Question

Of the 400 eighth-graders at pascal middle school, 117 take algebra, 109 take advanced computer, and 114 take industrial technology. furthermore, 70 take both algebra and advanced computer, 34 take both algebra and industrial technology, and 29 take both advanced computer and industrial technology. finally, 164 students take none of these courses. how many students take all three courses

Answer 1

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.

Answer 2

Number of students that take all three courses is 29 students

Further explanation

A set is a clearly defined collection of objects.

To declare a set can be done in various ways such as:

• With words or the nature of membership

• With set notation

• By registering its members

• With Venn diagrams

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 :

$p + q + r + x + y + z + w + t = 400$

$(13 + x) + (51 + x) + ( 10 + x ) + x + (34 - x) + (29 - x) + (70 - x) + 164 = 400$

$371 + x = 400$

$x = 400 - 371$

$\large {\boxed{x = 29} }$

Learn more

• Mean , Median and Mode : brainly.com/question/2689808
• Centers and Variability : brainly.com/question/3792854
• Subsets of Set : brainly.com/question/2000547

Answer details

Grade: High School

Subject: Mathematics

Chapter: Sets

Keywords: Sets , Venn , Diagram , Intersection , Union , Mean , Median , Mode