Divide Rs. , `800` amongst `A,B,C` and `D` so that `A` and `B` receives three times of the part of `C` and `D` both and `B` gets `4` times the share of `C` and `C` gets `1-(1)/(2)` times of `D'` s part , then the share of `A` is .

Options

A. `1170`
B. `780`
C. `800`
D. `120`

Explanation

To solve the problem of dividing Rs. 800 among A, B, C, and D based on the given conditions, we can follow these steps:

### Step 1: Define Variables
Let the shares of A, B, C, and D be represented as:
- A = a
- B = b
- C = c
- D = d

### Step 2: Set Up the Equations Based on Given Conditions
1. A and B receive three times the part of C and D combined:
\[ a + b = 3(c + d) \quad \text{(Equation 1)} \]

2. B gets four times the share of C:
\[ b = 4c \quad \text{(Equation 2)} \]

3. C gets \(1 - \frac{1}{2}\) times the part of D, which simplifies to:
\[ c = \frac{1}{2}d \quad \text{(Equation 3)} \]

### Step 3: Substitute Equation 3 into Equation 2
From Equation 3, we can express d in terms of c:
\[ d = 2c \]
Substituting this into Equation 2:
\[ b = 4c \quad \text{(remains the same)} \]

### Step 4: Substitute Equations 2 and 3 into Equation 1
Now substitute \(b = 4c\) and \(d = 2c\) into Equation 1:
\[ a + 4c = 3(c + 2c) \]
This simplifies to:
\[ a + 4c = 3(3c) \quad \Rightarrow \quad a + 4c = 9c \]
Rearranging gives:
\[ a = 9c - 4c = 5c \quad \text{(Equation 4)} \]

### Step 5: Express All Shares in Terms of c
Now we have:
- A = 5c
- B = 4c
- C = c
- D = 2c

### Step 6: Set Up the Total Share Equation
The total amount shared is Rs. 800:
\[ a + b + c + d = 800 \]
Substituting the expressions from above:
\[ 5c + 4c + c + 2c = 800 \]
This simplifies to:
\[ 12c = 800 \]

### Step 7: Solve for c
Now, divide both sides by 12:
\[ c = \frac{800}{12} = \frac{200}{3} \approx 66.67 \]

### Step 8: Calculate Shares of A, B, C, and D
Now substitute back to find the shares:
- C = \(c = \frac{200}{3}\)
- D = \(d = 2c = \frac{400}{3}\)
- B = \(b = 4c = \frac{800}{3}\)
- A = \(a = 5c = \frac{1000}{3}\)

### Step 9: Calculate A's Share
To find A's share:
\[ a = 5c = 5 \times \frac{200}{3} = \frac{1000}{3} \approx 333.33 \]

### Final Answer
Thus, the share of A is approximately Rs. 333.33.

---