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, we will follow these steps:

### Step 1: Determine the total area of the walls to be papered.

Given:
- Cost of papering = Rs. 240
- Cost per square meter = 75 paise = Rs. 0.75

To find the total area of the walls, we can use the formula:
\[ \text{Total Area} = \frac{\text{Total Cost}}{\text{Cost per square meter}} \]

Calculating the total area:
\[ \text{Total Area} = \frac{240}{0.75} = 320 \text{ square meters} \]

### Step 2: Use the formula for the surface area of the four walls.

The formula for the surface area of the four walls of a room is:
\[ \text{Area} = 2h(l + b) \]
where \( h \) is the height of the room, \( l \) is the length, and \( b \) is the breadth.

Given:
- Height \( h = 5 \) meters
- Area \( = 320 \) square meters

Substituting the values into the formula:
\[ 320 = 2 \times 5 \times (l + b) \]
\[ 320 = 10(l + b) \]
\[ l + b = \frac{320}{10} = 32 \text{ meters} \]

### Step 3: Set up the ratio of length and breadth.

We know that the length and breadth are in the ratio \( 5:3 \). Let:
- Length \( l = 5x \)
- Breadth \( b = 3x \)

### Step 4: Substitute the expressions for length and breadth into the equation.

Substituting \( l \) and \( b \) into the equation \( l + b = 32 \):
\[ 5x + 3x = 32 \]
\[ 8x = 32 \]
\[ x = \frac{32}{8} = 4 \]

### Step 5: Calculate the length and breadth.

Now substituting \( x \) back to find \( l \) and \( b \):
\[ l = 5x = 5 \times 4 = 20 \text{ meters} \]
\[ b = 3x = 3 \times 4 = 12 \text{ meters} \]

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

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