The cost of papering the four walls of a room at 75 paise per square metre is Rs 240. The height of the room is 5 metres. Find the length and the breadth of the room, if they are in the ratio `5: 3`.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Calculate the Area of the Walls
Given:
- Cost of papering per square meter = 75 paise = 0.75 rupees
- Total cost of papering = Rs 240

To find the area of the walls, we use the formula:
\[ \text{Area} = \frac{\text{Total Cost}}{\text{Cost per square meter}} \]
Substituting the values:
\[ \text{Area} = \frac{240}{0.75} = 320 \text{ square meters} \]

### Step 2: Set Up the Ratio for Length and Breadth
Let the length (L) and breadth (B) of the room be in the ratio of 5:3. We can express them as:
\[ L = 5x \quad \text{and} \quad B = 3x \]

### Step 3: Use the Formula for the Surface Area of the Four Walls
The surface area of the four walls of the room can be calculated using the formula:
\[ \text{Surface Area} = 2H(L + B) \]
Where H is the height of the room. Given that the height (H) is 5 meters:
\[ \text{Surface Area} = 2 \times 5 \times (L + B) = 10(L + B) \]

### Step 4: Substitute Length and Breadth into the Surface Area Formula
Substituting L and B:
\[ \text{Surface Area} = 10(5x + 3x) = 10(8x) = 80x \]

### Step 5: Set the Surface Area Equal to the Calculated Area
We know from Step 1 that the area is 320 square meters:
\[ 80x = 320 \]

### Step 6: Solve for x
Dividing both sides by 80:
\[ x = \frac{320}{80} = 4 \]

### Step 7: Calculate Length and Breadth
Now that we have the value of x, we can find the length and breadth:
- Length:
\[ L = 5x = 5 \times 4 = 20 \text{ meters} \]
- Breadth:
\[ B = 3x = 3 \times 4 = 12 \text{ meters} \]

### Final Answer
The length of the room is 20 meters, and the breadth of the room is 12 meters.

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