In a class of 38 students, each student

studies at least one of the following subjects:
French, Mathematics and History. The following
table gives further details of the subject studied.
French 9
Mathematics only 8
History only 1
French & Math. 7
French & History 9
History & Math. 8
a) If x is the number of students who
studies all three subjects, illustrate the
information on a Venn diagram.
b) Write an equation in x involving the
number of students in the region of the
diagram.
c) Find the number of students who stdy
i) All the 3 subjects
ii) French

I an answer

I need the answers

Answer

The number of students who study French is 23 and 2 students study all three

Solution

a) To illustrate the information on a Venn diagram, we need to consider three sets: French, Mathematics, and History.

Let's start by drawing three overlapping circles to represent each subject. Label them as F (French), M (Mathematics), and H (History).

In the middle section, where all three circles overlap, we will place the variable x to represent the number of students who study all three subjects.

According to the information given, we know:
- French: 9 students
- Mathematics only: 8 students
- History only: 1 student
- French & Math: 7 students
- French & History: 9 students
- History & Math: 8 students

Place the numbers in the corresponding sections of the Venn diagram:
- In the section labeled F, but not in the other sections, place 9 (French only).
- In the section labeled M, but not in the other sections, place 8 (Mathematics only).
- In the section labeled H, but not in the other sections, place 1 (History only).
- In the overlapping parts, where two subject areas intersect, place the numbers accordingly. For example, in the overlapped section of F and M, place 7 (French & Math).

b) To write an equation involving the number of students in the region of the Venn diagram, we can use the principle of inclusion-exclusion.

The total number of students studying at least one of the three subjects is given as 38. This can be written as:
(F) + (M) + (H) - (F ∩ M) - (F ∩ H) - (H ∩ M) + (F ∩ M ∩ H) = 38

This equation accounts for counting each student only once and removes the double counts when students study more than one subject.

c) To find the number of students who study:
i) All three subjects (x): Look at the overlapped section representing F ∩ M ∩ H, which is the value of x.
ii) French: Add up the number of students in all sections involving French: (F) + (F ∩ M) + (F ∩ H) + (F ∩ M ∩ H)

In this case, the values will depend on the placement of the numbers in the Venn diagram. Count the number of students in each appropriate section to find the answers.

c) ii) you asked how many study French, but your first data information says that 9 study French

Either you meant 9 to be "French only" , or else the question is bogus.
Confirm.