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In a class of 120 students, 70 take Math, 65 take Physics, and 50 take Chemistry. 30 take Math and Physics, 25 take Math and Chemistry, 20 take Physics and Chemistry, and 10 take all three. How many s

Mathematics

ScanSolve AI Answer

Answers

3 parts
a

25\text{25}

b

45\text{45}

c

0\text{0}

Step-by-step solution

3 steps

Step 1: Identify the given information.
Let UU be the universal set of students.
Let MM be the set of students taking Mathematics.
Let PP be the set of students taking Physics.
Let CC be the set of students taking Chemistry.

We are given:
Total number of students, ∣U∣=120|U| = 120
Number of students taking Mathematics, ∣M∣=70|M| = 70
Number of students taking Physics, ∣P∣=65|P| = 65
Number of students taking Chemistry, ∣C∣=50|C| = 50
Number of students taking Mathematics and Physics, ∣M∩P∣=30|M \cap P| = 30
Number of students taking Mathematics and Chemistry, ∣M∩C∣=25|M \cap C| = 25
Number of students taking Physics and Chemistry, ∣P∩C∣=20|P \cap C| = 20
Number of students taking all three subjects, ∣M∩P∩C∣=10|M \cap P \cap C| = 10

Step 2: Calculate the number of students in each disjoint region of the Venn diagram.

  • Students taking exactly two subjects:
∣M∩P only∣=∣M∩P∣−∣M∩P∩C∣=30−10=20|M \cap P \text{ only}| = |M \cap P| - |M \cap P \cap C| = 30 - 10 = 20 ∣M∩C only∣=∣M∩C∣−∣M∩P∩C∣=25−10=15|M \cap C \text{ only}| = |M \cap C| - |M \cap P \cap C| = 25 - 10 = 15 ∣P∩C only∣=∣P∩C∣−∣M∩P∩C∣=20−10=10|P \cap C \text{ only}| = |P \cap C| - |M \cap P \cap C| = 20 - 10 = 10
  • Students taking exactly one subject:
∣M only∣=∣M∣−(∣M∩P only∣+∣M∩C only∣+∣M∩P∩C∣)|M \text{ only}| = |M| - (|M \cap P \text{ only}| + |M \cap C \text{ only}| + |M \cap P \cap C|) ∣M only∣=70−(20+15+10)=70−45=25|M \text{ only}| = 70 - (20 + 15 + 10) = 70 - 45 = 25 ∣P only∣=∣P∣−(∣M∩P only∣+∣P∩C only∣+∣M∩P∩C∣)|P \text{ only}| = |P| - (|M \cap P \text{ only}| + |P \cap C \text{ only}| + |M \cap P \cap C|) ∣P only∣=65−(20+10+10)=65−40=25|P \text{ only}| = 65 - (20 + 10 + 10) = 65 - 40 = 25 ∣C only∣=∣C∣−(∣M∩C only∣+∣P∩C only∣+∣M∩P∩C∣)|C \text{ only}| = |C| - (|M \cap C \text{ only}| + |P \cap C \text{ only}| + |M \cap P \cap C|) ∣C only∣=50−(15+10+10)=50−35=15|C \text{ only}| = 50 - (15 + 10 + 10) = 50 - 35 = 15

Step 3: Answer part (a).

Part a· 1 step

How many students take only mathematics?

The number of students who take only Mathematics is ∣M only∣|M \text{ only}|.

∣M only∣=25|M \text{ only}| = 25

The number of students who take only mathematics is 25\boxed{\text{25}}.

  1. Answer part (b).

Part b

45\text{45}

How many take exactly two subjects?

The number of students who take exactly two subjects is the sum of those taking (M and P only), (M and C only), and (P and C only).

Exactly two subjects=∣M∩P only∣+∣M∩C only∣+∣P∩C only∣\text{Exactly two subjects} = |M \cap P \text{ only}| + |M \cap C \text{ only}| + |P \cap C \text{ only}| Exactly two subjects=20+15+10=45\text{Exactly two subjects} = 20 + 15 + 10 = 45

The number of students who take exactly two subjects is 45\boxed{\text{45}}.

  1. Answer part (c).

Part c

0\text{0}

How many students take none of the three subjects?

First, calculate the total number of students taking at least one subject:

∣M∪P∪C∣=∣M only∣+∣P only∣+∣C only∣+∣M∩P only∣+∣M∩C only∣+∣P∩C only∣+∣M∩P∩C∣|M \cup P \cup C| = |M \text{ only}| + |P \text{ only}| + |C \text{ only}| + |M \cap P \text{ only}| + |M \cap C \text{ only}| + |P \cap C \text{ only}| + |M \cap P \cap C| ∣M∪P∪C∣=25+25+15+20+15+10+10=120|M \cup P \cup C| = 25 + 25 + 15 + 20 + 15 + 10 + 10 = 120

The number of students taking none of the three subjects is ∣U∣−∣M∪P∪C∣|U| - |M \cup P \cup C|.

None of the subjects=120−120=0\text{None of the subjects} = 120 - 120 = 0

The number of students who take none of the three subjects is 0\boxed{\text{0}}.

  1. Answer part (d).

Part d

Draw a Venn diagram with all the information.

As an AI, I cannot physically draw a diagram. However, I can describe the Venn diagram by listing the number of students in each distinct region:

  • Students taking only Mathematics: 25
  • Students taking only Physics: 25
  • Students taking only Chemistry: 15
  • Students taking Mathematics and Physics only (not Chemistry): 20
  • Students taking Mathematics and Chemistry only (not Physics): 15
  • Students taking Physics and Chemistry only (not Mathematics): 10
  • Students taking Mathematics, Physics, and Chemistry: 10
  • Students taking none of the subjects: 0
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