The ratio of the area of a square to the area of a rectangle is $3:2$. The side of the square is equal to the length of the rectangle. If the perimeter of the rectangle is $40$ meters, what is the breadth of the rectangle?

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**Solution:**

1. Let the length of the rectangle be $l$ meters and the breadth be $b$ meters.

2. Since the side of the square equals the length of the rectangle, the side of the square is also $l$ meters.

3. Given the ratio of the areas:
   $$\frac{	ext{Area of square}}{	ext{Area of rectangle}} = \frac{3}{2}$$
   $$\Rightarrow \frac{l^2}{l 	imes b} = \frac{3}{2}$$

4. Simplify the ratio:
   $$\frac{l}{b} = \frac{3}{2}$$
   This implies:
   $$l = 3x, \quad b = 2x$$ for some positive $x$.

5. The perimeter of the rectangle is given as $40$ meters:
   $$2(l + b) = 40$$
   Substitute $l$ and $b$:
   $$2(3x + 2x) = 40$$
   $$2(5x) = 40$$
   $$10x = 40$$
   $$x = 4$$

6. Therefore, the breadth of the rectangle is:
   $$b = 2x = 2 	imes 4 = 8 	ext{ meters}$$

**Answer:** $8 	ext{ m}$

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