P invests ₹ 6,000 for X months while Q invests ₹ 8,000 for 9 months in a scheme. The profit share of Q is t ₹ 24,000 out of total profit ₹ 42,000. Then find the value of X?

Options

A. 6 months
B. 9 months
C. 8 months
D. 7 months

Explanation

To solve the problem step by step, we will follow the logic presented in the video transcript:

### Step 1: Identify the given values
- P's investment = ₹ 6,000
- Q's investment = ₹ 8,000
- Q's investment duration = 9 months
- Total profit = ₹ 42,000
- Q's profit = ₹ 24,000

### Step 2: Calculate P's profit
To find P's profit, we subtract Q's profit from the total profit:
\[ \text{P's profit} = \text{Total profit} - \text{Q's profit} = 42,000 - 24,000 = ₹ 18,000 \]

### Step 3: Set up the profit-sharing ratio
The profit-sharing ratio is based on the product of the investment and the time for which the investment is made. We can express this as:
\[ \frac{\text{Q's profit}}{\text{P's profit}} = \frac{\text{Q's investment} \times \text{Q's time}}{\text{P's investment} \times \text{P's time}} \]
Substituting the known values:
\[ \frac{24,000}{18,000} = \frac{8,000 \times 9}{6,000 \times X} \]

### Step 4: Simplify the profit ratio
We can simplify the left side:
\[ \frac{24}{18} = \frac{4}{3} \]
So, we have:
\[ \frac{4}{3} = \frac{8,000 \times 9}{6,000 \times X} \]

### Step 5: Cross-multiply to solve for X
Cross-multiplying gives us:
\[ 4 \times (6,000 \times X) = 3 \times (8,000 \times 9) \]
This simplifies to:
\[ 24,000X = 216,000 \]

### Step 6: Solve for X
Now, divide both sides by 24,000:
\[ X = \frac{216,000}{24,000} = 9 \]

### Conclusion
Thus, the value of \( X \) is 9 months.

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