Proportional Reasoning

One of the most important skills in “numerical literacy” is proportional reasoning.

I was reminded of this earlier today when I went out to buy a couple of pies for this evening’s family gathering. The pies came in many flavours, and two sizes, 8 inch diameter and 10 inch diameter. There was never any question about which size I was going to get, but I wondered whether the two sizes of pies were priced reasonably.

Assuming that the cost of two pies of the same flavour depend only on the volume of the pie, and that they have the same thickness, the price should depend on the area of the top. The area of a circle is proportional to the square of its diameter, so the ratio of the costs of the pies should be equal to the square of the ratio of the diameters, which is (10/8)2 = 25/16, which is about 1.56. Thus, the larger pie ought to cost about 50% more than the smaller one. Is it surprising that a 25% increase in diameter should lead to a more than 50% increase in area, and therefore price?

The kinetic energy of a moving object is proportional to the square of its speed. The work needed to stop the object is proportional to its kinetic energy. For a moving car on a slippery road, the stopping distance is therefore proportional to the square of the speed. That is, doubling the speed quadruples the stopping distance. Impact forces are also (approximately) proportional to the square of the speed, and impact forces are dangerous to a car’s passengers. So, increasing the speed from 100 km/h to 123 km/h increases the impact forces (and stopping distances) by 50%. Increasing from 100 km/h to 142 km/h doubles the impact forces and stopping distances. Moral: increasing speeds are far more dangerous that we would expect based on our experiences of the linear increases of many phenomena.

The volume of an animal is proportional to the cube of its “size,” where the size is some average linear dimension. Assuming a fairly constant density, the mass of an animal is proportional to its volume, and so the mass is proportional to the cube of the animal’s size. However, the strength of the leg bones is proportional to their cross-sectional area, which depends on the square of the leg bones’ diameters. This explains why there is a limit to the size of land animals on earth: the mass grows faster than the strength of the supporting leg bones as the size of the animal increases, and a point will be reached when the leges are not strong enough to support the animal. (Unless the number of legs increases … maybe a super-mastodon shaped like a giant centipede would work?) Sea creatures are not so constrained, because they are supported throughout their bodies, not by legs. So maybe a giant land snake would be possible?

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 22 January 2021.)