A' began business with `₹ 45,000` and was later joined by 'B' with `₹ 54,000`. When did B join if the profit at the end of the year were divided in the ratio `2:1`?

Options

A. 5 months after
B. 10 months after
C. 7 months after
D. 12 months after

Explanation

To solve the problem step by step, we need to determine when B joined the business based on the profit-sharing ratio.

### Step 1: Understand the Profit Sharing Ratio
The profit is shared between A and B in the ratio of 2:1. This means that for every 2 parts of profit A receives, B receives 1 part.

### Step 2: Calculate A's Contribution
A started the business with ₹45,000 and worked for the entire year (12 months). Therefore, A's contribution to the profit can be calculated as:
\[ \text{A's Contribution} = 45,000 \times 12 \]

### Step 3: Calculate B's Contribution
B joined later with ₹54,000. Let’s assume B joined the business after \( x \) months. Therefore, B's contribution to the profit can be calculated as:
\[ \text{B's Contribution} = 54,000 \times (12 - x) \]

### Step 4: Set Up the Equation Based on the Profit Ratio
According to the problem, the ratio of A's profit to B's profit is 2:1. We can express this as:
\[ \frac{A's Contribution}{B's Contribution} = \frac{2}{1} \]
Substituting the contributions we calculated:
\[ \frac{45,000 \times 12}{54,000 \times (12 - x)} = \frac{2}{1} \]

### Step 5: Cross Multiply to Solve for \( x \)
Cross multiplying gives us:
\[ 45,000 \times 12 = 2 \times 54,000 \times (12 - x) \]
This simplifies to:
\[ 540,000 = 108,000 \times (12 - x) \]

### Step 6: Simplify and Solve for \( x \)
Now, divide both sides by 108,000:
\[ \frac{540,000}{108,000} = 12 - x \]
Calculating the left side:
\[ 5 = 12 - x \]
Rearranging gives:
\[ x = 12 - 5 \]
Thus,
\[ x = 7 \]

### Step 7: Conclusion
B joined the business 7 months after A started.