Explanation:

Let’s break the problem down step by step:

- Let the side of the square be x (natural number).

- Perimeter of square = 4x

- Perimeter of rectangle = 2(l + b) = 2(l + 5)

- Given: Rectangle’s perimeter is 45% more than square’s ?

2(l + 5) = 4x × 1.45 = 5.8x

- Areas:

Area of square = x²

Area of rectangle = l × 5

Difference = x² – 5l = 20 ? x² – 5l = 20

Now solve:

- 2(l + 5) = 5.8x ? l + 5 = 2.9x ? l = 2.9x – 5

- From area difference: x² – 5l = 20 ? x² – 5(2.9x – 5) = 20 ? x² – 14.5x + 25 = 20 ? x² – 14.5x + 5 = 0

Test natural numbers for x:

- x = 14 ? x² – 14.5x + 5 = 196 – 203 + 5 = -2 (not zero)

- x = 15 ? 225 – 217.5 + 5 = 12.5 (not zero)

- x = 10 ? 100 – 145 + 5 = -40 (not zero)

But also check for x = 5:

- l = 2.9*5 – 5 = 14.5 – 5 = 9.5 (not natural number)

Try x = 8:

- l = 2.9 × 8 – 5 = 23.2 – 5 = 18.2

Try x = 10:

- l = 2.9 × 10 – 5 = 29 – 5 = 24

Check area difference:

- x² – 5l = 100 – 5×24 = 100 – 120 = -20 (should be +20 though, so let’s swap order)

- 5l – x² = 120 – 100 = 20 (so the difference works if rectangle area is greater!)

So,

Length of rectangle = 24 cm.

Option:2 is correct.

- Option 2 (24 cm): Matches all conditions.

- Option 1 (10 cm): l = 10, perimeter = 30, x = 6.03 (not natural number).

- Option 3 (36 cm), Option 4 (15 cm), Option 5 (32 cm): Don’t satisfy all constraints.