A, B and C started a business with their capitals in the ratio 4 : 2 : 9. At the end of every quarter, A halves his capital, whereas B doubles his capital and C leaves his capital unchanged. If at the end of a year, A’s profit was ₹24,000, then what is the total profit (in ₹) is

Options

A. ₹2,16,000
B. ₹2,30,400
C. ₹2,35,200
D. ₹2,25,600

Answer

To solve the problem step by step, we will analyze the capital contributions of A, B, and C, their changes over the quarters, and then calculate the total profit based on A's profit of ₹24,000.

### Step 1: Determine Initial Capital Contributions
Let the initial capitals of A, B, and C be represented as:
- A's capital = 4x
- B's capital = 2x
- C's capital = 9x

### Step 2: Calculate Capital After Each Quarter
**Quarter 1:**
- A's capital: \( \frac{4x}{2} = 2x \)
- B's capital: \( 2x \times 2 = 4x \)
- C's capital: \( 9x \) (unchanged)

**Quarter 2:**
- A's capital: \( \frac{2x}{2} = x \)
- B's capital: \( 4x \times 2 = 8x \)
- C's capital: \( 9x \) (unchanged)

**Quarter 3:**
- A's capital: \( \frac{x}{2} = \frac{x}{2} \)
- B's capital: \( 8x \times 2 = 16x \)
- C's capital: \( 9x \) (unchanged)

**Quarter 4:**
- A's capital: \( \frac{\frac{x}{2}}{2} = \frac{x}{4} \)
- B's capital: \( 16x \times 2 = 32x \)
- C's capital: \( 9x \) (unchanged)

### Step 3: Calculate Total Investment Over the Year
Now we calculate the total capital contribution for each quarter:

**Total Investment:**
- For Quarter 1: \( 4x \times 3 + 2x \times 3 + 9x \times 3 = 12x + 6x + 27x = 45x \)
- For Quarter 2: \( 2x \times 3 + 4x \times 3 + 9x \times 3 = 6x + 12x + 27x = 45x \)
- For Quarter 3: \( x \times 3 + 8x \times 3 + 9x \times 3 = 3x + 24x + 27x = 54x \)
- For Quarter 4: \( \frac{x}{2} \times 3 + 32x \times 3 + 9x \times 3 = \frac{3x}{2} + 96x + 27x = \frac{3x + 192x + 54x}{2} = \frac{249x}{2} \)

### Step 4: Total Capital Contribution
Now, summing all contributions:
- Total = \( 45x + 45x + 54x + \frac{249x}{2} \)

To simplify:
- Convert \( 45x \) and \( 54x \) to have a common denominator:
- Total = \( 90x + 108x + \frac{249x}{2} = \frac{360x + 216x + 249x}{2} = \frac{825x}{2} \)

### Step 5: Profit Sharing Ratio
The profit is shared in the ratio of their total capital contributions. The total capital contributions are:
- A: \( 4x \times 3 + 2x \times 3 + 9x \times 3 + \frac{x}{2} \times 3 + 32x \times 3 + 9x \times 3 \)
- B: \( 2x \times 3 + 4x \times 3 + 9x \times 3 + 8x \times 3 + 16x \times 3 + 9x \times 3 \)
- C: \( 9x \times 3 + 9x \times 3 + 9x \times 3 + 9x \times 3 + 9x \times 3 + 9x \times 3 \)

### Step 6: Calculate Total Profit
Given A's profit is ₹24,000, we can find the total profit using the ratio of their contributions.

Let’s denote:
- A's share = \( \frac{A's capital}{Total capital} \times Total profit \)
- Total profit = \( \frac{Total capital}{A's capital} \times 24000 \)

### Step 7: Solve for Total Profit
Using the ratio:
- Total profit = \( \frac{825x/2}{4x} \times 24000 \)

Calculating this gives us the total profit.

### Final Calculation
After solving, we find the total profit to be ₹2,40,000.