n 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 assume that you meant 9 study French only. Otherwise, it's impossible to have 9 who study French and also 9 who study French and History, and x>0 who study all three.
That said, the Venn diagram makes it clear that
9+7+8+9+1+9+x = 38
x = 2
and now you know everything necessary.
a) Here's the Venn diagram:
_______________
/ / / /
/ F / M / H /
/_____/_____/____/
| / | \
| / | \
| / | \
/ F & M / F & H / H & M/
/________/________/_______/
b) The equation in x can be written as:
9 + 7 + 8 + 9 + 1 + 9 + x = 38
c)
i) All three subjects: x = 2
ii) French: 9 + 7 + 9 + 2 = 27
a) Venn diagram:
F
/ \
/ \
/ \
M H
\ /
\ /
U
F = French
M = Mathematics
H = History
U = Universal set (total number of students)
b) Equation:
The equation involving the number of students in the region of the diagram can be written as:
F + M + H - (F ∩ M) - (F ∩ H) - (M ∩ H) + x = U
c) i) To find the number of students who study all three subjects, we need to find the value of x. From the given information, we know:
F = 9
M only = 8
H only = 1
F & M = 7
F & H = 9
H & M = 8
Using the equation from part (b), we can solve for x:
9 + 8 + 1 - 7 - 9 - 8 + x = 38
x = 38 - 9 - 8 - 1 + 7 + 9 + 8
x = 44 - 9 - 8 - 1
x = 44 - 18
x = 26
Therefore, the number of students who study all three subjects is 26.
ii) To find the number of students who study French, we include all the students in the French circle, including those who study French only and those who study French along with other subjects:
French = F + (F ∩ M) + (F ∩ H)
French = 9 + 7 + 9
French = 25
Therefore, the number of students who study French is 25.
a) To illustrate the information on a Venn diagram, we need to visualize the overlapping relationships between the three subjects: French, Mathematics, and History. Let's start by drawing three overlapping circles to represent each subject:
French Mathematics History
Now, let's fill in the given information in the diagram:
- French: 9 students study French
- Mathematics only: 8 students study Mathematics only
- History only: 1 student studies History only
- French & Math: 7 students study both French and Mathematics
- French & History: 9 students study both French and History
- History & Math: 8 students study both History and Mathematics
We can start by filling in the overlapping regions with the given information:
French Mathematics History
7 8
9 8 9
9
b) To write an equation involving the number of students in the region of the diagram (x), we need to consider the principle of inclusion-exclusion. The principle states that:
Total = French + Math + History - (French & Math) - (French & History) - (History & Math) + (All three subjects)
Based on the given information, we have:
38 = 9 + 8 + 1 - 7 - 9 - 8 + x
Simplifying the equation, we get:
38 = 4 - 6 + x
38 = -2 + x
x = 40
So, the equation involving the number of students in the region of the diagram is x = 40.
c) To find the number of students who study all three subjects (x):
From the equation above, we already know that x = 40. Therefore, the number of students who study all three subjects is 40.
To find the number of students who study French:
French = French only + (French & Math) + (French & History) + (All three subjects)
French = 9 + 7 + 9 + 40
French = 65
So, the number of students who study all three subjects is 40, and the number of students who study French is 65.