Henry Ernest Dudeney/Puzzles and Curious Problems/74 - The Omnibus Ride/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $74$

The Omnibus Ride
George treated his best girl to a ride on a motor omnibus,
but on account of his limited resources it was necessary that they should walk back.
Now, if the bus goes at the rate of nine miles an hour and they walk at the rate of three miles an hour,
how far can they ride so they may be back in eight hours?


Solution

$18$ miles.


Proof

Let $D$ be the distance they can travel.

Let $T_1$ and $T_2$ be the lengths of time taken for the outward and return journeys respectively.

We have:

\(\ds D\) \(=\) \(\ds 9 T_1\) if the bus goes at the rate of nine miles an hour
\(\ds D\) \(=\) \(\ds 3 T_2\) and they walk at the rate of three miles an hour,
\(\ds T_1 + T_2\) \(=\) \(\ds 8\) how far can they ride so they may be back in eight hours?
\(\ds \leadsto \ \ \) \(\ds T_2\) \(=\) \(\ds 8 - T_1\)
\(\ds \leadsto \ \ \) \(\ds 9 T_1\) \(=\) \(\ds 3 \paren {8 - T_1}\)
\(\ds \leadsto \ \ \) \(\ds 24\) \(=\) \(\ds 12 T_1\)
\(\ds \leadsto \ \ \) \(\ds T_1\) \(=\) \(\ds 2\)
\(\ds \leadsto \ \ \) \(\ds T_2\) \(=\) \(\ds 6\)
\(\ds \leadsto \ \ \) \(\ds D\) \(=\) \(\ds 2 \times 9\)
\(\ds \) \(=\) \(\ds 18\)

$\blacksquare$


Sources

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