Math Problem Statement
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?
Solution
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:
- = Number of students who used package A
- = Number of students who used package B
- = Number of students who used package C
We will also define:
- = Number of students who used both packages A and C but not B
- = 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:
- (students who used all three packages)
- The region but not : 9 students (given)
- The region but not : students (unknown)
- The region but not : 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:
Let's express and in terms of and :
Given the condition , we get the equation:
Step iii: Solve the Simultaneous Equations
We also have another equation from the total number of students who used at least one package:
Simplifying this:
Now, the system of simultaneous equations is:
Solving:
From the first equation:
Substitute into the second equation:
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:
If instead:
Let's substitute: 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?
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Math Problem Analysis
Mathematical Concepts
Set theory
Simultaneous equations
Venn diagrams
Formulas
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Theorems
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Suitable Grade Level
Advanced High School
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