Analysis of an approximation claim
You case (i) analysis is wrong, consider the input $60 + 60 + 60 + 20 = 200$. The most even split you can do is $60 + 60$ and $60 + 20$ which has a makespan of $120$, thus there is at least one feasible input for which professor Smart's algorithm returns the optimal result.

For case (ii) we know each job size is at most 10. Professor Smart's algorithm returns a makespan of $120$, which means one machine has a sum of jobs of $120$ and thus the other $200 - 120 = 80$.

But since we know each job has at most a size of $10$ we could move jobs over from the bigger instance to the smaller instance at least until the bigger instance hits $110$ and the smaller $90$ while being guaranteed that each job moved reduces the makespan. But $110 \cdot 1.05 &lt; 120$ thus the algorithm is not a $1.05$ approximation algorithm.