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:

  • A|A| = Number of students who used package A
  • B|B| = Number of students who used package B
  • C|C| = Number of students who used package C

We will also define:

  • xx = Number of students who used both packages A and C but not B
  • yy = 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:

  • ABC=8|A \cap B \cap C| = 8 (students who used all three packages)
  • The region ABA \cap B but not CC: 9 students (given)
  • The region ACA \cap C but not BB: xx students (unknown)
  • The region BCB \cap C but not AA: yy 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×A|B| = 2 \times |A|

Let's express A|A| and B|B| in terms of xx and yy:

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

Given the condition B=2×A|B| = 2 \times |A|, we get the equation: 38+y=2(25+x)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=658 + 21 + 17 + 9 + x + y + 8 = 65 Simplifying this: 63+x+y=6563 + x + y = 65 x+y=2x + y = 2

Now, the system of simultaneous equations is:

  1. x+y=2x + y = 2
  2. 38+y=50+2x38 + y = 50 + 2x

Solving:

From the first equation: y=2xy = 2 - x

Substitute into the second equation: 38+(2x)=50+2x38 + (2 - x) = 50 + 2x 40x=50+2x40 - x = 50 + 2x 4050=3x40 - 50 = 3x 10=3x-10 = 3x x=103x = -\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=2x + y = 2
  2. 38+y=50+2x38 + y = 50 + 2x

If 38+y=2(25+x)38 + y = 2(25 + x) instead: y=2xy = 2x

Let's substitute: x+2x=2x=23,y=43x + 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?

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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