Question

The area of a rectangle decreases by 10cm² if the length is decreased by 5cm and the breadth is increased by 3cm. If the length is increased by 5cm and the breadth is increased by 2cm, then the area increases by 80cm². Find the perimeter of the rectangle.

Answer 1

Here, let the original length be x cm and the breadth be y cm.

According to the given conditions,

(1) Area decreases by 10cm² when:

1. Length is decreased by 5cm = x − 5
2. Breadth is increased by 3cm = y + 3

(x − 5)(y + 3) = xy − 10

x(y + 3) − 5(y + 3) = xy − 10

xy + 3x − 5y − 15 = xy − 10

xy − xy + 3x − 5y = −10 + 15

3x − 5y = 5     ...(i)

(2) Area decreases by 80cm² when:

1. Length is decreased by 5cm = x + 5
2. Breadth is increased by 2cm = y + 2

(x + 5)(y + 2) = xy + 80

x(y + 2) + 5(y + 2) = xy + 80

xy + 2x + 5y + 10 = xy + 80

xy − xy + 2x + 5y = 80 − 10

2x + 5y = 70     ...(ii)

Now, adding equations (i) and (ii):

(3x − 5y) + (2x + 5y) = 5 + 70

3x − 5y + 2x + 5y = 75

3x + 2x = 75

5x = 75

x = `75/5`

∴ x = 15

Substitute x = 15 in equation (i):

3(15) − 5y = 5

45 − 5y = 5

5y = 40

y = `40/5`

∴ y = 8

Perimeter of the rectangle = 2(x + y)

= 2(15 + 8)

= 2 × 23

= 46

Hence, the perimeter of the rectangle is 46cm.