Question

the financial aid department at a college surveyed 44 students, asking if they receive any type of financial aid. the results of the survey are summarized in the table below.

financial aid	number of students
student loans	11
private grants	17
scholarships and loans	6
loans and grants	7
scholarships and grants	8
scholarships, loans, and grants	4

part: 0 / 4
part 1 of 4
(a) construct a venn diagram illustrating these results.

Explanation:

Step1: Identify 3 sets: S(scholarships), L(loans), G(grants)

We first define the three core sets for the Venn diagram:

• $S$: Students with Scholarships
• $L$: Students with Student loans
• $G$: Students with Private grants

Step2: Calculate all 3-set overlap

The number of students with all three aids is given directly:
$n(S \cap L \cap G) = 4$

Step3: Calculate pairwise only overlaps

Scholarships and Loans only:

Subtract the 3-set overlap from the total of scholarships and loans:
$n(S \cap L \text{ only}) = 6 - 4 = 2$

Loans and Grants only:

Subtract the 3-set overlap from the total of loans and grants:
$n(L \cap G \text{ only}) = 7 - 4 = 3$

Scholarships and Grants only:

Subtract the 3-set overlap from the total of scholarships and grants:
$n(S \cap G \text{ only}) = 8 - 4 = 4$

Step4: Calculate single-set only counts

Scholarships only:

Subtract all overlapping groups from total scholarships:
$n(S \text{ only}) = 14 - 2 - 4 - 4 = 4$

Loans only:

Subtract all overlapping groups from total loans:
$n(L \text{ only}) = 11 - 2 - 3 - 4 = 2$

Grants only:

Subtract all overlapping groups from total grants:
$n(G \text{ only}) = 17 - 4 - 3 - 4 = 6$

Step5: Calculate no financial aid

Subtract all aid recipients from total surveyed students:
$n(\text{No aid}) = 44 - (4 + 2 + 6 + 2 + 3 + 4 + 4) = 44 - 25 = 19$

Answer:

The Venn diagram will have three overlapping circles (Scholarships, Loans, Grants) with the following values:

• Scholarships only: 4
• Loans only: 2
• Grants only: 6
• Scholarships and Loans only: 2
• Loans and Grants only: 3
• Scholarships and Grants only: 4
• All three (Scholarships, Loans, Grants): 4
• Outside all circles (no financial aid): 19