A confusion relates to Lawgic and Sufficent Assumption Questions

RiemannZ

I am very confused with a specific relationship between universal quantifiers and existential quantifiers. This confusion becomes annoying in Assumption Questions. Please help! So, basically this is it:
1. "A-->C + A -->B"
2. "A-->C + A -most->B"
3. "A-->C + A some B"
For each of three given premises, we can conclude the same "B some C" relationship. Though the first part is the same "A-->C", the second part is different. I thought that this difference is understandable, because "A-->B" implies "A-most->B" and "A some B". So, we should have the same conclusion for "B some C". But the problem often arises.

For example, PT 24, LR2, Section 3, Question 19. Sufficient Assumption.

"Every student who walks to school goes home for lunch. It follows that some students who have part-time jobs do not walk to school."
The conclusion of the argument follows logically if which one of the following is assumed?
Premise: Walks to schools-->Goes home for lunch.
Conclusion: Part-time jobs (some) Do not walk to school.
Take the contrapositive of the premise, we have "Do not go home for Lunch--> Do not walk to School"

Now, it becomes clear that he Sufficient Assumption to bridge the gap could be:
1. "Do not go home for lunch (some) Part-time jobs". This is the correct answer choice (d).
(d). Some students who do not go home for lunch have part-time jobs.
2. Do not go home for lunch -most-> Part-time jobs.
3. Do not go home for lunch --> Part-time jobs. (conditional)

If we take the contrapositive of 3, we have "No part-time jobs-->Go home for lunch". The contrapostive is logically equivalent to the original. Now, "No part-time jobs-->Go home for lunch" implies
"No part-time jobs -most-> Go home for lunch." and also implies
"No part-time jobs (some) go home for lunch." (This is exactly what the wrong answer choice A says.)
(a). some students who do not have part-time jobs go home for lunch.

Please help me clear this confusion. Is there anything I misunderstood? I really appreciate your help.