A starts a certain business with rs4000. Three months from the start of the business, B joins with an amount which is rs500 more than that invested by A. Nine months from the start of the business, B leaves and C joins, with an amount which is rs1500 more than that invested by B. If B's share from the annual profit is rs18000, what is the annual profit earned?

Options

A. rs64,000
B. rs2,000
C. rs62000
D. rs54,000

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the investments of A, B, and C.
- A starts the business with Rs. 4000.
- B joins 3 months later with Rs. 500 more than A. Therefore, B's investment is:
\[ B = A + 500 = 4000 + 500 = 4500 \]
- C joins 9 months after the start of the business with Rs. 1500 more than B. Therefore, C's investment is:
\[ C = B + 1500 = 4500 + 1500 = 6000 \]

### Step 2: Calculate the time each partner's investment was active.
- A invests for 12 months.
- B invests for 9 months (since he joins after 3 months).
- C invests for 3 months (since he joins after 9 months).

### Step 3: Calculate the capital contribution of each partner.
- A's contribution:
\[ \text{A's contribution} = 4000 \times 12 = 48000 \]
- B's contribution:
\[ \text{B's contribution} = 4500 \times 9 = 40500 \]
- C's contribution:
\[ \text{C's contribution} = 6000 \times 3 = 18000 \]

### Step 4: Calculate the total capital contribution.
\[ \text{Total capital} = \text{A's contribution} + \text{B's contribution} + \text{C's contribution} = 48000 + 40500 + 18000 = 106500 \]

### Step 5: Calculate the ratio of their contributions.
- The ratio of A, B, and C's contributions is:
\[ \text{A : B : C} = 48000 : 40500 : 18000 \]
- To simplify, we can divide each term by 3000:
\[ \text{A : B : C} = 16 : 13.5 : 6 \]
- To avoid decimals, multiply by 2:
\[ \text{A : B : C} = 32 : 27 : 12 \]

### Step 6: Find the value of each part of the profit.
- Given that B's share of the profit is Rs. 18000, which corresponds to the 27 parts in the ratio:
\[ \text{Value of one part} = \frac{18000}{27} = 666.67 \]

### Step 7: Calculate the total annual profit.
- The total number of parts is:
\[ 32 + 27 + 12 = 71 \]
- Therefore, the total annual profit is:
\[ \text{Total annual profit} = 71 \times 666.67 \approx 47333.33 \]

### Step 8: Conclusion
The total annual profit is approximately Rs. 47333.33.