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.

The correct option is 12.

Step 1: Define the initial dimensions of the rectangle

Let the ratio constant be $x$.

According to the question, the ratio of the length to the breadth of the rectangle is $3:2$.

Therefore, we can write:

Original Length, $L=3x$

Original Breadth, $B=2x$

Step 2: Calculate the original area of the rectangle

The area of a rectangle is given by the product of its length and breadth:

$\text{Original Area}=L\times B$

$\text{Original Area}=3x\times 2x=6x^2$

Step 3: Determine the new dimensions and new area

When the length is increased by 25% and the breadth remains the same:

New Length, $L^{\prime }=3x+25\%\text{of}3x=3x\times 1.25=3.75x$

New Breadth, $B^{\prime }=2x$

Now, calculate the new area:

$\text{New Area}=L^{\prime }\times B^{\prime }$

$\text{New Area}=3.75x\times 2x=7.5x^2$

Step 4: Set up the equation using the given increase in area

We are given that the area of the rectangle increases by 24 m².

$\text{Increase in Area}=\text{New Area}-\text{Original Area}$

$7.5x^2-6x^2=24$

$1.5x^2=24$

$x^2=\frac{24}{1.5}$

$x^2=16$

$x=\sqrt{16}=4$

Step 5: Find the original length of the rectangle

Now substitute the value of $x=4$ back into the expression for original length:

$\text{Original Length}=3x=3\times 4=12\text{meters}$

Thus, the original length of the rectangle is 12 meters.