If you fold a rectangular piece of paper in half

Question

If you fold a rectangular piece of paper in half and the resulting rectangles have the same aspect ratio as the original rectangle, then what is the aspect ratio of the rectangles?

Answer 1

$$ \frac{\text{width}}{\text{lengh}}=\sqrt{2} $$

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Answer 2

Assume the original rectangle is $x$ wide and $a \cdot x$ long, where a is the requested aspect ratio. If we fold the paper in half (lengthwise), the resulting rectangles will be $\frac{a \cdot x}{2}$ wide and $x$ long. Equating these ratios, we find that

$$a = \frac{1}{a / 2} = \frac{2}{a}$$

Solving this simple equation yields

$$a=\sqrt{2}$$