If 164 of the 400 eighth-graders take none of these courses, then 400-164=236 students take some courses.
Let x students take all three courses, then:
1. 70-x take only both algebra and advanced computer,
2. 34-x take only both algebra and industrial technology,
3. 29-x take only both advanced computer and industrial technology.
Now let's count how many student take only one course:
1. 117-(x+34-x+70-x)=13+x take algebra,
2. 109-(x+29-x+70-x)=10+x take advanced computer,
3. 114-(x+29-x+34-x)=51+x take industrial technology.
Now count all students that take some courses:
51+x+10+x+13+x+29-x+34-x+70-x+x=236,
x=236-51-10-13-29-34-70,
x=29.
Answer: all three courses take 29 students.
Number of students that take all three courses is 29 students
A set is a clearly defined collection of objects.
To declare a set can be done in various ways such as:
Multiplying set A x B is by pairing each member of set A with each member of set B.
Example:
A = {1, 2, 3}
B = {a, b}
Then
A x B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}
Union of set A and B ( A ∪ B ) is rewriting each member A and combined with each member B.
Intersection of set A and B ( A ∩ B ) is to find the members that are both in Set A and Set B.
Example:
A = {1, 2, 3, 4}
B = {3, 4, 5}
A ∪ B = {1, 2, 3, 4, 5}
A ∩ B = {3, 4}
Let us now tackle the problem!
To solve this problem, it is better to draw the Venn diagram as shown in the picture in the attachment.
Let :
x → students take all three courses
y = 34 - x → students only take both algebra and industrial technology
z = 29 - x → students only take both computer and industrial technology
w = 70 - x → students only take both algebra and computer
p = 117 - w - x - y = 13 + x → students only take algebra
q = 114 - x - y - z = 51 + x → students only take industrial technology
r = 109 - x - z - w = 10 + x → students only take computer
t = 164 → students take none of these courses
Because the total number of students is 400 people, then :
[tex]p + q + r + x + y + z + w + t = 400[/tex]
[tex](13 + x) + (51 + x) + ( 10 + x ) + x + (34 - x) + (29 - x) + (70 - x) + 164 = 400[/tex]
[tex]371 + x = 400[/tex]
[tex]x = 400 - 371[/tex]
[tex]\large {\boxed{x = 29} }[/tex]
Grade: High School
Subject: Mathematics
Chapter: Sets
Keywords: Sets , Venn , Diagram , Intersection , Union , Mean , Median , Mode
