Math CentralQuandaries & Queries


Question from Anna, a student:

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?

Hi Anna,

The aspect ratio of a rectangle is the ratio of the long side to the short side. I am thinking of the rectangle in landscape mode so I am going to call the long side $L$ for length and the short side $H$ for height. If you fold the rectangle in half by folding the left edge to the right edge then the resulting rectangle is in portrait mode an the length of the long side is $H$ and the length of the short side is $ L/2.$ For the original rectangle the aspect ratio is $L:H$ and for the smaller rectangle the aspect ratio is $H:L/2.$ Since these aspect ratios are equal $L:H = H:L/2.$

You might find it easier to work with fraction and the above equation can be written

\[\frac{L}{H} = \frac{H}{L/2}.\]

Manipulate this equation to express $L/H$ and the ratio of two known numbers and then change the fractions back to ratios.



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