In a school, 100 students have access to three software packages A, B and C. 28 did not use any software , 8 used only packages A 26 used only packages B , 7 used only packages C. 10 used all three packages. , 13 used both A and B.1. Draw a Venn diagram with all sets enumerated as for as possible. Label the two subsets which cannot be enumerated as x and y in any order.2. If twice as many students used package B as package A, write down a pair of simultaneous equations in x and y.3. Solve the equations to find x and y. 4. How many students used package C?
Answers
Given : 100 students have access to three software packages A, B and C. 28 did not use any software , 8 used only packages A 26 used only packages B , 7 used only packages C. 10 used all three packages. , 13 used both A and B
Label the two subsets which cannot be enumerated as x and y in any order
twice as many students used package B as package A,
To Find : Draw a Venn diagram
Find x and y
Number of students used package C
Solution:
A only = 8
B only = 26
C only = 7
All three = 10
both A and B = 13
both A and B but not C = 13 - 10 = 3
both A and C but not B = x
both B and C but not A = y
None = 28
Total = 100
8 + 26 + 7 + 10 + 3 + x + y + 28 = 100
82 + x + y = 100
=> x + y = 18
twice as many students used package B as package A,
B used = 26 + 13 + y = 39 + y
A used = 8 + 13 + x = 21 + x
=> 39 + y = 2(21 + x)
=> 39 + y = 42 + 2x
=> y = 2x + 3
x + y = 18
=> x + 2x + 3 = 18
=> 3x = 15
=> x = 5
Hence y = 13
C used = 7 + 10 + x + y = 17 + 18 = 35
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