Question

The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that,

➢ Length of a rectangle be x units.

➢ Breadth of rectangle be y units.

➢ So, Area of rectangle = xy

According to first condition

The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units.

Thus,

➢ Length of a rectangle = x - 5 units.

➢ Breadth of rectangle = y + 3 units.

So, Area of rectangle = (x - 5)(y + 3) square units

Also, Area of rectangle = xy - 9 square units

Thus, we have

$\rm :\longmapsto\:(x - 5)(y + 3) = xy - 9$

$\rm :\longmapsto\:xy + 3x - 5y - 15 = xy - 9$

$\rm :\longmapsto\:3x - 5y - 15 = - 9$

$\rm :\longmapsto\:3x - 5y = - 9 + 15$

$\rm :\longmapsto\:3x - 5y = 6 - - - - (1)$

According to second condition

If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units.

So,

➢ Length of a rectangle = x + 3 units.

➢ Breadth of rectangle = y + 2 units.

So, Area of rectangle = (x + 3)(y + 2) square units

Also, Area of rectangle = xy + 67 square units

Thus, we have

$\rm :\longmapsto\:(x + 3)(y + 2) = xy + 67$

$\rm :\longmapsto\:2x + 3y + xy + 6 = xy + 67$

$\rm :\longmapsto\:2x + 3y + 6 = 67$

$\rm :\longmapsto\:2x + 3y = 67 - 6$

$\rm :\longmapsto\:2x + 3y = 61 - - - - (2)$

On multiply equation (1) by 2 and equation (2) by 3, we get

$\rm :\longmapsto\:6x - 10y = 12 - - - - (3)$

and

$\rm :\longmapsto\:6x + 9y = 183 - - - - (4)$

On Subtracting equation (3) from equation (4), we get

$\rm :\longmapsto\:19y = 171$

$\rm :\longmapsto\:y = 9$

On substituting the value of y in equation (1), we get

$\rm :\longmapsto\:3x - 5(9)= 6$

$\rm :\longmapsto\:3x - 45= 6$

$\rm :\longmapsto\:3x = 6 + 45$

$\rm :\longmapsto\:3x = 51$

$\rm :\longmapsto\:x = 17$

Hence,

Length of a rectangle = 19 units.

Breadth of rectangle = 17 units.