Question

At what point in its motion is the kinetic energy of the end of a pendulum greatest? At what point is its potential energy greatest when it's kinetic energy is half its greatest value how much potential energy did it gain

Answer 1

-- The kinetic energy of any object is greatest when it's moving the fastest.
For the weight at the end of a pendulum, that's the instant when it passes
through the bottom of the arc.  That's the same place it would be if it were
not swinging at all.

-- An object's potential energy is greatest when it's in the highest position
of all.  For the weight at the end of a pendulum, that's the instant when it's
at either end of the arc, momentarily standing still and ready to reverse its
direction.  That's the height to which you lifted it, just before you let it go
to start swinging.

-- The potential and kinetic energy of an ideal pendulum keep trading off,
and their sum is always the same total.  It's exactly the amount of energy
you put into the pendulum when you lifted the weight up to some height
and let it go. 

As it falls from that height, it loses some potential energy because it's
not as high, and it gains some kinetic energy because it's moving faster. 
The amount it gains and the amount it loses are equal, and the sum of
the potential and kinetic energy never changes. 

Whenever the kinetic energy has reached half of its greatest value, the
potential energy has lost half of its greatest value.  That means that this
must happen when the weight is at half of its greatest height, midway (in
height) between the bottom and the ends of the arc.

Answer 2

The kinetic energy reaches a max once the mass returns to its equilibrium position. At this point all of the potential energy which is greatest at its maximum displacement from the center, turns into kinetic energy.