\documentclass[12pt]{article}
%\setlength{\topmargin}{-.25in}5
%\setlength{\oddsidemargin}{-.3in}
%\setlength{\textheight}{9in}
%\setlength{\textwidth}{7.2in}


\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amsthm}
\usepackage{amstext}
\usepackage{amssymb}


\newcommand{\ra}{\rightarrow}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\E}{\mathbb{E}}

\newtheorem{definition}{Definition}
\newtheorem{formula}[definition]{Formula}

\begin{document}

\begin{center}
\textbf{Units - January 28, 2002}
\end{center}

I hope this helps explain how to convert units.
Rather then give lots of theory, I thought I would put a few examples down.

\paragraph{Feet to Meters:}
\begin{enumerate}
\item  First determine how many feet are equal to a meter.
	You can find this in a chart (I found many charts on the internet for example).
	For feet and meters:
	\[
	3.28 ft = 1 m
	\]
	While it is not true that $3.28=1$, with the meters and feet in the equation, this is a true equation.
\item  We now take our equation and divide.
	There are two ways to do this:
	\[
	1= \frac{3.28 ft}{1 m} \quad \mbox{or} \quad 1 = \frac{1 m}{3.28 ft}
	\]
	Even though $1=3.28/1$ is definitely a false equation, $1=3.28ft/1m$ is a true equation.
\item  Lets convert $10$ feet into meters.  
	We first take what we are given and we want to multiply it by $1$.
	Here is one way to multiply by one:
	\[
	10 ft (1) = 10 ft \left(\frac{3.28 ft}{1m} \right) = \frac{32.8 ft \cdot ft}{1m}
	\]
	Not too useful.
	Lets try the other equation for $1$:
	\[
	10 ft (1) = 10 ft \left( \frac{1m}{3.28} \right) = \frac{ 10 ft \cdot m}{3.28 ft} = 3.05 m
	\]
	Notice how this time the feet canceled out.
\item  Some for you to try.  Try to do these carefully like I did above.
	\begin{enumerate}
	\item  Convert $25 m$ to feet.
	\item  Convert $98 ft$ to meters.
	\item  Determine your height in both feet and meters.
	\end{enumerate}
\end{enumerate}


\paragraph{Miles per hour to feet per second:}
\begin{enumerate}
\item  Well, this is a bit tougher.  First we know the following:
	\[
	1 mi = 5280 ft \quad 60 secs = 1 min \quad 60 min = 1 hr
	\]
	In your head or on the paper, make these equations into fractions equal to $1$.
\item  Now, lets convert $80$ miles per hour into feet per second.
	$80 mph$ is the same as $80 mi/hr$ (thats what it means to be per hour).
	Think about which of the fractions above we can use.
	\[
	\left( \frac{80 mi}{hr} \right) \left( \frac{5280 ft}{1 mi} \right) = \frac{422400 ft}{hr}
	\]
	This gives us feet per hour.  Notice how the miles canceled out.
	Lets keep going:
	\[
	\left( \frac{80 mi}{hr} \right) = \left( \frac{422400 ft}{hr} \right) \left( \frac{1 hr}{60 min} \right)
		= \frac{7040 ft}{min}
	\]
	OK, now we have feet per minute.  One more step.
	\[
	\left( \frac{80 mi}{hr} \right) = \left( = \frac{7040 ft}{min} \right) \left( \frac{1 min}{60 sec} \right) 
		= \frac{117.3 ft}{sec}
	\]
	And, there it is, $80mph = 117.3 ft/s$.
\item  Try some on your own.  Again, work carefully and neatly.
	\begin{enumerate}
	\item  What is $100 ft/s$ in mph?
	\item  What is $50 mph$ in meters per second?
	\item  What is $50 mph$ in kilometers per second?
	\end{enumerate}
\end{enumerate}

\paragraph{More for you to try:}
\begin{enumerate}
\item  Convert 2.5 gallons into liters (1 liter is .264 gallons).
\item  Convert 10 radians into degrees (as we know, $\pi$ radians are 180 degrees).
\item  Convert 100 inches per second into meters per day.
\end{enumerate}



\end{document}


