A started business with ₹ 45000 and B joined after ward with ₹ 30000. If the profits at the end of one year were divided in the ratio 2:1 respectively, then B would have joined A for business after:

Options

A. 1 month
B. 2 months
C. 3 months
D. 4 months

Answer

To solve the problem step by step, we need to determine how long B was in the business with A, given the profits were divided in the ratio of 2:1.

### Step 1: Understand the Investment
A started the business with ₹45,000 and B joined later with ₹30,000.

### Step 2: Define the Time Periods
Let’s denote the time A invested in the business as 12 months (1 year). If B joined after x months, then B's investment period is (12 - x) months.

### Step 3: Calculate the Effective Capital Contribution
The effective capital contribution can be calculated by multiplying the amount invested by the time period for which it was invested.

- A's contribution = ₹45,000 * 12 months = ₹540,000
- B's contribution = ₹30,000 * (12 - x) months = ₹30,000 * (12 - x)

### Step 4: Set Up the Ratio
According to the problem, the profits are divided in the ratio of 2:1. This means:

\[ \frac{A's \ Contribution}{B's \ Contribution} = \frac{2}{1} \]

Substituting the contributions into the ratio gives us:

\[ \frac{540,000}{30,000 \times (12 - x)} = \frac{2}{1} \]

### Step 5: Cross-Multiply to Solve for x
Cross-multiplying gives:

\[ 540,000 = 2 \times 30,000 \times (12 - x) \]

This simplifies to:

\[ 540,000 = 60,000 \times (12 - x) \]

### Step 6: Divide Both Sides by 60,000
Dividing both sides by 60,000:

\[ 9 = 12 - x \]

### Step 7: Solve for x
Rearranging gives:

\[ x = 12 - 9 = 3 \]

### Conclusion
B joined A for business after 3 months.