Question

the financial aid department at a college surveyed 80 students, asking if they receive any type of financial aid. the results of the survey table below.
financial aid number of students
scholarships 16
student loans 25
private grants 20
scholarships and loans 9
loans and grants 6
scholarships and grants 7
scholarships, loans, and grants 4
part: 0 / 4
part 1 of 4
(a) construct a venn diagram illustrating these results.

Explanation:

Step1: Calculate only scholarships

Only scholarships = Total scholarships - (Scholarships+loans + Scholarships+grants - All three)
$\text{Only scholarships} = 16 - (9 + 7 - 4) = 16 - 12 = 4$

Step2: Calculate only student loans

Only loans = Total loans - (Scholarships+loans + Loans+grants - All three)
$\text{Only loans} = 25 - (9 + 6 - 4) = 25 - 11 = 14$

Step3: Calculate only private grants

Only grants = Total grants - (Scholarships+grants + Loans+grants - All three)
$\text{Only grants} = 20 - (7 + 6 - 4) = 20 - 9 = 11$

Step4: Calculate pairwise only overlaps

Scholarships & loans only = 9 - 4 = 5
Loans & grants only = 6 - 4 = 2
Scholarships & grants only = 7 - 4 = 3

Step5: Calculate no financial aid

No aid = Total students - (Only scholarships + Only loans + Only grants + Scholarships&loans only + Loans&grants only + Scholarships&grants only + All three)
$\text{No aid} = 80 - (4 + 14 + 11 + 5 + 2 + 3 + 4) = 80 - 43 = 37$

Step6: Map to Venn diagram

Draw 3 overlapping circles labeled Scholarships, Student Loans, Private Grants:

• Place 4 in the non-overlapping part of Scholarships
• Place 14 in the non-overlapping part of Student Loans
• Place 11 in the non-overlapping part of Private Grants
• Place 5 in the overlap of Scholarships and Student Loans only
• Place 2 in the overlap of Student Loans and Private Grants only
• Place 3 in the overlap of Scholarships and Private Grants only
• Place 4 in the triple overlap of all three circles
• Write 37 outside all circles (students with no aid)

Answer:

A Venn diagram with 3 overlapping circles has the following values:

• Only Scholarships: 4
• Only Student Loans: 14
• Only Private Grants: 11
• Scholarships + Student Loans only: 5
• Student Loans + Private Grants only: 2
• Scholarships + Private Grants only: 3
• All three (Scholarships, Loans, Grants): 4
• Students with no financial aid (outside circles): 37