Rs.5625 is to be divided among Kailash, Raj and Pushap so that Kailash may receive `1/2` as much as Raj and Pushap together recive. Raj receives `1/4` of what Kailash and Pushap together receive. The share of Kailash is more than that of Raj by:

Options

A. 1125
B. 1875
C. 2625
D. None

Answer

To solve the problem, we need to set up equations based on the information given about the shares of Kailash, Raj, and Pushap.

### Step 1: Define Variables
Let:
- \( K \) = share of Kailash
- \( R \) = share of Raj
- \( P \) = share of Pushap

### Step 2: Set Up Equations
From the problem, we have the following relationships:
1. Kailash receives half as much as Raj and Pushap together:
\[ K = \frac{1}{2}(R + P) \]
2. Raj receives one fourth of what Kailash and Pushap together receive:
\[ R = \frac{1}{4}(K + P) \]

### Step 3: Express \( K \) and \( R \) in Terms of \( P \)
From the first equation, we can express \( K \):
\[ K = \frac{1}{2}(R + P) \quad \text{(1)} \]

From the second equation, we can express \( R \):
\[ R = \frac{1}{4}(K + P) \quad \text{(2)} \]

### Step 4: Substitute Equation (1) into Equation (2)
Substituting \( K \) from equation (1) into equation (2):
\[ R = \frac{1}{4}\left(\frac{1}{2}(R + P) + P\right) \]
Simplifying:
\[ R = \frac{1}{4}\left(\frac{1}{2}R + \frac{1}{2}P + P\right) \]
\[ R = \frac{1}{4}\left(\frac{1}{2}R + \frac{3}{2}P\right) \]
Multiplying through by 4:
\[ 4R = \frac{1}{2}R + 3P \]
Rearranging:
\[ 4R - \frac{1}{2}R = 3P \]
\[ \frac{8R - R}{2} = 3P \]
\[ \frac{7R}{2} = 3P \]
Thus,
\[ R = \frac{6P}{7} \quad \text{(3)} \]

### Step 5: Substitute \( R \) from Equation (3) into Equation (1)
Now substituting \( R \) from equation (3) back into equation (1):
\[ K = \frac{1}{2}\left(\frac{6P}{7} + P\right) \]
Converting \( P \) to a fraction:
\[ K = \frac{1}{2}\left(\frac{6P}{7} + \frac{7P}{7}\right) \]
\[ K = \frac{1}{2}\left(\frac{13P}{7}\right) \]
\[ K = \frac{13P}{14} \quad \text{(4)} \]

### Step 6: Total Amount Equation
The total amount distributed among Kailash, Raj, and Pushap is Rs. 5625:
\[ K + R + P = 5625 \]
Substituting equations (3) and (4):
\[ \frac{13P}{14} + \frac{6P}{7} + P = 5625 \]
Finding a common denominator (which is 14):
\[ \frac{13P}{14} + \frac{12P}{14} + \frac{14P}{14} = 5625 \]
Combining the fractions:
\[ \frac{39P}{14} = 5625 \]
Multiplying both sides by 14:
\[ 39P = 5625 \times 14 \]
Calculating:
\[ 39P = 78750 \]
Dividing by 39:
\[ P = \frac{78750}{39} = 2019.23 \quad \text{(approximately)} \]

### Step 7: Calculate \( R \) and \( K \)
Using \( P \) to find \( R \) and \( K \):
\[ R = \frac{6 \times 2019.23}{7} \approx 1732.24 \]
\[ K = \frac{13 \times 2019.23}{14} \approx 1843.23 \]

### Step 8: Find the Difference Between \( K \) and \( R \)
Now, we find the difference between Kailash's and Raj's shares:
\[ K - R = 1843.23 - 1732.24 \approx 110.99 \]

### Final Answer
The share of Kailash is more than that of Raj by approximately Rs. 111.