In a class of 70 students, 45 offer Mathematics, 37 offer Chemistry and 43 offer
Physics. Five (5) offer all the three subjects, 20 Physics and Chemistry only, three (3)
offer Physics only. Each student offers at least one of the three subjects. Illustrate the
information in a Venn diagram. Find the number of students who offer:
(a) Chemistry only, (b) only one subject, (c) only two subjects.

We are given a class of 70 students where students are enrolled in Mathematics, Chemistry, and Physics. The aim is to organize the data into a Venn diagram and find specific quantities regarding students offering only one or two subjects.

Let:
- \( n(M) \) represent the number of students offering Mathematics.
- \( n(C) \) represent the number of students offering Chemistry.
- \( n(P) \) represent the number of students offering Physics.
- \( n(M \cap C \cap P) = 5 \) represent the number of students offering all three subjects.
- The total number of students is 70, and each student offers at least one subject.

### Given Data:
- \( n(M) = 45 \)
- \( n(C) = 37 \)
- \( n(P) = 43 \)
- 5 students offer all three subjects.
- 20 students offer Physics and Chemistry only.
- 3 students offer Physics only.

We can now organize this information and solve step by step.

#### Step 1: Define regions
Let’s define the number of students offering various combinations of subjects:
- \( x_1 \) = Students offering only Mathematics
- \( x_2 \) = Students offering only Chemistry
- \( x_3 \) = Students offering only Physics
- \( x_4 \) = Students offering both Mathematics and Chemistry (but not Physics)
- \( x_5 \) = Students offering both Mathematics and Physics (but not Chemistry)
- \( x_6 \) = Students offering both Physics and Chemistry (but not Mathematics)

From the information provided:
- \( x_3 = 3 \) (students offering only Physics)
- \( x_6 = 20 \) (students offering both Physics and Chemistry only)
- \( x_7 = 5 \) (students offering all three subjects)

#### Step 2: Calculate for other unknowns using given totals

1. Total students offering Physics:  
   \( n(P) = x_3 + x_5 + x_6 + x_7 = 43 \)  
   Substituting values:
   \[
   3 + x_5 + 20 + 5 = 43
   \]
   Solving:
   \[
   x_5 = 15 \quad \text{(students offering both Mathematics and Physics but not Chemistry)}
   \]

2. Total students offering Chemistry:  
   \( n(C) = x_2 + x_4 + x_6 + x_7 = 37 \)  
   Substituting values:
   \[
   x_2 + x_4 + 20 + 5 = 37
   \]
   Solving:
   \[
   x_2 + x_4 = 12
   \]

3. Total students offering Mathematics:  
   \( n(M) = x_1 + x_4 + x_5 + x_7 = 45 \)  
   Substituting values:
   \[
   x_1 + x_4 + 15 + 5 = 45
   \]
   Solving:
   \[
   x_1 + x_4 = 25
   \]

#### Step 3: Solve for \( x_4 \), \( x_1 \), and \( x_2 \)

- From \( x_2 + x_4 = 12 \) and \( x_1 + x_4 = 25 \), subtract the first equation from the second:
  \[
  (x_1 + x_4) - (x_2 + x_4) = 25 - 12
  \]
  \[
  x_1 - x_2 = 13
  \]
  Therefore, \( x_1 = x_2 + 13 \).

- Substitute into \( x_1 + x_4 = 25 \):
  \[
  (x_2 + 13) + x_4 = 25
  \]
  \[
  x_2 + x_4 = 12
  \]
  Solving gives \( x_4 = 12 - x_2 \). Substitute into \( x_1 + x_4 = 25 \) to get \( x_1 + 12 - x_2 = 25 \).

####

In a class of 70 students, 45 offer Mathematics, 37 offer Chemistry and 43 offer Physics. Five (5) offer all the three subjects, 20 Physics and Chemistry only, three (3) offer Physics only. Each student offers at least one of the three subjects. Illustrate the information in a Venn diagram. Find the number of students who offer: (a) Chemistry only, (b) only one subject, (c) only two subjects.

Learn how to use Venn diagrams to illustrate the distribution of 70 students across three subjects: Mathematics, Chemistry, and Physics. Find the number of students offering only one or two subjects using set theory and algebraic principles. Suitable for Grades 10-12.

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