Question

If the length of a rectangle is increased by 12 cm and the width is decreased by 8 cm, the area is unchanged. If the original length is increased by 5 cm and the original width is decreased by 4 cm, also the area remains the same. Find the original dimensions.

Answer 1

Here, let the original length = x cm,

The original width = y cm,

And the original area = x × y

Given:

(1) When the length is increased by 12 cm and the width is decreased by 8 cm, the area remains the same:

(x + 12)(y − 8) = xy     ...(i)

(2) When the length is increased by 5 cm and the width is decreased by 4 cm, the area remains the same:

(x + 5)(y − 4) = xy     ...(ii)

Expanding both equations,

From equation (i):

(x + 12)(y − 8) = xy

x(y − 8) + 12(y − 8) = xy

xy − 8x + 12y − 96 = xy

−8x + 12y = 96     ...(iii)

From equation (ii):

(x + 5)(y − 4) = xy

x(y − 4) + 5(y − 4) = xy

xy − 4x + 5y − 20 = xy

−4x + 5y = 20     ...(iv)

Let’s multiply (iv) by 2:

2(−4x + 5y) = 2(20)

−8x + 10y = 40     ...(v)

Now subtracting equation (v) from equation (iii):

(−8x + 12y) − (−8x + 10y) = 96 − 40

−8x + 12y + 8x − 10y = 56

12y − 10y = 56

2y = 56

y = `56/2`

∴ y = 28

Substitute y = 28 in equation (iv):

−4x + 5(28) = 20

−4x + 140 = 20

−4x = 20 − 140

−4x = −120

x = `(-120)/-4`

∴ x = 30

Hence, Original Length = 30cm and Original Width = 28cm.