P, Q, and R enter into a partnership. P invests some money at the beginning, Q invests twice the amount after 6 months, and R invests thrice the amount after 8 months. If the annual profit earned is Rs. 9,000, what is the share of Q?

Let's solve this step by step.

### Assumptions:
- Let the amount of money P invests initially be \( x \).
- Q invests twice the amount of P, which is \( 2x \), after 6 months.
- R invests thrice the amount of P, which is \( 3x \), after 8 months.

### Step 1: Calculate the time each partner's money is invested.
- **P**: P invests \( x \) from the beginning for the whole year, i.e., for 12 months.
- **Q**: Q invests \( 2x \) after 6 months, so the money is invested for 6 months.
- **R**: R invests \( 3x \) after 8 months, so the money is invested for 4 months.

### Step 2: Calculate the effective capital for each partner.
The effective capital is the product of the invested amount and the time it remains invested.

- For **P**: \( x \times 12 = 12x \)
- For **Q**: \( 2x \times 6 = 12x \)
- For **R**: \( 3x \times 4 = 12x \)

### Step 3: Determine the ratio of their investments.
The total investment ratio is proportional to their effective capital contributions:
\[
\text{Ratio of P:Q:R} = 12x : 12x : 12x = 1:1:1
\]

### Step 4: Calculate Q's share of the profit.
Since the ratio of investments is \( 1:1:1 \), the total profit will be divided equally among P, Q, and R.

The total profit is Rs. 9,000. Therefore, Q's share is:
\[
\text{Q's Share} = \frac{9000}{3} = 3000 \text{ Rs.}
\]

Thus, Q's share of the profit is Rs. 3,000.

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Do you want any further details or have any questions? 

Here are some related questions:
1. How is the effective capital ratio calculated in partnership?
2. How would the result change if R invested earlier than 8 months?
3. What would happen if P, Q, and R invested different amounts for the same time periods?
4. How is profit distributed in a partnership if no information on time or capital is provided?
5. What is the general formula for calculating profit share in a partnership?

**Tip**: In partnership problems, both the amount of investment and the time for which the investment is made are crucial in determining the profit share.

Learn how to calculate profit sharing in a partnership where partners invest varying amounts at different times. This example explores how P, Q, and R share an annual profit of Rs. 9,000 based on their time-adjusted investments. Suitable for students in Grades 9-11.

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