I am overthinking my working when calculating person-time - is it right to subtract total cases from cases in the beginning? Is this correct?

The question is:

An epidemiological study follows 153 disease-free individuals over 12 months. During the follow up five individuals developed the disease after two months. Nine individuals were diagnosed at the end of six months (or at mid-point of the follow up) while three additional cases were recorded at the end of follow up.

1. Please calculate the Person-time in months. Calculate Incidence rate & provide answer as per 1000 person-months

(5 x 2 months) +(9 x 6 months) +(3 x 12 months) = 100 person-months

Total new cases= 5+9+3=17

153-17 = 136

136 x 12 months = 1632 person-months

1632+100

=

1732 person-months

(17/1732) x 1000 = 9.815

IR= 9.82 cases per 1000 person-months