Which responsibility is one that would be shared by the adults and older children in the family?
a. education
b. grocery shopping
c. healthcare
d. shelter

I would get more than one answer on this, but here's my thoughts on that.  Shelter & healthcare definitely should be the sole responsibility of the adult.  Grocery shopping would depend on the age of the older children and if they are old enough to drive.  Education seems like the best answer to me because both have to assume some kind of role in that.  The adults are responsible for guidance with the education and the older children need to assume some personal responsibility as well.  So education seems like the best choice.

It depends what families see as a responsibility. Mainly, the adults are seen as the 'carers' of the children, however because the children are older, this makes responsibilities to start to fall on their shoulders. They can all be shared by the parents and children because you are constantly learning throughout life.

It is less likely that the shoe will slip while someone is walking

Answerfarmer's market; (b) would be my second choice