Couriers' Meeting
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Classic Problem
The Holy Father sent a courier from Rome to Venice, commanding him that he reach Venice in $7$ days.
The Most Illustrious Signorina of Venice sent a courier from Venice to Rome, directing him to reach Rome in $9$ days.
It is $250$ miles between Rome and Venice.
It so happened that both couriers started at exactly the same time.
In how many days do they meet?
Solution
- $3 \frac {15} {16}$ days.
Proof
Let $A$ and $B$ denote the couriers starting from Rome and Venice respectively.
Let $t$ be the time in days after they set out when they meet.
Let $x$ be the number of miles from Rome where that happens.
We have that:
- $A$ travels at $\dfrac {250} 7$ miles a day
- $B$ travels at $\dfrac {250} 9$ miles a day.
Thus we have:
\(\ds \dfrac {250} 7 t\) | \(=\) | \(\ds x\) | ||||||||||||
\(\ds \dfrac {250} 9 t\) | \(=\) | \(\ds 250 - x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 250 \paren {\dfrac 1 7 + \dfrac 1 9} t\) | \(=\) | \(\ds x + \paren {250 - x}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 250 \paren {\dfrac {16} {63} } t\) | \(=\) | \(\ds 250\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds t\) | \(=\) | \(\ds \dfrac {63} {16}\) |
Hence the result.
$\blacksquare$
Sources
- 1478: Anonymous: Treviso Arithmetic
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): The Couriers Meeting: $94$