In this circuit, we determine that the emf needs to be reversed and that its value is 108.75 V.
My confusion is that I thought the voltage between any two points must be the same irrespective of the path taken.
For instance, in the parallel group of 15.0 and 20.0 resistors, let's call the point on the right a and left b. From the top branch, V_ab = 108.75 - 15(4.25) = +45V. From the bottom branch, V_ab = (-2.25)20 = -45V.
I see how this is true from Kirchoff's loop rule, but it violates my previous intuition.
(A copy of the problem can be found in (PPT file): physics.wku.edu/~womble/phys260/ch26exp1.ppt)