The ratio of the perimeters of two similar squares is 5 to 4. If the area of the smaller square is 32 square units, what is the area of the larger square?

Since both the length of sides and perimeter are linear

the perimeters would be in that same ratio as the sides.
So the sides are in the ratio of 5:4

The ratio of areas is equal to the square of their corresponding sides.
so
A/32 = 5^2 / 4^2
A = 32(25)/16 = 50

If Smaller square = 32

Perimeter of Small Square = 4x
Perimeter of Larger Square = 5x
Perimeter Of Small Square = 4L
4x = 32
x = 32 / 4
x = 8
Then
5x = 5(8)
5x = 40

To find the area of the larger square, we first need to determine the length of its side. Since the ratio of the perimeters of the two squares is given as 5 to 4, we can set up the following equation:

(perimeter of larger square) / (perimeter of smaller square) = 5/4

Let's assume the side length of the smaller square is "s". Then the perimeter of the smaller square would be 4s, and the perimeter of the larger square would be 5s.

So we have:

5s / 4s = 5/4

Cross-multiplying, we get:

4 * 5s = 5 * 4s
20s = 20s

This means that the ratio of the perimeters is true for any value of "s". And since the area is directly proportional to the square of the side length, we can conclude that the ratio of the areas of the two squares is also 5 to 4.

Now, we know that the area of the smaller square is 32 square units. So let's set up the following equation:

(area of larger square) / (area of smaller square) = 5/4

Let's call the area of the larger square "A". We can rewrite the equation as:

A / 32 = 5/4

Cross-multiplying, we have:

4A = 32 * 5
4A = 160

Dividing both sides by 4, we get:

A = 40

Therefore, the area of the larger square is 40 square units.