A4 paper ratio
The A series of paper sizes (and the B series come to that) have the property that when you fold a sheet in half with the fold parallel to the short side, the ratio of the sides remains the same. This means that the sides must be in the ratio 1:1.414. The algebra is sketched below (on a 5 by 3 inch record card scanned at 75 dpi).
This is one of the most accessible situations I know of where a quadratic equation (a very simple one) can be set up and solved.