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%TCIDATA{Created=Wednesday, September 21, 2005 19:23:38}
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\begin{document}


\begin{center}
\bigskip {\Large Puzzle 11 - SOLUTION}
\end{center}

\bigskip

\begin{problem}
\bigskip A bus makes a roundtrip between towns A and B. From A to B, the bus
completes the trip with an average speed of $60$ $\dfrac{\text{mi}}{\text{hr}%
}$. From B to A, it travels with an average velocity of $40$ $\dfrac{\text{mi%
}}{\text{hr}}$. What is the average speed of the bus for the entire
roundtrip? (Hint: it is NOT the average of $40$ and $60$)
\end{problem}

\bigskip

\begin{solution}
Suppose the towns are $120$ miles apart. We can use any numbers, but $120$
is 'nice' because it is divisible by $40$ and $60.$

The time it took to travel from A to B was%
\[
t=\dfrac{\text{distance}}{\text{velocity}}=\dfrac{120\text{ mi}}{60\dfrac{%
\text{mi}}{\text{hr}}}=2\text{hr} 
\]

The trip from B to A:%
\[
t=\dfrac{\text{distance}}{\text{velocity}}=\dfrac{120\text{ mi}}{40\dfrac{%
\text{mi}}{\text{hr}}}=3\text{hr} 
\]

The average velocity for the round trip is%
\[
v=\dfrac{\text{total distance}}{\text{total time}}=\dfrac{240\text{ mi}}{5%
\text{ hr}}=48\dfrac{\text{mi}}{\text{hr}} 
\]

Thus the average velocity is $48$ $\dfrac{\text{mi}}{\text{hr}}.$

Note: We can obtain these results without making up a distance. Denote it by 
$s$. Then the computation works like this.

The time it took to travel from A to B was%
\[
t=\dfrac{\text{distance}}{\text{velocity}}=\dfrac{s}{60} 
\]

The trip from B to A:%
\[
t=\dfrac{\text{distance}}{\text{velocity}}=\dfrac{s}{40} 
\]

The average velocity for the round trip is%
\[
v=\dfrac{\text{total distance}}{\text{total time}}=\dfrac{s+s}{\dfrac{s}{60}+%
\dfrac{s}{40}}=\dfrac{2s}{\dfrac{2s}{120}+\dfrac{3s}{120}}=\dfrac{2s}{\left( 
\dfrac{5s}{120}\right) }=2s\cdot \dfrac{120}{5s}=\dfrac{240s}{5s}=48 
\]

Thus the average velocity is $48$ $\dfrac{\text{mi}}{\text{hr}}.$
\end{solution}

\end{document}