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Question

Three major sources of financial aid are government grants, private scholarships, and the colleges themselves. Susan, the director of a small private college, surveyed the records of 100 sophomores and found

49 receive government grants 55 receive private sholarships 43 receive aid from the college 23 receive government grants and private scholarships 18 receive government grants and aid from the college 28 receive private scholarships and aid from the college 8 receive help from all three sources

How many of the students in the survey (e) receive no financial aid from any of these sources?

Answer 1

This is a problem dealing with sets of students.  

49 = G = government grant

55 = P = private scholarship

43 = C = college aid

23 = G ∩ P

18 = G ∩ C

28 = P ∩ C

8 = G ∩ P ∩ C

The easiest way to solve this problem is to draw a Venn Diagram with a total population of 100, 3 circles (one for each type of finaid, with 3 binary intersections, and 1 triple intersection.  

 

For example with 8 students having all 3 forms of aid, 23-8 = 15 students have exactly P and G,

18-8 = 10 students have exactly G and C

28-8 - 20 students have exactly P and C

those students having only G = 49 - 15 - 8 - 10 = 16

those student shaving only P = 55 - 15 - 8 - 20 = 12

those students having only C = 43 - 20 - 8 - 10 = 5

Having found all the students with exactly 1, 2, and 3 finaid sources, we can add them and find that this totals to 86

'This then leads to 100 - 86 = 4 students with no finaid at all.