Each side of a rectangle is doubled. Find the ratio between :

(i) perimeters of the original rectangle and the resulting rectangle

(ii) areas of the original rectangle and the resulting rectangle

(i) Given:

Each side of a rectangle is doubled.

Let the length and breadth of the rectangle be l and b.

Thus, the new length and breadth of the rectangle are 2l and 2b.

As we know, the perimeter of a rectangle = 2 (length + breadth)

The perimeter of the original rectangle = 2 (l + b)

The perimeter of the new rectangle = 2 (2l + 2b)

= 4 (l + b)

Thus, the ratio of the perimeters of the original rectangle and the resulting rectangle = $\dfrac{2 (l + b)}{4 (l + b)}$

= $\dfrac{2 \cancel{(l + b)}}{4 \cancel{(l + b)}}$

= $\dfrac{2}{4}$

= $\dfrac{1}{2}$

Hence, the ratio of the perimeters of the original rectangle and the resulting rectangle is $\dfrac{1}{2}$.

(ii) As we know, the area of a rectangle = length x breadth

The area of the original rectangle = l x b

= lb

The area of the new rectangle = 2l x 2b

= 4 lb

Thus, the ratio of the areas of the original rectangle and the resulting rectangle = $\dfrac{lb}{4lb}$

= $\dfrac{\cancel{lb}}{4 \cancel{lb}}$

= $\dfrac{1}{4}$

Hence, the ratio of areas of the original rectangle and the resulting rectangle is $\dfrac{1}{4}$.