P and Q started business with Rs 600 and Rs 500 respectively. After 4 months. R replaces Q with `x%` of Q's capital. After 1 year R's share out of the toatal profit 48000 is 11200. Find the value of x.

Options

A. 60
B. 70
C. 75
D. 66

Explanation

To solve the problem step by step, let's break it down:

### Step 1: Determine the investment and time for P and Q
- P's investment = Rs 600 for 12 months.
- Q's investment = Rs 500 for 4 months (since Q left after 4 months).

### Step 2: Determine R's investment
- R replaces Q with `x%` of Q's capital, which is `x%` of Rs 500.
- Therefore, R's investment = \( \frac{x}{100} \times 500 = 5x \) (for 8 months, since R invests after Q leaves).

### Step 3: Calculate the effective capital contribution
- P's effective capital contribution = \( 600 \times 12 = 7200 \)
- Q's effective capital contribution = \( 500 \times 4 = 2000 \)
- R's effective capital contribution = \( 5x \times 8 = 40x \)

### Step 4: Set up the ratio of investments
The total effective capital contributions can be expressed as:
\[ \text{Total contributions} = P : Q : R = 7200 : 2000 : 40x \]

### Step 5: Simplify the ratio
To simplify the ratio, we can divide each term by 2000:
\[ P : Q : R = \frac{7200}{2000} : \frac{2000}{2000} : \frac{40x}{2000} = 3.6 : 1 : 0.02x \]
To eliminate the decimal, we can multiply through by 10:
\[ P : Q : R = 36 : 10 : 0.2x \]

### Step 6: Find R's share of profit
Given that R's share of the total profit (Rs 48,000) is Rs 11,200, we can set up the equation:
\[ \frac{R's \, share}{Total \, profit} = \frac{11200}{48000} \]
This can be simplified to:
\[ \frac{R's \, share}{Total \, share} = \frac{11200}{48000} = \frac{7}{30} \]

### Step 7: Express R's share in terms of the ratio
From the ratio \( P : Q : R = 36 : 10 : 0.2x \), we can express R's share as:
\[ R's \, share = \frac{0.2x}{36 + 10 + 0.2x} \times 48000 \]

### Step 8: Set up the equation
Setting the two expressions for R's share equal gives:
\[ \frac{0.2x}{46 + 0.2x} \times 48000 = 11200 \]

### Step 9: Solve for x
Cross-multiplying gives:
\[ 0.2x \times 48000 = 11200 \times (46 + 0.2x) \]
\[ 9600x = 515200 + 2240x \]
\[ 9600x - 2240x = 515200 \]
\[ 7360x = 515200 \]
\[ x = \frac{515200}{7360} = 70 \]

### Conclusion
The value of \( x \) is 70.