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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.

A rectangle with area 125 cm² has sides in the ratio 4:5. What is the perimeter of the rectangle?

Learn how to find the perimeter of a rectangle with an area of 125 cm² and a side ratio of 4:5. This geometry problem covers key concepts like area, perimeter, and ratio-based dimensions. Suitable for students in Grades 7-9.

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