4.1: Population Scaling

Last updated
Save as PDF

Page ID: 24100

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

An everyday example of proportional reasoning often happens when cooking for a dinner party. When I prepare fish curry, which I normally cook for our family of four, I buy 250 grams of fish. But today another family of four will join us.

How much fish do I need?

I need 500 grams. As a general relation,

\[\textrm{new amount} = \textrm{old amount} \times \frac{\textrm{new number of diners}}{\textrm{usual numbers of diners}}.\]

Another way to state this relation is that the amount of fish is proportional to the number of diners. In symbols,

\[m_{fish} \propto N_{diners}.\]

where the \(\propto\) symbol is read "is proportional to."

But where in this analysis is the quantity that does not change?

Another way to write the proportionality relation is

\[\frac{\textrm{new amount of fish}}{\textrm{new number of diners}} = \frac{\textrm{old amount of fish}}{\textrm{old number of diners}}.\]

Thus, even when the number of diners changes, the quotient

\[\frac{\textrm{amount of fish}}{\textrm{number of diners}}\]

does not change.

For an analogous application of proportional reasoning, here’s one way to estimate the number of gas stations in the United States. Following the principle of using human-sized numbers, which we discussed in Section 1.4, I did not try to estimate this large number directly. Instead, I started with my small hometown of Summit, New Jersey. It had maybe 20,000 people and maybe five gas stations; the “maybe” indicates that these childhood memories may easily be a factor of 2 too small or too large. If the number of gas stations is proportional to the population (N_stations \(alpha\) N_people), then

\[N_{US stations} = N_{Summit stations} \times \frac{3 \times 10^{8} \: [N_{US people}]}{2 \times 10^{4} \: [N_{Summit people}]}.\]

The population ratio is roughly 15 000. Therefore, if Summit has five gas stations, the United States should have 75,000. We can check this estimate. The US Census Bureau has an article (from 2008) entitled “A Gas Station for Every 2,500 People”; its title already indicates that an estimate of roughly 10⁵ gas stations is reasonably accurate: Summit, in my reckoning, had 4000 people per gas station. Indeed, the article gives the total as 116,855 gas stations—as close to the estimate as we can expect given the uncertainties in childhood memories!

Exercise \(\PageIndex{1}\): Homicide rates

The US homicide rate (in 2011) was roughly 14,000 per year. The UK rate in the same year was roughly 640. Which is the more dangerous country (per person), and by what factor?