It seems like you're asking for two separate things at once:
- An angle that has one ray common with another angle, and whose measure is some fraction of the other angle
- A point on the new angle that is the same fractional distance between the defining points on the original angle
You can do this if:
- Point $B$ is the center of a circle on which $A$ and $C$ lie, and
- Point $D$ also lies on this circle.
In this case, point $D$ will be the same fraction of the way from $B$ to $C$ as the ratio of $\angle ABD$ to $\angle ABC$. In other words, the ratio of the arc lengths is the same as the ratio of the angles.
However, if $B, C, D$ aren't so nicely constrained, in general you won't get both conditions met at once. Absent a circle, the most intuitive way to interpret the distance from $B$ to $C$ would be a straight line, and if $D$ doesn't lie on the line defined by $B$ and $C$, what does "three-fifths of the way from $B$ to $C$" really mean, anyway?
Even constraining $B,C,D$ to be collinear won't get you both conditions at once, because $\angle ABD$ doesn't change proportionally with the distance along $AC$.