\chapter{Ratios, Percents, Proportions}

\specialsection{Introduction}
Measurements sometimes have meanings in isolation.  For instance, the fact that 
the day was sunny and the high temperature in Worcester, MA on June 21, 2017 as I was writing 
this was $81^\circ$ F meant that this was a quite comfortable early summer
day(!)  However, there are also times when we want to \emph{compare} measurements or other
quantities in various ways.  Here is an everyday example that will illustrate some of
the key points we want to make in this chapter.  

According to a news story published in the New York Daily News on January 4, 2017\footnote{
Archived at {\tt http://www.nydailynews.com/new-york/nyc-crime/nyc-historically-number-shootings-2016-article-1.2933098}, 
consulted June 21, 2017.}
there were $352$ homicides reported in New York City during the calendar year 2015 versus $335$ homicides during the 
calendar year 2016.  On the other hand,\footnote{According to {\tt https://www.neighborhoodscout.com/ar/crime}, 
consulted June 21, 2017.} there were 181 homicides in the state of Arkansas in calendar year 2016.  

In comparing the homicides in New York City, the most basic comparison would be to say 
$335 - 352 = -17$, so the number of homicides \emph{decreased by $17$}.  But note we could 
ask more about this situation!  Is that a big decrease or a small decrease?  One way to answer that
would be to compute what is called the \emph{percent decrease}.  The idea here is that 
we look at the decrease \emph{as a fraction of the number of homicides in the previous year}.
The computation would be 
$${\rm percent\ decrease} = \frac{335 - 352}{352} \times 100\% \doteq -4.8\%.$$
A decrease of $4.8\%$ would usually be interpreted as a ``medium size'' decrease--not huge, but 
not negligibly small either.   Crime figures were moving in the right
direction in that year.

In a similar way, looking at the New York City homicide figure versus the Arkansas state
homicide figure, it might be tempting to think that New York City was measurably less safe
and more crime-ridden than Arkansas.  After all, $335$ is only a bit less than $2 \times 181$, 
and the $181$ is the figure for a whole state, not just one city!  

But we have to ask whether this is a reasonable way to make the comparison.  Note that we
are leaving out one key piece of information(!)  In 2016, the population of Arkansas was
approximately $2.98$ million people, while the population of New York City was approximately
$8.55$ million people.  If we take that difference into account by computing the 
\emph{per capita homicide rates}:
$${\rm New\ York\ per\ capita\ homicide\ rate} = \frac{335}{8.55 \times 10^6} \doteq .000039,$$
and
$${\rm Arkansas\ per\ capita\ homicide\ rate} = \frac{181}{2.98\times 10^6} \doteq .000063.$$
One way to interpret these numbers is that in New York City, if you picked a person
at random from the population, then there would be a $.0039\%$ chance that person was
murdered in 2016.  Similarly, the chance was $.0063\%$ chance a randomly chosen Arkansan
was murdered.  Although these are both quite small numbers, the chance in Arkansas 
was actually
$$\frac{.0063 - .0039}{.0039} \times 100\% = 61.5\%$$
higher than the chance in New York City.
By this measure, New York City was actually quite a bit less dangerous than
Arkansas! 


\endinput
