P and Q started a business by investing in ratio 5:6 .After 6 months R 
joined them who invested equal amount as Q. At the end of the year 20% 
profit is gained and P received Rs. 3400 then find capital invested by P.  

Let's break down the problem step by step:

### Given Data:
1. **P and Q's investment ratio**: 5:6
2. **R joins after 6 months** and invests the same amount as Q.
3. **P's share of profit**: Rs. 3400
4. **Total profit at the end of the year**: 20%

We are tasked with finding **the capital invested by P**.

### Step 1: Assume Capital
Let P's capital investment be \( 5x \) and Q's capital investment be \( 6x \), as per the given ratio.

When R joins after 6 months, he invests the same as Q, i.e., \( 6x \).

### Step 2: Time-weighted shares of investment
- P invested \( 5x \) for the entire year, so his time-weighted investment is:
  \[
  5x \times 12 = 60x
  \]
- Q invested \( 6x \) for the entire year, so his time-weighted investment is:
  \[
  6x \times 12 = 72x
  \]
- R invested \( 6x \), but only for 6 months, so his time-weighted investment is:
  \[
  6x \times 6 = 36x
  \]

### Step 3: Total time-weighted investment
The total time-weighted investment for P, Q, and R is:
\[
60x + 72x + 36x = 168x
\]

### Step 4: Profit Sharing
The profit is shared in the ratio of their time-weighted investments. Hence, the profit share of P is in the ratio:
\[
\frac{60x}{168x} = \frac{5}{14}
\]

### Step 5: P's Profit
We know P's profit share is Rs. 3400. Let the total profit be \( P_{\text{total}} \).

So, \( \frac{5}{14} \) of the total profit is equal to Rs. 3400:
\[
\frac{5}{14} \times P_{\text{total}} = 3400
\]
Solving for \( P_{\text{total}} \):
\[
P_{\text{total}} = \frac{3400 \times 14}{5} = 9520
\]

### Step 6: Total Capital Invested
Since the total profit is 20% of the total capital invested, we have:
\[
\text{Total capital} \times 0.20 = 9520
\]
Solving for total capital:
\[
\text{Total capital} = \frac{9520}{0.20} = 47600
\]

### Step 7: P's Capital
Since P's investment was \( 5x \), and the total capital invested was \( 5x + 6x + 6x = 17x \), we have:
\[
17x = 47600
\]
Solving for \( x \):
\[
x = \frac{47600}{17} = 2800
\]
Therefore, P's capital is \( 5x = 5 \times 2800 = 14000 \).

### Final Answer:
P invested **Rs. 14,000**.

---

Would you like more details or have any questions?

Here are some related questions:
1. What if R had joined after 8 months instead of 6?
2. How would the solution change if P received Rs. 4000 instead of Rs. 3400?
3. What if the profit percentage was 25% instead of 20%?
4. How would the calculation change if Q's investment was doubled after 6 months?
5. What if P, Q, and R all invested the same amount?

**Tip**: When dealing with profit-sharing problems, focus on calculating time-weighted investments to distribute profits correctly.

P and Q started a business by investing in ratio 5:6. After 6 months, R joined them who invested an equal amount as Q. At the end of the year, 20% profit is gained, and P received Rs. 3400. Find the capital invested by P.

Solve a business problem where P and Q invest in a 5:6 ratio, and R joins after 6 months. Learn how to calculate P's capital using time-weighted investments and profit-sharing formulas, suitable for Grades 9-12.

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