Carburizing Calculation The problem is stated as follows:
The surface of a steel gear made of 1022 steel (0.22 wt% C) is to be gas-carburized at $927$ °C. Calculate the time necessary to increase the carbon content to 0.30 wt% at 0.030 in below the surface of the gear. Assume the carbon content of the surface to be 1.20 wt% and $D_{927^{\circ}C}=1.28 \times 10^{-11}\frac{m^{2}}{s}$

The formula is $\frac{C_{s}-C_{x}}{C_{s}-C_{0}}=ERF(\frac{x}{2\sqrt{Dt}})$

The solution manual is setting $C_{x}=0.30$, but should it not be $C_{x}=0.30+0.22=0.52$, since the initial carbon content in the steel is $C_{0}=0.22$?
No.

In the equation $C_x$ simply means the concentration at depth x. It does not mean the change of concentration. There is no need to add its initial concentration. 

Besides, concentration cannot be linearly added. (Does adding a 1M solution to a 2M solution give 3M?)