Two gear system in static equilibrium

[Jage]

Homework Statement

Two solid steel shafts are coupled together using meshed gears. Determine the angle of twist of end A. Shaft DC is fixed at D.

Homework Equations

angle of twist = torque * shaft length / ( polar moment of inertia * shear modulus)

balance of forces where the two gears mesh:
150mm * F_ab = 0.75mm * F_cd

The Attempt at a Solution

I know the solution, but am puzzled about what is wrong with the following approach:

The sum of forces and moments on the system must equal zero since the system is at static equilibrium. Since torque exerted on the system at A is 45Nm, the torque exerted on the system at D must therefore be -45 Nm.

However, balancing forces at where the two gears mesh shows that torque at D is really -22.5 Nm.

 

[Simon Bridge]

To balance torques, they must be about the same axis.