A,B, C and D purchased a cine-multiplex for ₹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 contributed 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 define the contributions of A, B, C, and D in terms of units and then calculate the amount contributed by each person.

### Step 1: Define the contributions
Let:
- A's contribution = A
- B's contribution = B
- C's contribution = C
- D's contribution = D

According to the problem, the total contribution is:
\[ A + B + C + D = 56 \text{ lakhs} \]

### Step 2: Set up the equations based on the given percentages
1. The contribution of B, C, and D together is 460% of A's contribution:
\[ B + C + D = 4.6A \]

2. The contribution of A, C, and D together is 366.66% of B's contribution:
\[ A + C + D = 3.6666B \quad \text{(which can be simplified to } \frac{11}{3}B\text{)} \]

3. The contribution of C is 40% of A, B, and D together:
\[ C = 0.4(A + B + D) \]

### Step 3: Express B, C, and D in terms of A
From the first equation:
\[ B + C + D = 4.6A \implies C + D = 4.6A - B \]

From the second equation:
\[ A + C + D = \frac{11}{3}B \implies C + D = \frac{11}{3}B - A \]

Setting the two expressions for \(C + D\) equal:
\[ 4.6A - B = \frac{11}{3}B - A \]

### Step 4: Solve for B in terms of A
Rearranging gives:
\[ 4.6A + A = \frac{11}{3}B + B \]
\[ 5.6A = \frac{14}{3}B \]
Multiplying through by 3:
\[ 16.8A = 14B \implies B = \frac{16.8}{14}A = 1.2A \]

### Step 5: Substitute B back to find C and D
Now substituting \(B\) back into the equations:
Using \(B = 1.2A\) in \(C + D = 4.6A - B\):
\[ C + D = 4.6A - 1.2A = 3.4A \]

Using \(C = 0.4(A + B + D)\):
Substituting \(B\):
\[ C = 0.4(A + 1.2A + D) = 0.4(2.2A + D) \]

### Step 6: Solve for C and D
Now we have two equations:
1. \(C + D = 3.4A\)
2. \(C = 0.4(2.2A + D)\)

Substituting \(C\) from the second equation into the first:
\[ 0.4(2.2A + D) + D = 3.4A \]
\[ 0.88A + 0.4D + D = 3.4A \]
\[ 0.88A + 1.4D = 3.4A \]
\[ 1.4D = 3.4A - 0.88A \]
\[ 1.4D = 2.52A \implies D = \frac{2.52}{1.4}A = 1.8A \]

### Step 7: Substitute back to find C
Now substituting \(D\) back into \(C + D = 3.4A\):
\[ C + 1.8A = 3.4A \implies C = 3.4A - 1.8A = 1.6A \]

### Step 8: Find the total contributions
Now we have:
- \(A\) = A
- \(B\) = 1.2A
- \(C\) = 1.6A
- \(D\) = 1.8A

Adding these contributions gives:
\[ A + 1.2A + 1.6A + 1.8A = 56 \]
\[ 5.6A = 56 \implies A = 10 \text{ lakhs} \]

### Step 9: Calculate contributions
Now we can find the contributions:
- \(B = 1.2 \times 10 = 12 \text{ lakhs}\)
- \(C = 1.6 \times 10 = 16 \text{ lakhs}\)
- \(D = 1.8 \times 10 = 18 \text{ lakhs}\)

### Final Answer
The amount contributed by D is:
\[ \boxed{18 \text{ lakhs}} \]