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1. In a class of 60 students, some study at least one of the following subjects: Mathematics, Economics, and Accounting. Eight students study none of them. The following table gives further details of the subjects studied.

\begin{tabular}{|l|l|}
\hline Mathematics only & 6 \\
\hline Economics only & 1 \\
\hline Accounting only & 5 \\
\hline All three subjects & 7 \\
\hline Mathematics and Accounting & 18 \\
\hline Economics and Accounting & 17 \\
\hline
\end{tabular}

a. Illustrate the data on a Venn diagram.

b. Find the number of students who study:
i. Mathematics or Accounting or both, but not Economics.
ii. Economics.



Answer :

GinnyAnswer
Absolutely, let's go through the solution step-by-step.

### a. Venn Diagram Illustration

1. Total Number of Students: There are 60 students in total.
2. Students Studying None of the Subjects: 8 students do not study any of the subjects.

With this information, we are left with [tex]\(60 - 8 = 52\)[/tex] students who study at least one subject.

3. Details Provided:
- Mathematics only: 6 students
- Economics only: 1 student
- Accounting only: 5 students
- All three subjects: 7 students
- Mathematics and Accounting: 18 students
- Economics and Accounting: 17 students

We will use three overlapping circles to represent the students studying Mathematics (M), Economics (E), and Accounting (A).

- All three subjects (intersection of all three circles): 7 students
- Mathematics and Accounting but not Economics: The 18 students studying both Mathematics and Accounting include the 7 students who study all three subjects. So, the number of students studying only Mathematics and Accounting is [tex]\(18 - 7 = 11\)[/tex].
- Economics and Accounting but not Mathematics: The 17 students studying both Economics and Accounting include the 7 students who study all three subjects. So, the number of students studying only Economics and Accounting is [tex]\(17 - 7 = 10\)[/tex].

Here's what we have:
- [tex]\(A = 5\)[/tex] (Accounting only)
- [tex]\(B = 1\)[/tex] (Economics only)
- [tex]\(C = 6\)[/tex] (Mathematics only)
- [tex]\(D = 7\)[/tex] (All three subjects)
- [tex]\(E = 11\)[/tex] (Mathematics and Accounting but not Economics)
- [tex]\(F = 10\)[/tex] (Economics and Accounting but not Mathematics)

The Venn Diagram would have:
- 6 in Mathematics only
- 1 in Economics only
- 5 in Accounting only
- 11 in both Mathematics and Accounting only
- 10 in both Economics and Accounting only
- 7 in all three subjects

### b. Finding the Numbers:

#### i. Number of Students Who Study Mathematics or Accounting or Both but Not Economics

From the Venn Diagram, the students who study Mathematics or Accounting or both but not Economics are those who study:
- Mathematics only (6 students)
- Accounting only (5 students)
- Mathematics and Accounting but not Economics (11 students)

So, the total number of students who study Mathematics or Accounting or both but not Economics is:
[tex]\[ 6 + 5 + 11 = 22 \][/tex]

#### ii. Number of Students Who Study Economics

Students who study Economics can be found by summing up all the segments where Economics is involved. This includes:
- Economics only: 1 student
- Economics and Accounting but not Mathematics: 10 students
- All three subjects: 7 students

Thus, the number of students who study Economics is:
[tex]\[ 1 + 10 + 7 = 18 \][/tex]

### Conclusion
1. Venn Diagram: Drawn showing the distribution of students across subjects.
2. Mathematics or Accounting or Both but Not Economics: [tex]\(22\)[/tex] students.
3. Economics: [tex]\(18\)[/tex] students.

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