Question

The widths of two similar rectangles are 10 cm and 4 cm. What is the ratio of their perimeters? Of their areas?

Answer 1

The widths of two similar rectangles are 45 yd and 35 yd. What is the ratio of their perimeters? Of their areas?

Answer 2

Since the rectangles are similar, their corresponding sides are proportional.
P = 2W + 2L
The widths are 10 cm and 4 cm, or a ratio of 10/4 = 5/2
The lengths are in the same ratio. Let's call them 5a/2a

Large rectangle: P = 2(10) + 2(5a) = 20 + 10a
Small rectangle: P = 2(4) + 2(2a) = 8 + 4a
The ratio of the perimeters is (20 + 10a) / (8 + 4a) = ( 10(2 + a) ) / ( 4(2 + a) ) = 5/2
The ratio of the perimeters is the same as the ratio of the sides, 5/2

The area of the large rectangle is A = LW
A = 5a(10) = 50a
The area of the small rectangle is A = LW
A = 2a(4) = 8a
The ratio of the areas is 50a/8a = 50/8 = 25/4
The ratio of the areas is the square of the ratio of the sides and perimeters
25/4 = (5/2)^2

Answer 3

Now you know that the ratio of the perimeters is the same as the ratio of two sides. The ratio of the areas is the square of the ratio of the sides or perimeters.
This will make the next problem much easier.