A, B ,And c started a bussiness. A invested `33(1)/(3)%` of the total capital. B invests `33(1)/(3)%` of the remaining capital and c, the remaining.If the total profit at the end of the year was Rs. 20,250 then the profit of c excceds the profit of B by

Options

A. Rs. 5,200
B. Rs. 4,500
C. Rs. 6,700
D. Rs. 2,700

Answer

To solve the problem, let's break it down step by step.

### Step 1: Determine A's Investment
A invested \( 33\frac{1}{3}\% \) of the total capital. This percentage can be converted into a fraction:
\[ 33\frac{1}{3}\% = \frac{100}{3\cdot100} = \frac{1}{3} \]
So, A's investment is:
\[ A's\ investment = \frac{1}{3} \times Total\ Capital \]

### Step 2: Calculate Remaining Capital After A's Investment
After A's investment, the remaining capital is:
\[ Remaining\ Capital = Total\ Capital - A's\ investment = Total\ Capital - \frac{1}{3} \times Total\ Capital = \frac{2}{3} \times Total\ Capital \]

### Step 3: Determine B's Investment
B invested \( 33\frac{1}{3}\% \) of the remaining capital. Using the same fraction as before:
\[ B's\ investment = \frac{1}{3} \times Remaining\ Capital = \frac{1}{3} \times \frac{2}{3} \times Total\ Capital = \frac{2}{9} \times Total\ Capital \]

### Step 4: Calculate Remaining Capital After B's Investment
After B's investment, the remaining capital is:
\[ Remaining\ Capital = Remaining\ Capital - B's\ investment = \frac{2}{3} \times Total\ Capital - \frac{2}{9} \times Total\ Capital \]
To perform this subtraction, we need a common denominator:
\[ \frac{2}{3} = \frac{6}{9} \]
So,
\[ Remaining\ Capital = \frac{6}{9} \times Total\ Capital - \frac{2}{9} \times Total\ Capital = \frac{4}{9} \times Total\ Capital \]

### Step 5: Determine C's Investment
C invested the remaining capital:
\[ C's\ investment = \frac{4}{9} \times Total\ Capital \]

### Step 6: Calculate the Total Investment
Now we can summarize the investments:
- A's investment: \( \frac{1}{3} \times Total\ Capital = \frac{3}{9} \times Total\ Capital \)
- B's investment: \( \frac{2}{9} \times Total\ Capital \)
- C's investment: \( \frac{4}{9} \times Total\ Capital \)

### Step 7: Calculate the Profit Shares
The profit is distributed according to the amount invested:
- Total profit = Rs. 20,250
- Total shares = \( 3 + 2 + 4 = 9 \)

Profit per share:
\[ Profit\ per\ share = \frac{20,250}{9} = 2,250 \]

### Step 8: Calculate Individual Profits
Now we can calculate the profits for B and C:
- B's profit: \( 2 \times 2,250 = 4,500 \)
- C's profit: \( 4 \times 2,250 = 9,000 \)

### Step 9: Find the Difference Between C's and B's Profit
The difference between C's profit and B's profit is:
\[ C's\ profit - B's\ profit = 9,000 - 4,500 = 4,500 \]

### Final Answer
Thus, the profit of C exceeds the profit of B by Rs. 4,500.

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