The ratio of difference between area of a rectangle obtained in two cases, first when length of a rectangle is decreased by 4 cm and second when breadth of the original rectangle is increased by 4 cm to the area of rectangle is 4 : 9. Find ratio of numerical value perimeter to area of rectangle.

The correct option is 2 : 9.

Let us break down the logical reasoning and mathematical steps to find the ratio of the numerical value of the perimeter to the area of the rectangle.

Step 1: Define the variables and original formulas

Let the length of the rectangle be $L$ cm and the breadth of the rectangle be $B$ cm.

The area of the original rectangle is:

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

The perimeter of the original rectangle is:

$\text{Perimeter}=2(L+B)$

Step 2: Calculate the areas for both cases

Case 1: Length is decreased by 4 cm while breadth remains unchanged.

$\text{New Length}=L-4$

${\text{Area}}_1=(L-4)B=LB-4B$

Case 2: Breadth is increased by 4 cm while length remains unchanged.

$\text{New Breadth}=B+4$

${\text{Area}}_2=L(B+4)=LB+4L$

Step 3: Find the difference between the areas obtained in the two cases

The difference between the area in Case 2 and Case 1 is calculated as:

$\text{Difference}={\text{Area}}_2-{\text{Area}}_1$

$=(LB+4L)-(LB-4B)$

$=LB+4L-LB+4B$

$=4L+4B=4(L+B)$

Step 4: Formulate the equation using the given ratio

We are given that the ratio of this difference to the original area of the rectangle is 4 : 9.

$\frac{4(L+B)}{LB}=\frac{4}{9}$

Dividing both sides of the equation by 4:

$\frac{L+B}{LB}=\frac{1}{9}$

Step 5: Determine the ratio of Perimeter to Area

The ratio of the numerical value of the perimeter to the area of the rectangle is:

$\frac{\text{Perimeter}}{\text{Area}}=\frac{2(L+B)}{LB}$

Substituting $\frac{L+B}{LB}=\frac{1}{9}$ into the expression:

$\frac{\text{Perimeter}}{\text{Area}}=2\times \frac{1}{9}=\frac{2}{9}$

Therefore, the required ratio of perimeter to area is 2 : 9.