Solving Problems with Venn Diagrams - Free Printable

To solve the problem using the given Venn diagram and data, we will follow a step-by-step approach. Let's break it down:

Given Data:

- Total students interviewed: \( 100 \)
- Students who took:
- PE: \( 28 \)
- BIO: \( 31 \)
- ENG: \( 42 \)
- Students who took combinations of subjects:
- PE and BIO: \( 9 \)
- PE and ENG: \( 10 \)
- BIO and ENG: \( 6 \)
- All three subjects (PE, BIO, and ENG): \( 4 \)

Venn Diagram Information:

From the Venn diagram:
- The intersection of all three circles (PE, BIO, and ENG) is \( 4 \).
- The intersection of PE and BIO only (excluding ENG) is \( 9 - 4 = 5 \).
- The intersection of PE and ENG only (excluding BIO) is \( 10 - 4 = 6 \).
- The intersection of BIO and ENG only (excluding PE) is \( 6 - 4 = 2 \).

Step 1: Calculate the number of students who took only one subject.

#### Only PE:
Students who took only PE are those who took PE but not BIO or ENG. This can be calculated as:
\[
\text{Only PE} = \text{Total PE} - (\text{PE and BIO} + \text{PE and ENG} - \text{All three})
\]
\[
\text{Only PE} = 28 - (9 + 10 - 4) = 28 - 15 = 13
\]

#### Only BIO:
Students who took only BIO are those who took BIO but not PE or ENG. This can be calculated as:
\[
\text{Only BIO} = \text{Total BIO} - (\text{PE and BIO} + \text{BIO and ENG} - \text{All three})
\]
\[
\text{Only BIO} = 31 - (9 + 6 - 4) = 31 - 11 = 20
\]

#### Only ENG:
Students who took only ENG are those who took ENG but not PE or BIO. This can be calculated as:
\[
\text{Only ENG} = \text{Total ENG} - (\text{PE and ENG} + \text{BIO and ENG} - \text{All three})
\]
\[
\text{Only ENG} = 42 - (10 + 6 - 4) = 42 - 12 = 30
\]

Step 2: Calculate the total number of students who took at least one subject.

The total number of students who took at least one subject is the sum of:
- Only PE
- Only BIO
- Only ENG
- PE and BIO only
- PE and ENG only
- BIO and ENG only
- All three subjects

\[
\text{Total taking at least one subject} = (\text{Only PE}) + (\text{Only BIO}) + (\text{Only ENG}) + (\text{PE and BIO only}) + (\text{PE and ENG only}) + (\text{BIO and ENG only}) + (\text{All three})
\]
\[
\text{Total taking at least one subject} = 13 + 20 + 30 + 5 + 6 + 2 + 4 = 80
\]

Step 3: Calculate the number of students who took none of the three subjects.

The number of students who took none of the three subjects is the total number of students interviewed minus the number of students who took at least one subject:
\[
\text{None of the three subjects} = \text{Total students} - \text{Total taking at least one subject}
\]
\[
\text{None of the three subjects} = 100 - 80 = 20
\]

Step 4: Answer the specific questions.

1. How many students took none of the three subjects?
\[
\boxed{20}
\]

2. How many students took PE, but not BIO or ENG?
From the calculation above:
\[
\boxed{13}
\]

3. How many students took BIO and PE but not ENG?
From the Venn diagram, the number of students who took BIO and PE but not ENG is:
\[
\boxed{5}
\]

Final Answers:

1. \( \boxed{20} \)
2. \( \boxed{13} \)
3. \( \boxed{5} \)