### Instruction: A $4$-foot by $8$-foot rectangular piece of plywood will be cut into $4$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perimeter of a single piece?### Response: I need to find the possible ways to divide the plywood into $4$ congruent rectangles.
One way is to cut it along the middle of the length and the width, forming $4$ rectangles that are each $2$ feet by $4$ feet.
The perimeter of each rectangle is $2 + 4 + 2 + 4 = 12$ feet.
Another way is to cut it along the middle of the length and then cut one of the halves into two equal parts along the width, forming $4$ rectangles that are each $1$ foot by $8$ feet.
The perimeter of each rectangle is $1 + 8 + 1 + 8 = 18$ feet.
A third way is to cut it along the middle of the width and then cut one of the halves into two equal parts along the length, forming $4$ rectangles that are each $4$ feet by $2$ feet.
The perimeter of each rectangle is the same as the first way, $12$ feet.
I can't think of any other ways to divide the plywood into $4$ congruent rectangles, since any other cut would either leave some wood left over or make the rectangles not congruent.
Therefore, the greatest possible perimeter of a single piece is $18$ feet and the least possible perimeter of a single piece is $12$ feet.
The positive difference between them is $18 - 12 = 6$ feet.