Fly puzzle

The two cyclists are headed to each other until collision. The time frame until collision is also the time frame the bird travels. The time frame until collision is

$\text{d = Vt}$

$\text{10 miles = (10 mph)(t)}$

$\text{t = 1 hour}$

The total travel distance of the fly is

$d_{fly} = V_{fly}t = (15 mph)(1 hour) = \boxed{\text{15 miles}}$

Both travel at 10 mph so in one hour the both will reach the center and collide. This means the fly has one hour for its travels. In that time it travels 15 miles.

Total distance $=20$ miles

Speed of Cycle $_1 = 10$ mph

Speed of Cycle $_2 = 10$ mph

Since the cycles are moving in opposite directions,

Relative speed $= 10$ mph $+ 10$ mph $=20$ mph

Relative speed $=20$ mph

Total distance $=20$ miles

Time taken $=$ Total distance $\div$ Relative speed $= 1$ hour

The fly runs around for $1$ hour.

Speed of fly $=15$ mph

Hence, it travels $\boxed{15}$ miles.