A wire of length 20 m is to be folded in the form of a rectangle. How many rectangles can be formed by folding the wire if the sides are positive integers in metres?

To find out how many rectangles can be formed by folding a wire of length 20 meters into a rectangle with sides as positive integers in meters, we can start by considering the possible combinations of side lengths. Let's denote the length of one side of the rectangle as xx meters and the length of the other side as yy meters. Since the wire has a length of 20 meters, the perimeter of the rectangle formed by folding the wire must be equal to 20 meters. Therefore, we have the equation: 2x+2y=202x+2y=20 Simplifying this equation, we get: x+y=10x+y=10 Now, we need to find pairs of positive integers xx and yy that satisfy this equation and represent the possible side lengths of the rectangle. The pairs of positive integers xx and yy that satisfy x+y=10x+y=10 are: (1,9),(2,8),(3,7),(4,6),(5,5),(6,4),(7,3),(8,2),(9,1)(1,9),(2,8),(3,7),(4,6),(5,5),(6,4),(7,3),(8,2),(9,1) Each of these pairs represents a unique rectangle. However, notice that for each pair (x,y)(x,y), we also have the pair (y,x)(y,x), which represents the same rectangle but with the sides swapped. Since the order of the sides doesn't matter for a rectangle, we can cut the number of distinct rectangles in half by considering only one of the pairs for each unique combination of side lengths. Therefore, the total number of distinct rectangles that can be formed by folding the wire into a rectangle with sides as positive integers is half the number of pairs, which is 92=4.529=4.5. However, since we cannot have half a rectangle, we need to consider only the integer part of 4.5, which is 4. So, 4 rectangles can be formed by folding the wire if the sides are positive integers in meters. read less