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.

To find the original length of the rectangle, we can set up an algebraic solution based on the ratio of its dimensions.

Step 1: Express the original dimensions and area

The ratio of the length to the breadth of the rectangle is 3 : 2. Let the original length and breadth be:

$\text{Length}\left(L\right)=3x$

$\text{Breadth}\left(B\right)=2x$

where $x$ is a positive variable.

The original area of the rectangle is calculated as:

$\text{Original Area}\left(A_1\right)=L\times B=3x\times 2x=6x^2$

Step 2: Express the new dimensions and area

The length is increased by 25%, which means the new length is:

$L^{\prime }=3x\times (1+0.25)=3.75x$

The breadth remains the same:

$B^{\prime }=2x$

The new area of the rectangle is:

$\text{New Area}\left(A_2\right)=L^{\prime }\times B^{\prime }=3.75x\times 2x=7.5x^2$

Step 3: Solve for $x$ using the increase in area

We are given that the area increases by 24 m². Therefore:

$A_2-A_1=24$

Substitute the expressions for $A_1$ and $A_2$:

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

$1.5x^2=24$

Solve for $x^2$ by dividing both sides by 1.5:

$x^2=\frac{24}{1.5}=16$

Since $x$ must be positive, we take the square root of 16:

$x=4$

Step 4: Find the original length

The original length is $3x$ meters:

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