Rohit started a business with Rs. 75000 and after some months Simran joined him with Rs. 60000. If the profit at the end of the year is divided in the ratio 3:1, then after how many months did Simran join Rohit?

Options

A. 7
B. 6
C. 8
D. 4

Answer

To solve the problem step by step, we will analyze the information given and apply the concept of ratio and proportion.

### Step-by-Step Solution:

1. **Identify the Investment and Time**:
- Rohit started the business with Rs. 75,000 and invested for the entire year (12 months).
- Simran joined later with Rs. 60,000, and we need to find out after how many months she joined, which we will denote as \( X \) months.

2. **Understanding Profit Sharing**:
- The profit is shared in the ratio of 3:1 between Rohit and Simran.
- This means Rohit's share of the profit corresponds to 3 parts and Simran's share corresponds to 1 part.

3. **Setting Up the Equation**:
- The profit sharing ratio can be expressed in terms of their investments and the time they invested.
- The formula for profit sharing based on capital and time is:
\[ \text{Rohit's investment} \times \text{Rohit's time} : \text{Simran's investment} \times \text{Simran's time} \]
- This translates to:
\[ 75000 \times 12 : 60000 \times X = 3 : 1 \]

4. **Cross-Multiplying to Solve for \( X \)**:
- From the ratio, we can set up the equation:
\[ \frac{75000 \times 12}{60000 \times X} = \frac{3}{1} \]
- Cross-multiplying gives:
\[ 75000 \times 12 = 60000 \times X \times 3 \]

5. **Simplifying the Equation**:
- First, simplify \( 75000 \times 12 \):
\[ 75000 \times 12 = 900000 \]
- Now the equation becomes:
\[ 900000 = 180000 \times X \]

6. **Solving for \( X \)**:
- Divide both sides by 180000:
\[ X = \frac{900000}{180000} = 5 \]
- So, Simran invested for 5 months.

7. **Finding When Simran Joined**:
- Since the total time is 12 months, we find out when Simran joined:
\[ \text{Months after which Simran joined} = 12 - X = 12 - 5 = 7 \]

### Final Answer:
Simran joined Rohit after **7 months**.

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