Couriers' Meeting

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: $94$