Your reasoning is faulty . . .
Six machines, working at the same constant rate together can complete a certain job in 12 days.
How many additional machines, each working at the same constant rate, are needed to complete the job in 8 days?
I baby-talked my way through the problem like this . . .
That final fraction equals one whole job.
. . Hence:-
Nine machines are required.
Therefore, 3 additional machines are needed.