In a class of 80 students,53 study Art,60 study Biology,36 study biology and Art,34 study Art and chemistry and 6 study Biology only
a.Illastrate the information on a Venn diagram
b.Determin the number of the students who study
I.Art only
ii.Chemistry
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26. In a class of 80 students, 53 study Art, 60 st - Gauthmath
You will see solution and explanation.
a. To illustrate the information on a Venn diagram, we can create three overlapping circles representing Art, Biology, and Chemistry.
First, we know that 53 students study Art, so we can place that number inside the circle representing Art.
Next, we know that 60 students study Biology, so we can place that number inside the circle representing Biology.
Then, we know that 36 students study both Biology and Art, so we place that number in the overlap section between the Art and Biology circles.
Similarly, we know that 34 students study both Art and Chemistry, so we place that number in the overlap section between the Art and Chemistry circles.
Finally, we know that 6 students study only Biology, so we subtract the number of students who study both Biology and Art (36) from the total number of students studying Biology (60) to find that 24 students study Biology only.
The Venn diagram representing this information would look like this:
```
Art
/ \
/ \
/ \
/ \
Biology Chemistry
\ /
\ /
\ /
\ /
```
b. To determine the number of students who study:
i. Art only: In the Venn diagram, the Art-only section is the students within the Art circle but outside the overlap sections.
To calculate the number of students who study Art only, we subtract the number of students who study both Biology and Art (36) and the number of students who study both Art and Chemistry (34) from the total number of students who study Art (53):
53 - 36 - 34 = 17 students study Art only.
ii. Chemistry: In the Venn diagram, the Chemistry-only section is the students within the Chemistry circle but outside the overlap sections.
To calculate the number of students who study Chemistry only, we subtract the number of students who study both Art and Chemistry (34) from the total number of students who study Chemistry:
34 students study Chemistry only.
sketch 3 intersecting circles, label them A, B, C
We don't know how many take all 3, so label the inner intersection as x
36 go into B and A, but we have already accounted for x of those,
so label the "B and A only" as 36-x, similarly
label the "A and C only" as 34-x
enter 6 in the "B only"
Now look at the B circle, we are missing the "B and C only" , call it y
then y + 6 + 36-x + x = 60
y = 18 , enter 18 in the "B and C only" section
Back to the A circle, we know the total is 53, and we don't know the "A only"
call it z
z + 36-x + x + 34-x = 53
z = x - 17
The only part not filled in is the "C only" , let's call that c
We know the total of all the entries should be 80, assuming that everbody
takes at least one of the subjects.
all of A + 6+18 + c = 80
53 + 6 + 18 + c = 80
c = 3 , fill that in
We have now filled in all the sections, but still don't know what x is
the only thing we know about x is that
17 < x < 34 , for x to be positive
We do know that C contains a total of 55
If we pick any value of x within the domain above, the Venn diagram entries
will be consistent, try it with x = 20, it will work out
Did you miss some additional data? e.g. did you know how many
took all 3 ??