How do pure social science ang applied social science alike with each other

Applied social science is more practically oriented towards activity in the community, that helps people. Fields like social work or applied anthropology are examples of fields that are practically focused but draw from social theory. "Pure" social science is more academically oriented to generating theoretical insights for a scholarly or academic audience primarily, not the general public.

Explanation:

Applied fields in the social sciences tend to work with organizations and in communities and they attempt to apply the theories that have been developed in academia regarding inequality, for instance, by recognizing that structural inequalities exist and that disadvantaged populations like single mothers or migrants might need assistance in overcoming the barriers and understanding how best to position themselves in society. Applied anthropologists, for example, tend to work with NGOs and non-profit organizations and study sustainable development initiatives and the viewpoints of the communities in remote parts of the world or in areas where there is great urban poverty. They write reports for government agencies and their organizations that are shared with other analysts and policymakers. Academic social scientists will use grant money and their university influence to conduct research that is then usually published in academic journals and books by university presses.  

d. i assume the underlined word is "role." reformat if it isn't.

explanation:

role is closest to part because when replacing words, "playing a vital decision," "playing a vital activity," and "playing a vital dollar" don't make logical sense. the only meaningful choice is "playing a vital part."

this fits one of merriam-webster's definitions of role: "a function or part performed especially in a particular operation or process."