To solve the problem, we need to determine the cost of painting the four walls of a rectangular hall, given the dimensions of the hall and the cost per square meter. We will analyze the two statements provided to see if they give us enough information to answer the question.
### Step-by-Step Solution:
1. **Understand the Question**: We need to find the cost of painting the four walls of a rectangular hall at a rate of ₹135 per square meter. The hall has a door measuring 3.5 m x 1.5 m and no windows.
2. **Calculate the Area of the Four Walls**:
- The formula for the area of the four walls of a rectangular hall is:
\[
\text{Area of four walls} = 2 \times (L + B) \times H
\]
where \(L\) is the length, \(B\) is the breadth, and \(H\) is the height of the hall.
3. **Calculate the Area of the Door**:
- The area of the door is given by:
\[
\text{Area of door} = \text{height} \times \text{width} = 3.5 \, \text{m} \times 1.5 \, \text{m} = 5.25 \, \text{m}^2
\]
4. **Final Area to be Painted**:
- The area to be painted will be:
\[
\text{Area to be painted} = \text{Area of four walls} - \text{Area of door}
\]
5. **Cost Calculation**:
- The total cost of painting will be:
\[
\text{Cost} = \text{Area to be painted} \times 135
\]
6. **Analyze Statement I**:
- Statement I tells us that the perimeter of the floor of the hall is equal to the perimeter of a square field with a side of 12 m.
- The perimeter of the square field is:
\[
\text{Perimeter} = 4 \times 12 = 48 \, \text{m}
\]
- Since the length and width are in the ratio of 5:1, we can let \(L = 5x\) and \(B = x\).
- The perimeter of the rectangular hall is given by:
\[
2(L + B) = 48 \implies L + B = 24
\]
- Substituting \(L\) and \(B\):
\[
5x + x = 24 \implies 6x = 24 \implies x = 4
\]
- Therefore, \(L = 20 \, \text{m}\) and \(B = 4 \, \text{m}\).
7. **Analyze Statement II**:
- Statement II states that the perimeter of a smaller wall is 15 m.
- The smaller wall can either be the wall with length \(L\) or breadth \(B\). Since we have already determined the dimensions from Statement I, we can find the height \(H\) using the perimeter formula:
\[
2(B + H) = 15 \implies B + H = 7.5
\]
- Substituting \(B = 4\):
\[
4 + H = 7.5 \implies H = 3.5 \, \text{m}
\]
8. **Final Calculation**:
- Now we have \(L = 20 \, \text{m}\), \(B = 4 \, \text{m}\), and \(H = 3.5 \, \text{m}\).
- Calculate the area of the four walls:
\[
\text{Area of four walls} = 2 \times (20 + 4) \times 3.5 = 2 \times 24 \times 3.5 = 168 \, \text{m}^2
\]
- Subtract the area of the door:
\[
\text{Area to be painted} = 168 - 5.25 = 162.75 \, \text{m}^2
\]
- Finally, calculate the cost:
\[
\text{Cost} = 162.75 \times 135 = 21971.25
\]
### Conclusion:
The cost of painting the four walls of the hall is ₹21,971.25.
