There are two squares. Their lengths at the ratio 2:3, what is the simplified ratio between their areas?
a1 = side length of first square
a2 = side length of second square
A1 = area of first square
A2 = Area of second square
a1 / a2 = 2 / 3 Multiply both sides by a2
a1 = 2 a2 / 3 = ( 2 / 3 ) a2
A1 / A2 = a1² / a2² =
[ ( 2 / 3 ) a2 ]² / a2² =
( 4 / 9 ) a2² / a2² = 4 / 9
To find the simplified ratio between the areas of two squares given their lengths, you need to square the lengths and then simplify the resulting ratio.
Let's assume the lengths of the two squares are 2x and 3x (since the ratio is given as 2:3).
The area of a square is equal to the length of one side squared. So, the areas of the two squares can be represented as (2x)^2 and (3x)^2.
Simplifying these expressions, we get:
Area of the first square: (2x)^2 = 4x^2
Area of the second square: (3x)^2 = 9x^2
Now, we can form the simplified ratio between their areas:
Ratio = Area of the first square : Area of the second square
= 4x^2 : 9x^2
To simplify this ratio, we can divide both terms of the ratio by the highest common factor, which is x^2:
Ratio = (4x^2 / x^2) : (9x^2 / x^2)
= 4 : 9
Therefore, the simplified ratio between the areas of the two squares is 4:9.