, 2015

Since the perimeter of the square is $P$ and the 4 sides of a square are equal in length, then each side of the square has length $\frac{1}{4}P$. We now work backward to determine the width and length of the rectangle.

The width of the rectangle was doubled to produced the side of the square with length $\frac{1}{4}P$.

Therefore, the width of the rectangle is half of the side length of the square, or $\frac12\times\frac{1}{4}P=\frac{1}{8}P$.

The length of the rectangle was halved to produce the side of the square with length $\frac{1}{4}P$.
Therefore, the length of the rectangle is twice the side length of the square, or $2\times\frac{1}{4}P=\frac{1}{2}P$. Finally, we determine the perimeter of the rectangle having width $\frac{1}{8}P$ and length $\frac{1}{2}P$, obtaining $2\times\left(\frac{1}{8}P+\frac{1}{2}P\right)=2\times\left(\frac{1}{8}P+\frac{4}{8}P\right)=2\times\left(\frac{5}{8}P\right)=\frac{5}{4}P$.