A started a business with an investment of Rs 54000 and after some -months ,B joined him with investing Rs 45000 .At the end of the year , total profit was Rs 35700 and share of A in the total profit is Rs 22950. Find after how many months B has joined the business.

Options

A. 5 months
B. 6 months
C. 4 months
D. 2 months

Answer

To solve the problem step by step, we will follow the information provided in the question and the calculations as outlined in the video transcript.

### Step 1: Understand the Investments
- A started the business with an investment of Rs 54,000.
- B joined later with an investment of Rs 45,000.

### Step 2: Determine the Time of Investment
- A invested for the entire year, which is 12 months.
- Let’s assume B joined after \( x \) months. Therefore, B invested for \( 12 - x \) months.

### Step 3: Calculate the Total Profit and A's Share
- The total profit at the end of the year is Rs 35,700.
- A's share of the profit is Rs 22,950.
- To find B's share, we subtract A's share from the total profit:
\[ B's \, share = Total \, profit - A's \, share = 35700 - 22950 = 12750 \]

### Step 4: Set Up the Profit Sharing Ratio
- The profit is shared in the ratio of the product of investment and time.
- For A, the profit can be expressed as:
\[ Profit_A = Investment_A \times Time_A = 54000 \times 12 \]
- For B, the profit can be expressed as:
\[ Profit_B = Investment_B \times Time_B = 45000 \times (12 - x) \]

### Step 5: Write the Profit Ratio
- The ratio of A's profit to B's profit is:
\[ \frac{Profit_A}{Profit_B} = \frac{22950}{12750} \]
- Simplifying this gives:
\[ \frac{22950}{12750} = \frac{22.95}{12.75} = \frac{3}{1.5} = \frac{2}{1} \]

### Step 6: Set Up the Investment-Time Ratio
- The ratio of their investments multiplied by the time they invested is:
\[ \frac{54000 \times 12}{45000 \times (12 - x)} = \frac{2}{1} \]

### Step 7: Cross Multiply and Solve for x
- Cross-multiplying gives:
\[ 54000 \times 12 = 45000 \times (12 - x) \times 2 \]
\[ 648000 = 90000 \times (12 - x) \]
\[ 648000 = 1080000 - 90000x \]
\[ 90000x = 1080000 - 648000 \]
\[ 90000x = 432000 \]
\[ x = \frac{432000}{90000} = 4.8 \]

### Step 8: Conclusion
- Since \( x \) represents the number of months after which B joined, we round it to the nearest whole number. Thus, B joined after approximately **4 months**.

### Final Answer
B joined the business after **4 months**.