A and B enter into a partnership. A puts in the whole capital of Rs 45,000 on the condition that the profits will be equally divided after which B will pay A interest on half the capital at 10% p.a. and receive Rs 60 per month from A for carrying on the concern. What is the yearly profit, if B's income is half of A's income?

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Partnership Agreement
A and B are partners. A invests the entire capital of Rs. 45,000. The profits will be equally divided between A and B. However, B will pay A interest on half of the capital at 10% per annum and receive Rs. 60 per month for managing the business.

### Step 2: Calculate the Interest on Half the Capital
Since A's capital is Rs. 45,000, half of this capital is:
\[ \text{Half of Capital} = \frac{45000}{2} = 22500 \]
The interest on this amount at 10% per annum is calculated as:
\[ \text{Interest} = \frac{10}{100} \times 22500 = 2250 \]

### Step 3: Calculate B's Monthly Payment
B receives Rs. 60 per month for managing the business. Therefore, the total amount B receives in a year is:
\[ \text{Total Payment to B} = 60 \times 12 = 720 \]

### Step 4: Determine B's Income
According to the problem, B's income is half of A's income. Let the total profit be \( P \). Since profits are shared equally:
\[ \text{A's Income} = \frac{P}{2} \]
\[ \text{B's Income} = \frac{P}{2} \]
However, B also pays A the interest of Rs. 2250 and receives Rs. 720. Therefore, B's effective income can be expressed as:
\[ \text{B's Effective Income} = \frac{P}{2} - 2250 + 720 \]

### Step 5: Set Up the Equation Based on B's Income
Since B's income is half of A's income:
\[ \frac{P}{2} - 2250 + 720 = \frac{1}{2} \left(\frac{P}{2} + 2250\right) \]

### Step 6: Simplify the Equation
Expanding and simplifying the equation gives:
\[ \frac{P}{2} - 2250 + 720 = \frac{P}{4} + 1125 \]
Combine like terms:
\[ \frac{P}{2} - \frac{P}{4} = 2250 - 720 + 1125 \]
\[ \frac{P}{4} = 2250 - 720 + 1125 \]
Calculating the right side:
\[ \frac{P}{4} = 2250 - 720 + 1125 = 3655 \]

### Step 7: Solve for P
Multiplying both sides by 4 to find \( P \):
\[ P = 4 \times 3655 = 14620 \]

### Step 8: Conclusion
The yearly profit \( P \) is Rs. 14620.