A, B, C and D purchased a cine-multiplex for Rs. 56 lakhs. The contribution of B, C and D together is 460% that of A, alone. The contribution of A, C and D together is 366.66% that of B's contribution and the contribution of C is 40% that of A, B and D together. The amount contibuted by D is :

Options

A. 10 lakh
B. 12 lakh
C. 16 lakh
D. 18 lakh

Answer

To solve the problem step by step, we will denote the contributions of A, B, C, and D as \( a \), \( b \), \( c \), and \( d \) respectively.

### Step 1: Set up the equations based on the given information.

1. The total contribution of A, B, C, and D is:
\[ a + b + c + d = 56 \text{ lakhs} \quad \text{(1)} \]

2. The contribution of B, C, and D together is 460% of A:
\[ b + c + d = 4.6a \quad \text{(2)} \]

3. The contribution of A, C, and D together is 366.66% of B:
\[ a + c + d = \frac{11}{3}b \quad \text{(3)} \]

4. The contribution of C is 40% of A, B, and D together:
\[ c = 0.4(a + b + d) \quad \text{(4)} \]

### Step 2: Solve the equations.

From equation (1):
\[ a + b + c + d = 56 \]

From equation (2):
\[ b + c + d = 4.6a \]

Subtract equation (2) from equation (1):
\[ a = 56 - (4.6a) \implies a + 4.6a = 56 \implies 5.6a = 56 \implies a = \frac{56}{5.6} = 10 \text{ lakhs} \]

### Step 3: Substitute \( a \) back into the equations.

Now substituting \( a = 10 \) lakhs into equation (2):
\[ b + c + d = 4.6 \times 10 = 46 \text{ lakhs} \quad \text{(5)} \]

Substituting \( a = 10 \) lakhs into equation (3):
\[ 10 + c + d = \frac{11}{3}b \quad \text{(6)} \]

### Step 4: Use equation (4) to find \( c \).

Substituting \( a = 10 \) into equation (4):
\[ c = 0.4(10 + b + d) \implies c = 4 + 0.4b + 0.4d \quad \text{(7)} \]

### Step 5: Substitute \( c \) from equation (7) into equations (5) and (6).

Substituting \( c \) from equation (7) into equation (5):
\[ b + (4 + 0.4b + 0.4d) + d = 46 \]
\[ b + 4 + 0.4b + 0.4d + d = 46 \]
\[ 1.4b + 1.4d = 42 \implies b + d = 30 \quad \text{(8)} \]

Substituting \( c \) from equation (7) into equation (6):
\[ 10 + (4 + 0.4b + 0.4d) + d = \frac{11}{3}b \]
\[ 14 + 0.4b + 0.4d + d = \frac{11}{3}b \]
\[ 14 + 0.4b + 1.4d = \frac{11}{3}b \]
Multiply through by 3 to eliminate the fraction:
\[ 42 + 1.2b + 4.2d = 11b \]
Rearranging gives:
\[ 42 = 11b - 1.2b - 4.2d \implies 42 = 9.8b - 4.2d \quad \text{(9)} \]

### Step 6: Solve equations (8) and (9).

From equation (8):
\[ d = 30 - b \]

Substituting \( d \) into equation (9):
\[ 42 = 9.8b - 4.2(30 - b) \]
\[ 42 = 9.8b - 126 + 4.2b \]
\[ 42 + 126 = 14b \implies 168 = 14b \implies b = 12 \text{ lakhs} \]

### Step 7: Find \( d \).

Substituting \( b = 12 \) into equation (8):
\[ d = 30 - 12 = 18 \text{ lakhs} \]

### Step 8: Find \( c \).

Substituting \( b = 12 \) into equation (7):
\[ c = 4 + 0.4(12) + 0.4(18) = 4 + 4.8 + 7.2 = 16 \text{ lakhs} \]

### Conclusion

The contributions are:
- \( a = 10 \) lakhs
- \( b = 12 \) lakhs
- \( c = 16 \) lakhs
- \( d = 18 \) lakhs

Thus, the amount contributed by D is **18 lakhs**.