A began a business with X 2250 and was joined afterwards by B with X 2700. If the profit at the end of the year were divided in the ratio of `2:1` after how much time B joined the business ?

Options

A. 5 months
B. 6 months
C. 3 months
D. 7 months

Answer

To solve the problem step by step, we will follow the given information and apply the concept of partnership in business.

### Step 1: Understand the investment and profit sharing
A started the business with an investment of ₹2250 and B joined later with an investment of ₹2700. The profit is shared in the ratio of 2:1 between A and B.

### Step 2: Calculate A's investment in terms of time
Since A invested ₹2250 for the entire year (12 months), we can calculate A's contribution to the profit based on time and investment:
\[ \text{A's Contribution} = 2250 \times 12 = 27000 \]

### Step 3: Define B's investment time
Let’s assume B joined the business after \( x \) months. Therefore, B's investment of ₹2700 will be for \( x \) months. We can calculate B's contribution to the profit:
\[ \text{B's Contribution} = 2700 \times x \]

### Step 4: Set up the ratio of contributions
According to the profit sharing ratio:
\[ \frac{\text{A's Contribution}}{\text{B's Contribution}} = \frac{2}{1} \]

Substituting the contributions we calculated:
\[ \frac{27000}{2700 \times x} = \frac{2}{1} \]

### Step 5: Cross-multiply to solve for \( x \)
Cross-multiplying gives us:
\[ 27000 \times 1 = 2 \times (2700 \times x) \]
\[ 27000 = 5400x \]

### Step 6: Solve for \( x \)
To find \( x \), we divide both sides by 5400:
\[ x = \frac{27000}{5400} \]
\[ x = 5 \]

### Step 7: Determine how long A was in the business before B joined
Since B joined the business for \( x = 5 \) months, and A was in the business for the entire year (12 months), we can find out how long A was in the business before B joined:
\[ \text{Time before B joined} = 12 - 5 = 7 \text{ months} \]

### Final Answer
B joined the business after **7 months**.

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