The aspect ratio of a rectangle is the length of its longer side divided by the length of its shorter side. For example, a piece of paper $10\,\mathrm{cm}$ by $15\,\mathrm{cm}$ has an aspect ratio of $15\,\mathrm{cm}\div10\,\mathrm{cm} = 1.5$. Suppose you have a rectangular piece of paper, and you cut part of it off so that you get a square. If the leftover rectangle of paper has the same aspect ratio as the original piece of paper, what is the aspect ratio? Show hint

The width of a square is the same as its height.

We could assign variables to the width and height of the pieces of paper, but a neater way to work this out would be to call the short side of the original piece of paper $1$ (because it really doesn’t matter if it’s $12\,\mathrm{cm}$ or $8\,\mathrm{in}$ or whatever; we only care about the ratios). Then we can call the aspect ratio $r$, and work out all the lengths in terms of that. The long side of the original piece of paper is $r\times1 = r$.

At this point you may get slightly stuck and not know what to do next. It helps to ask yourself if you’ve used all the information that was in the question. We haven’t yet used the fact that we have a square – a rectangle with equal height and width. The height of the square is $1$, so its width is also $1$, so the cut-off rectangle has short side $r-1$. Its long side is $1$, so its aspect ratio is $1/(r-1)$. This must be the same as the original aspect ratio, $r$. \begin{align*} r &= \frac{1}{r-1}\\ r(r-1)&= 1\\ r^2 - r &= 1\\ r^2 - r - 1 &= 0. \end{align*} Now we have a standard quadratic equation and can apply the quadratic formula: \begin{align*} r &= \frac{-(-1) \pm \sqrt{(-1)^2 - 4\times1\times(-1)}}{2\times1}\\ &= \frac{1 \pm \sqrt{5}}{2}. \end{align*} Since $\sqrt5 \gt 1$, $(1-\sqrt5)/2$ will be negative, so we can ignore that solution. The aspect ratio is: \[\frac{1 + \sqrt5}{2} \approx 1.618.\] An equation as simple as $r=1/(r-1)$ can come up quite frequently, in many different situations, so this number gets a special name: the ‘Golden ratio’, and a symbol: $\phi$ (the Greek letter phi).