investment for B:
\[ \text{Total B} = 900 \times 12 + 4x \times 6 + 1y \times 3 \]

**For C:**
- Initial investment = 500 (for 12 months)
- Additional investment after 6 months = 10x (for 6 months)
- Additional investment after 9 months = 2y (for 3 months)

Total investment for C:
\[ \text{Total C} = 500 \times 12 + 10x \times 6 + 2y \times 3 \]

### Step 5: Profit Calculation
The total profit earned is Rs 17,500, and it is distributed in the ratio of their total investments.

Let’s calculate the total investment in terms of x and y.

### Step 6: Determine x and y
Since we need to find the profit of B, we can assume values for x and y based on the ratios given.

For simplicity, let’s assume:
- At the end of the second quarter, the total investment ratio is maintained, so we can set x = 100 (this is arbitrary for calculation purposes).

Then:
- A's additional investment = 500
- B's additional investment = 400
- C's additional investment = 1000

For the third quarter, we can assume:
- y = 100 (again, arbitrary).

Then:
- A's additional investment = 100
- B's additional investment = 100
- C's additional investment = 200

### Step 7: Total Investment Calculation
Now substituting x and y into the total investment formulas:

**For A:**
\[ \text{Total A} = 400 \times 12 + 500 \times 6 + 100 \times 3 = 4800 + 3000 + 300 = 8100 \]

**For B:**
\[ \text{Total B} = 900 \times 12 + 400 \times 6 + 100 \times 3 = 10800 + 2400 + 300 = 13400 \]

**For C:**
\[ \text{Total C} = 500 \times 12 + 1000 \times 6 + 200 \times 3 = 6000 + 6000 + 600 = 12600 \]

### Step 8: Total Proportions
Now we find the total proportions:
- Total investment = Total A + Total B + Total C = 8100 + 13400 + 12600 = 34100

### Step 9: Profit Distribution
Now we calculate B's share of the profit:
\[ \text{Profit of B} = \frac{\text{Total B}}{\text{Total Investment}} \times \text{Total Profit} \]
\[ \text{Profit of B} = \frac{13400}{34100} \times 17500 \]

Calculating this gives:
\[ \text{Profit of B} = \frac{13400 \times 17500}{34100} = 6750 \]

### Final Answer
The profit of B at the end of the year is Rs 6750.