What approximate value will come in place of the question mark (?) in the following question?
A and B started a business together in a partnership. B left the business after 8 months. At the end of year profit is Rs. 4000, out of which profit of B is Rs. 3000, the investment of B is how much percentage more than the investment of A?

Options

A. `450%`
B. `400%`
C. `350%`
D. `300%`

Answer

To solve the problem step by step, we will follow the information given in the question regarding A and B's investments and the profits they earned.

### Step 1: Understand the Profit Sharing
A and B started a business together. The total profit at the end of the year is Rs. 4000. Out of this, B's share of the profit is Rs. 3000.

### Step 2: Calculate A's Share of Profit
To find A's share of the profit, we subtract B's profit from the total profit:
\[ \text{A's Profit} = \text{Total Profit} - \text{B's Profit} = 4000 - 3000 = 1000 \text{ Rs.} \]

### Step 3: Determine the Time of Investment
- A invested for 12 months (the entire year).
- B invested for 8 months (since he left after 8 months).

### Step 4: Use the Profit Sharing Formula
The profit share is proportional to the investment multiplied by the time of investment. Thus, we can set up the following equations based on the profit shares:
\[ \text{A's Investment} \times 12 = \text{B's Investment} \times 8 \]

Let A's investment be \( x \) and B's investment be \( y \). We can express this as:
\[ x \times 12 = y \times 8 \]

### Step 5: Express B's Profit in Terms of Investment
From the profit shares:
- A's profit share: \( \frac{x \times 12}{x \times 12 + y \times 8} \times 4000 = 1000 \)
- B's profit share: \( \frac{y \times 8}{x \times 12 + y \times 8} \times 4000 = 3000 \)

### Step 6: Set Up the Ratios
From the profit shares, we can set up the ratio of their investments:
\[ \frac{1000}{3000} = \frac{x \times 12}{y \times 8} \]
This simplifies to:
\[ \frac{1}{3} = \frac{x \times 12}{y \times 8} \]

### Step 7: Cross Multiply and Solve for the Investment Ratio
Cross-multiplying gives:
\[ 1 \times (y \times 8) = 3 \times (x \times 12) \]
This simplifies to:
\[ 8y = 36x \implies y = \frac{36}{8}x = 4.5x \]

### Step 8: Calculate the Percentage Difference
Now we know that B's investment \( y \) is 4.5 times A's investment \( x \). To find out how much percentage more B's investment is than A's:
\[ \text{Percentage More} = \left( \frac{y - x}{x} \right) \times 100 \]
Substituting \( y = 4.5x \):
\[ \text{Percentage More} = \left( \frac{4.5x - x}{x} \right) \times 100 = (3.5) \times 100 = 350\% \]

### Final Answer
The investment of B is approximately **350%** more than the investment of A.

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