We might be misunderstanding each other.
In any case, the last step is not an equality. If it were, we could treat it like other equalities.
By multiplying both sides of the equation with 15, we would get:
1 mile / 1 minute = 1 minute / 1 mile
<=> 15 miles / 1 minute = 15 minutes / 1 mile
And surely, running 15 miles in 1 minute is much faster than taking 15 minutes to run 1 mile.