A survey conducted amongst 100 school students showed that:
35 did not use any word processing packages,
8 used all three of the packages A, B and C, 8 used only package A,
21 used only package B, 17 used only package C, 9 used both package A and B
i.	Draw a Venn diagram with all the sets enumerated as far as possible. Label the two subsets which cannot be enumerated as x and y in any order.
ii.	If twice as many students used package B as package A, write down a pair of simultaneous equations in x and y.
iii.	Solve these equations to find x and y.
iv.	How many students used package B?

### Given Information

- Total number of students surveyed: 100
- Number of students who did not use any word processing packages: 35
- Number of students who used all three packages (A, B, and C): 8
- Number of students who used only package A: 8
- Number of students who used only package B: 21
- Number of students who used only package C: 17
- Number of students who used both packages A and B (but not C): 9

Let's define the following:
- \( |A| \) = Number of students who used package A
- \( |B| \) = Number of students who used package B
- \( |C| \) = Number of students who used package C

We will also define:
- \( x \) = Number of students who used both packages A and C but not B
- \( y \) = Number of students who used both packages B and C but not A

### Step i: Draw a Venn Diagram

We need to enumerate the regions of the Venn diagram as follows:

- \( |A \cap B \cap C| = 8 \) (students who used all three packages)
- The region \( A \cap B \) but not \( C \): 9 students (given)
- The region \( A \cap C \) but not \( B \): \( x \) students (unknown)
- The region \( B \cap C \) but not \( A \): \( y \) students (unknown)
- The region where only A is used: 8 students
- The region where only B is used: 21 students
- The region where only C is used: 17 students
- Total number of students who used at least one package = 100 - 35 = 65 students

### Step ii: Write Down the Simultaneous Equations

Given that twice as many students used package B as package A:
\[ |B| = 2 \times |A| \]

Let's express \( |A| \) and \( |B| \) in terms of \( x \) and \( y \):

\[ |A| = 8 + 9 + x + 8 = 25 + x \]
\[ |B| = 21 + 9 + y + 8 = 38 + y \]

Given the condition \( |B| = 2 \times |A| \), we get the equation:
\[ 38 + y = 2(25 + x) \]

### Step iii: Solve the Simultaneous Equations

We also have another equation from the total number of students who used at least one package:

\[ 8 + 21 + 17 + 9 + x + y + 8 = 65 \]
Simplifying this:
\[ 63 + x + y = 65 \]
\[ x + y = 2 \]

Now, the system of simultaneous equations is:
1. \( x + y = 2 \)
2. \( 38 + y = 50 + 2x \)

### Solving:

From the first equation:
\[ y = 2 - x \]

Substitute into the second equation:
\[ 38 + (2 - x) = 50 + 2x \]
\[ 40 - x = 50 + 2x \]
\[ 40 - 50 = 3x \]
\[ -10 = 3x \]
\[ x = -\frac{10}{3} \]

However, since x cannot be negative in this context, let's review our calculations.

The calculation gives a non-feasible result, suggesting there might be an inconsistency in the assumptions. Let me recheck the situation.

---

Upon reviewing:
1. \( x + y = 2 \)
2. \( 38 + y = 50 + 2x \)

If \( 38 + y = 2(25 + x) \) instead:
\[ y = 2x \]

Let's substitute:
\[ x + 2x = 2 \Rightarrow x = \frac{2}{3}, y = \frac{4}{3} \]  which doesn't match.

Conclusion: Recheck assumptions.

Let's explore another approach.

### Conclusion: 
Mismatch in assumptions suggests more than twice used packages. Venn suggests 25 B, 12 used A, needing recalculation.

Would you like a recalculated Venn scenario or proceed?

Explore how to solve complex set theory problems using Venn diagrams and simultaneous equations. This example involves analyzing student survey data on word processing package usage, including detailed steps and calculations. Suitable for advanced high school students.

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