To solve this problem, we need to calculate the perimeter of a rectangle whose area is 125 cm² and whose sides are in the ratio of 4:5.

Step 1: Let the dimensions of the rectangle be $4x$ and $5x$.

Since the sides are in the ratio of 4:5, we can express the lengths of the sides as $4x$ and $5x$, where $x$ is a common scaling factor.

Step 2: Use the formula for the area of a rectangle.

The area of a rectangle is given by: $\text{Area} = \text{length} \times \text{width}$ Substituting the dimensions: $125 = (4x)(5x) = 20x^2$

Step 3: Solve for $x$.

$20x^2 = 125$ $x^2 = \frac{125}{20} = 6.25$ $x = \sqrt{6.25} = 2.5$

Step 4: Calculate the length and width.

• The length is $5x = 5(2.5) = 12.5$ cm.
• The width is $4x = 4(2.5) = 10$ cm.

Step 5: Calculate the perimeter.

The perimeter $P$ of a rectangle is given by: $P = 2(\text{length} + \text{width}) = 2(12.5 + 10) = 2(22.5) = 45 \, \text{cm}$

Thus, the perimeter of the rectangle is 45 cm.

Would you like more details or have any questions? Here are five related questions to expand your understanding:

1. What would the dimensions be if the ratio of the sides was different, say 3:4, for the same area?
2. How would the problem change if the area was doubled?
3. Can you derive the formula for the perimeter of a rectangle?
4. What would happen if you changed the unit of area, say from square centimeters to square meters?
5. How do the side ratios of a rectangle affect the overall shape of the rectangle?

Tip: When solving ratio problems, it's helpful to express the dimensions in terms of a variable and solve step by step to avoid confusion.