The length and breadth of a rectangle is 3 : 2 respectively. When length of the rectangle increased by 25% and the breadth remain same, then the area of the rectangle is increased by 24 m². Find original length (in meters) of the rectangle.

Correct Answer: Option 12

Let's solve the problem step-by-step:

Step 1: Define the original dimensions of the rectangle
The ratio of the length to the breadth of the rectangle is given as 3 : 2.
Let the original length of the rectangle be $3x$ meters.
Let the original breadth of the rectangle be $2x$ meters.

Step 2: Calculate the original area
The area of a rectangle is calculated as length multiplied by breadth:
$\text{Original Area}=3x\times 2x=6x^2{\text{m}}^2$

Step 3: Determine the new dimensions and the new area
The length of the rectangle is increased by 25%:
$\text{New Length}=3x+(25\%\text{of}3x)=3x\times 1.25=3.75x\text{meters}$
The breadth remains the same:
$\text{New Breadth}=2x\text{meters}$
Now, calculate the new area:
$\text{New Area}=3.75x\times 2x=7.5x^2{\text{m}}^2$

Step 4: Set up the equation using the change in area
We are given that the area increased by 24 m². Therefore:
$\text{New Area}-\text{Original Area}=24$
Substitute the expressions we found:
$7.5x^2-6x^2=24$
Simplify the equation:
$1.5x^2=24$
Divide both sides by 1.5:
$x^2=16$
Taking the square root (since dimensions must be positive):
$x=4$

Step 5: Find the original length of the rectangle
Now, we can find the original length:
$\text{Original Length}=3x=3\times 4=12\text{meters}$