Helping grandson with math. I would like to know how to do it before he does.
"How many different rectangles can be made with a perimeter of 20 units? (Remember, a square is also a rectangle.)"
I assume you mean with integer lengths
L + w = 10
L = 5, w = 5 as you said
L = 6, w = 4
L = 7, w = 3
L = 8, w = 2
L = 9, w = 1
the ones with w>L do not count because they are indistinguishable from the ones we have now. , So, 5.
To find the number of different rectangles that can be made with a perimeter of 20 units, you can follow a step-by-step approach:
Step 1: Understand the problem
The question asks for the number of different rectangles that can be formed with a perimeter of 20 units. It states that a square is also considered a rectangle.
Step 2: Recall the properties of a rectangle
A rectangle has two pairs of equal sides that are parallel to each other. The opposite sides have equal lengths. The perimeter of a rectangle is found by adding all four sides.
Step 3: Determine the possible side lengths
Since the perimeter is given as 20 units, we need to find all the possible combinations of side lengths that equal 20 when added together. So, let's list down the possible pairs of side lengths that can form a 20-unit perimeter:
- Sides: 1 unit & 9 units
- Sides: 2 units & 8 units
- Sides: 3 units & 7 units
- Sides: 4 units & 6 units
- Sides: 5 units & 5 units
Step 4: Calculate the number of rectangles for each pair of side lengths
For each pair of side lengths, there can be one or more rectangles that can be formed. To determine the number of rectangles, we need to consider that the lengths of the opposite sides can vary. However, their sum must be equal to one of the pairs of side lengths we listed.
- Sides: 1 unit & 9 units
- Rectangles: Only one rectangle can be obtained by combining these side lengths. It will be a long and narrow rectangle.
- Sides: 2 units & 8 units
- Rectangles: Two rectangles can be formed by combining these side lengths. One rectangle will have a shorter length (2 units) and a longer width (8 units); the other will have a longer length (8 units) and a shorter width (2 units).
- Sides: 3 units & 7 units
- Rectangles: Two rectangles can be obtained. The first rectangle will have a shorter length (3 units) and a longer width (7 units); the other will have a longer length (7 units) and a shorter width (3 units).
- Sides: 4 units & 6 units
- Rectangles: Two rectangles can be formed. The first rectangle will have a shorter length (4 units) and a longer width (6 units); the other will have the longer length (6 units) and a shorter width (4 units).
- Sides: 5 units & 5 units
- Rectangles: One square can be obtained when both side lengths are equal.
Step 5: Sum up the total number of different rectangles
Finally, add up the number of rectangles obtained for each pair of side lengths:
- For the pair of side lengths 1 unit & 9 units: 1 rectangle
- For the pair of side lengths 2 units & 8 units: 2 rectangles
- For the pair of side lengths 3 units & 7 units: 2 rectangles
- For the pair of side lengths 4 units & 6 units: 2 rectangles
- For the pair of side lengths 5 units & 5 units: 1 square (also counted as a rectangle)
By adding up all these numbers, we get a total of 8 different rectangles that can be formed with a perimeter of 20 units.