X and Y started a business investing Rs 12000 and Rs 9000 respectively. After 4 months, X left the business and Z joined the business by investing Rs 18000. At the end of the year, there was a profit of Rs 12500. What is the share (in Rs) of Z in the profit?

Options

A. 6000
B. 4500
C. 7200
D. 7500

Answer

To solve the problem step by step, we need to calculate the share of Z in the profit based on the capital invested and the time for which it was invested.

### Step 1: Determine the capital and time invested by each partner

- **X's Investment**: Rs 12,000 for 4 months
- **Y's Investment**: Rs 9,000 for 12 months
- **Z's Investment**: Rs 18,000 for 8 months (since Z joined after 4 months)

### Step 2: Calculate the capital contribution in terms of "capital-months"

- For X:
\[ \text{Capital-months} = 12000 \times 4 = 48000 \]

- For Y:
\[ \text{Capital-months} = 9000 \times 12 = 108000 \]

- For Z:
\[ \text{Capital-months} = 18000 \times 8 = 144000 \]

### Step 3: Calculate the total capital-months

Now, we sum up the capital-months of all partners:
\[ \text{Total Capital-months} = 48000 + 108000 + 144000 = 300000 \]

### Step 4: Determine the ratio of capital contributions

Now we can find the ratio of the capital contributions of X, Y, and Z:
- X's contribution: 48000
- Y's contribution: 108000
- Z's contribution: 144000

The ratio of their contributions is:
\[ X : Y : Z = 48000 : 108000 : 144000 \]

### Step 5: Simplify the ratio

To simplify, we can divide each term by 24000:
\[ X : Y : Z = 2 : 4.5 : 6 \]

To avoid decimals, we can multiply through by 2:
\[ X : Y : Z = 4 : 9 : 12 \]

### Step 6: Calculate the total parts in the ratio

The total parts in the ratio are:
\[ 4 + 9 + 12 = 25 \]

### Step 7: Calculate Z's share of the profit

The total profit is Rs 12,500. To find Z's share:
- Z's part in the ratio is 12 out of 25 parts.

Calculating Z's share:
\[ \text{Z's Share} = \left(\frac{12}{25}\right) \times 12500 \]
\[ \text{Z's Share} = 6000 \]

### Final Answer

Z's share in the profit is **Rs 6000**.

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