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The problem provided involves three individuals, A, B, and C, starting a business with their capital in a certain ratio. To determine C's share of the profit from a total of Rs. 81,000, we need to calculate their investments in terms of both capital and time, and then find the profit division accordingly.

### Given Data:

1. **Ratio of Capitals**:
   - A's capital : B's capital : C's capital = 1/2 : 1/3 : 1/6

2. **Ratio of Periods (Time)**:
   - A's time : B's time : C's time = 1/3 : 1/2 : 1/4

3. **Total Profit**:
   - Rs. 81,000.

### Steps to solve:

1. **Simplify the Capital Ratios**:
   - To eliminate fractions, multiply by the least common multiple (LCM) of 2, 3, and 6, which is 6:
     - A's capital = 1/2 × 6 = 3
     - B's capital = 1/3 × 6 = 2
     - C's capital = 1/6 × 6 = 1
   - Therefore, the capital ratio becomes A : B : C = 3 : 2 : 1.

2. **Simplify the Time Ratios**:
   - Similarly, eliminate fractions in the time ratio by multiplying by the LCM of 3, 2, and 4, which is 12:
     - A's time = 1/3 × 12 = 4
     - B's time = 1/2 × 12 = 6
     - C's time = 1/4 × 12 = 3
   - Therefore, the time ratio becomes A : B : C = 4 : 6 : 3.

3. **Calculate the Product of Capital and Time for each partner**:
   - A's product = 3 (capital) × 4 (time) = 12
   - B's product = 2 (capital) × 6 (time) = 12
   - C's product = 1 (capital) × 3 (time) = 3

   The total product = 12 + 12 + 3 = 27.

4. **Determine C's Share of the Profit**:
   - C's share = (C's product / Total product) × Total profit
   - C's share = (3 / 27) × 81,000
   - C's share = 1/9 × 81,000 = Rs. 9,000.

### Final Answer:
C's share of the total profit is **Rs. 9,000**.

Let me know if you need any further details or explanations.

Here are 5 related questions to expand the concept:
1. How would the result change if the total profit was Rs. 100,000?
2. What happens to the profit share if C had invested for a longer period?
3. How would the profits be divided if all partners had invested for the same period?
4. Can you solve for A's and B's shares of the profit from Rs. 81,000?
5. What if the capital ratio was in whole numbers like 2:3:5?

**Tip**: When dealing with ratios involving fractions, always simplify them by finding a common multiple to avoid complexity in calculations.

A, B, and C start a business with their capitals in the ratio of 1/2 : 1/3 : 1/6. The ratio of periods of their investment is 1/3 : 1/2 : 1/4. Find C's share out of a total profit of Rs. 81,000.

Learn how to solve a profit-sharing problem involving three partners A, B, and C, with given ratios of capital and time investments. Find out how to calculate C's share of the total profit of Rs. 81,000 by using proportion and ratio methods. Suitable for Grades 8-10.

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