Problem:

In a class of 50 students:

• 30 students study Mathematics.
• 25 students study Science.
• 10 students study both Mathematics and Science.

How many students:

1. Study only Mathematics?
2. Study only Science?
3. Study either Mathematics or Science?
4. Study neither Mathematics nor Science?

Use a Venn Diagram with two circles to represent this data and solve the questions.

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Solution:

Let's define:

• $M$: The set of students who study Mathematics.
• $S$: The set of students who study Science.
• $M \cap S$: The set of students who study both subjects.

From the problem:

• Total students = 50
• $|M| = 30$
• $|S| = 25$
• $|M \cap S| = 10$

Step 1: Finding Students in Each Category

1. Students who study only Mathematics:
   This group includes students in $M$ but not in $S$. $|M \text{ only}| = |M| - |M \cap S| = 30 - 10 = 20$

2. Students who study only Science:
   This group includes students in $S$ but not in $M$. $|S \text{ only}| = |S| - |M \cap S| = 25 - 10 = 15$

3. Students who study either Mathematics or Science:
   This includes students in $M \cup S$. $|M \cup S| = |M| + |S| - |M \cap S| = 30 + 25 - 10 = 45$

4. Students who study neither Mathematics nor Science:
   These students are outside both sets $M$ and $S$. $\text{Neither} = \text{Total students} - |M \cup S| = 50 - 45 = 5$

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Summary of Answers:

1. Only Mathematics: 20 students
2. Only Science: 15 students
3. Either Mathematics or Science: 45 students
4. Neither Mathematics nor Science: 5 students

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Would you like additional explanation, or have any questions?

Related Questions:

1. How many students study only one subject?
2. What percentage of students study both Mathematics and Science?
3. If 5 more students join Mathematics, how does that affect the Venn Diagram?
4. How would this change if we added a third subject?
5. What is the probability a randomly selected student studies both subjects?

Tip:

For Venn Diagrams with two sets, use the formula $|M \cup S| = |M| + |S| - |M \cap S|$ to find the union quickly.