The breadth of a rectangular hall is three-fourth of its length. If the area of the floor is 768 sq. m., then the difference between the length and breadth of the hall is :

Options

A. 8 metres
B. 12 metres
C. 24 metres
D. 32 metres

Answer

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

### Step 1: Define the variables
Let the length of the rectangular hall be \( L \) meters. According to the problem, the breadth \( B \) is three-fourths of the length. Therefore, we can express the breadth as:
\[ B = \frac{3}{4}L \]

### Step 2: Write the formula for the area
The area \( A \) of a rectangle is given by the formula:
\[ A = L \times B \]
Substituting the expression for breadth into the area formula, we get:
\[ A = L \times \left(\frac{3}{4}L\right) = \frac{3}{4}L^2 \]

### Step 3: Set up the equation using the given area
We know that the area of the floor is 768 sq. m. Therefore, we can set up the equation:
\[ \frac{3}{4}L^2 = 768 \]

### Step 4: Solve for \( L^2 \)
To eliminate the fraction, multiply both sides of the equation by 4:
\[ 3L^2 = 768 \times 4 \]
\[ 3L^2 = 3072 \]
Now, divide both sides by 3:
\[ L^2 = \frac{3072}{3} = 1024 \]

### Step 5: Solve for \( L \)
Now, take the square root of both sides to find \( L \):
\[ L = \sqrt{1024} = 32 \text{ meters} \]

### Step 6: Find the breadth \( B \)
Now that we have the length, we can find the breadth using the expression for breadth:
\[ B = \frac{3}{4}L = \frac{3}{4} \times 32 = 24 \text{ meters} \]

### Step 7: Calculate the difference between length and breadth
Finally, we find the difference between the length and the breadth:
\[ \text{Difference} = L - B = 32 - 24 = 8 \text{ meters} \]

### Final Answer
The difference between the length and breadth of the hall is **8 meters**.

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