A began a business with Rs 120000 and B joined after wards with Rs 60000. At the end of a year, the profit is divided in the ratio 4:1. When did B join?

Answer

To solve the problem step by step, we will analyze the information given and use the formula related to the ratio of profit sharing based on the capital invested and the time for which it was invested.

### Step 1: Understand the problem
A started a business with Rs 120,000, and B joined later with Rs 60,000. The profit at the end of the year was divided in the ratio of 4:1. We need to find out when B joined the business.

### Step 2: Define the variables
- Let \( T_1 \) be the time A invested in the business (in months).
- Let \( T_2 \) be the time B invested in the business (in months).
- A's capital \( C_1 = 120000 \)
- B's capital \( C_2 = 60000 \)
- The profit ratio \( P_1 : P_2 = 4 : 1 \)

### Step 3: Set up the equation
According to the formula for profit sharing based on capital and time:
\[ C_1 \times T_1 = C_2 \times T_2 \]
Substituting the known values:
\[ 120000 \times T_1 = 60000 \times T_2 \]

### Step 4: Determine the time A invested
Since A started the business and continued for the entire year, \( T_1 = 12 \) months.

### Step 5: Substitute \( T_1 \) into the equation
Now substituting \( T_1 \) into the equation:
\[ 120000 \times 12 = 60000 \times T_2 \]
This simplifies to:
\[ 1440000 = 60000 \times T_2 \]

### Step 6: Solve for \( T_2 \)
To find \( T_2 \), divide both sides by 60000:
\[ T_2 = \frac{1440000}{60000} = 24 \]

### Step 7: Interpret the result
The value of \( T_2 = 24 \) months means that B was in the business for 24 months. However, since A started the business 12 months before the end of the year, we need to find out when B joined.

### Step 8: Calculate when B joined
Since A started the business 12 months before the end of the year, and B was in the business for 6 months (because \( T_2 = 24 - 12 = 6 \)), B joined 6 months after A.

### Conclusion
B joined the business 6 months after A started.