On a bypass engine does the bypass air travel faster or slower than core air?
The simple way to think of it is the bypass engine is kind of part way between a pure jet and a turboprop.   In a nutshell, the pure jet moves a little bit of air a lot; the turboprop's propeller moves a lot of air a little, relatively.  In other words, small mass at high velocity vs large mass at low velocity, to get the same push.  

A bypass jet engine is somewhere in between.  It's just a fixed pitch non-geared turboprop with 30 or 40 blades, in a shroud, instead of 3 or 4 variable pitch blades in the open (geared turbofans get even closer to the "fixed pitch turboprop" thing).   

The package of air being accelerated is smaller than a propeller, but much larger than a pure jet, so fan's exhaust velocity is also in between; faster than the wash of a propeller, slower than the exhaust of the core.