Answer:
i) 28
ii) 19
Step-by-step explanation:
In a certain class, we are given that:
- 12 students study History.
- 13 students study Government.
- 15 students study Arts.
- Each student studies at least one of the three subjects.
- 6 students study two of the three subjects.
- 3 students study all three subjects.
To construct a Venn diagram to represent the given information, start by drawing a rectangle to represent the universal set.
Now, draw three overlapping circles within the rectangle. Label one set H to represent the set of students studying History, label one set G to represent the set of students studying Government, and label one set A to represent the set of students studying Arts.
The overlapping region of the three circles indicates the intersection of all three sets, representing the number of students who study all three subjects. Given that 3 students study all three subjects, place the number 3 in this overlapping region of the Venn diagram.
Let x be the number of students who study History and Government.
Let y be the number of students who study Government and Arts.
Let z be the number of students who study History and Arts.
Given that 6 students study two of the subjects, then:
[tex]x + y + z = 6[/tex]
Since 12 students study History, the number of students who study History only is:
[tex]\textsf{History only}=12 - x -3 -z\\\\\textsf{History only}=9-x-z[/tex]
Since 13 students study Government, the number of students who study Government only is:
[tex]\textsf{Government only}=13-x-3-y\\\\\textsf{Government only}=10-x-y[/tex]
Since 15 students study Arts, the number of students who study Arts only is:
[tex]\textsf{Arts only}=15-z-3-y\\\\\textsf{Arts only}=12-z-y[/tex]
Since each student in the class studies at least one of the three subjects, the total number of students is the sum of the values contained with the three circles, and can be represented by the equation:
[tex]\textsf{Total students}=\textsf{H only} +\textsf{G only}+\textsf{A only}+\textsf{2 subjects}+\textsf{3 subjects}\\\\\textsf{Total students}=(9-x-z)+(10-x-y)+(12-z-y)+(x+y+z)+3 \\\\ \textsf{Total students}=34-x-z-y \\\\ \textsf{Total students}=34-(x+y+z)[/tex]
As x + y + z = 6, then:
[tex]\textsf{Total number of students}=34-6\\\\ \textsf{Total number of students}=28[/tex]
Therefore, the total number of students in the class is 28.
The total number of students who study only one subject each can be represented by the equation:
[tex]\textsf{Students who study one subject}=\textsf{H only} +\textsf{G only}+\textsf{A only}\\\\\textsf{Students who study one subject}=(9-x-z)+(10-x-y)+(12-z-y) \\\\ \textsf{Students who study one subject}=31-2x-2y-2z \\\\ \textsf{Students who study one subject}=31-2(x+y+z)[/tex]
As x + y + z = 6, then:
[tex]\textsf{Students who study one subject}=31-2(6) \\\\\textsf{Students who study one subject}=31-12 \\\\\textsf{Students who study one subject}=19[/tex]
Therefore, the total number of students who each study only one subject each is 19.
