The key to solving this question is to recognise that time increments in the Poisson distribution are independent. Given a number of events $N_t$ occurring by time $t$, we want to solve $$ \text{E} \left[ N_{100} | N_{60} = 7 \right]. $$ Regardless of the number of events that have happened up to 60 seconds, the expected number of events in the next $t$ seconds is given by $$ \text{E} \left[ N_t \right] = \lambda t. $$ For the next 40 seconds then, the expected number of events is $$ \text{E}\left[ N_{40} \right] = 0.25 \times 40 = 10. $$ Therefore, the expected number of events at $t=100$ is 17.