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:

Hence the result.