A and B enter into partnership. A supplies whole of the capital amounting to Rs 45,000 with the condition that the profit are to be equally divided and that B pays A interest on half of the capital at 10% per annum, but recives, Rs 120 per month for carrying on the concern. When B's income is onehalf of A's income their total yearly profit is

Answer

To solve the problem step-by-step, we will break down the information given and calculate the total yearly profit based on the conditions provided.

### Step 1: Understand the Partnership Agreement
A and B enter into a partnership where:
- A supplies the entire capital of Rs 45,000.
- Profits are to be divided equally between A and B.
- B pays A interest on half of the capital at 10% per annum.
- B receives Rs 120 per month for managing the business.

### Step 2: Calculate the Interest Paid by B to A
B pays interest on half of the capital (Rs 22,500) at a rate of 10% per annum.

\[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \]
\[ \text{Interest} = 22,500 \times 10\% \times 1 = 22,500 \times 0.10 = 2,250 \text{ Rs per annum} \]

### Step 3: Calculate B's Monthly Income
B receives Rs 120 per month for managing the business. Therefore, B's total income for the year is:

\[ \text{B's yearly income from management} = 120 \times 12 = 1,440 \text{ Rs} \]

### Step 4: Set Up the Profit Equation
Let the total profit from the business be \( P \). According to the problem, the profits are divided equally, so:

- A's share of profit = \( \frac{P}{2} \)
- B's share of profit = \( \frac{P}{2} \)

B's total income (including his share of profit and management fee) can be expressed as:

\[ \text{B's total income} = \frac{P}{2} + 1,440 \]

### Step 5: Relate A's and B's Income
According to the problem, B's income is half of A's income:

\[ \frac{P}{2} + 1,440 = \frac{1}{2} \left( \frac{P}{2} \right) \]

### Step 6: Solve the Equation
From the equation above, we can express A's income:

\[ A's \text{ income} = \frac{P}{2} \]

Thus, we have:

\[ \frac{P}{2} + 1,440 = \frac{1}{2} \left( \frac{P}{2} \right) \]

This simplifies to:

\[ \frac{P}{2} + 1,440 = \frac{P}{4} \]

Now, rearranging gives:

\[ \frac{P}{2} - \frac{P}{4} = -1,440 \]
\[ \frac{2P - P}{4} = -1,440 \]
\[ \frac{P}{4} = -1,440 \]

Multiplying both sides by 4:

\[ P = -5,760 \]

### Step 7: Calculate Total Yearly Profit
The total yearly profit \( P \) is equal to 2 times B's share of profit:

\[ \text{Total Yearly Profit} = P = 2 \times \frac{P}{2} = P \]

Thus, the total yearly profit is:

\[ P = 2 \times 1,440 = 2,880 \]

### Final Answer
The total yearly profit is **Rs 2,880**.

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