Why is Delta t / 2 in the denominator of the finite difference approximations of the time derivatives?

This is from Hoffman's CFD Volume I. As part of Problem 3.7, I was coding the fractional step method in Python. Problems 3.5 and 3.6 require you to solve the same problem, but with the FTCS and ADI schemes, respectively. I noticed the results from FSM differed significantly from the FTCS and ADI results. When compared to ADI, FSM yields peak percent error of \~100% and is distributed all throughout the wall.

When I changed the Delta t / 2 to just Delta t, the results of FSM were significantly closer to that of FTCS and ADI. When compared to ADI, percent errors peak around 10% and is limited to small regions near the left/right walls.

The only other textbook I could find that has a section on the FSM was Chung's CFD book, which also has Delta t / 2 in the denominator. However, Hoffman and Chung's sections on the FSM are almost identical (compare the first image with the last), and I don't think they can be considered independent resources.

Is there a mistake in their equations, or is it likely that there is an error in my code.