In a class of 80 students 53 study art,60 study biology,36 study art and biology,34 study art and chemistry,6 study biology only and 18 study biology but not chemistry.Illustrate the information on a Venn diagram.Determine the number of student study (i)art only (ii)chemistry.
Make the Venn diagram. Let
a = # who study art only
b = # who study biology only
c = # who study chemistry only
x = # who study only art and biology
y = # who study only art and chemistry
z = # who study only biology and chemistry
n = # who study all three subjects
Now, we know that
a+x+y+n = 53
b+x+z+n = 60
x+n = 36
y+n = 34
b = 6
b+x = 18
Solving that, we get
a = 7
Note that nowhere does c get used. So, since the sum of all the other numbers is 77, if we assume that all 83 students study something, then 6 study only chemistry.
Well, I could draw you a Venn diagram, but I'm not exactly a Picasso. How about I describe it to you instead?
In the center of the Venn diagram, we'll have the region where all three subjects intersect. That's where the students who study art, biology, and chemistry will be. According to the information given, there are 36 students in this category.
To the left of the Venn diagram, we'll have the region exclusively for art students. Now, we know that 53 students study art in total. But since 36 of them are already in the intersection, we subtract 36 from 53 to find out how many students study art only. That leaves us with 17 students in this category.
On the right of the Venn diagram, we'll have the region exclusively for biology students. We know that 60 students study biology in total. However, 36 of them are already accounted for in the intersection and 6 study biology only. So, we subtract 36 + 6 from 60 to find out how many students study biology but not chemistry. That gives us 18 students in this category.
Now, you asked about the number of students studying art only and chemistry. Unfortunately, with the given information, we don't have a direct way to determine that. But hey, at least we got the numbers for art only and biology but not chemistry, so we're clowning in the right direction!
To illustrate the information on a Venn diagram, we will use three overlapping circles representing art, biology, and chemistry.
Let's label the circles: A for art, B for biology, and C for chemistry.
Now, let's fill in the known information:
- 53 students study art (A).
- 60 students study biology (B).
- 34 students study art and chemistry (A ∩ C).
- 6 students study biology only (B only).
- 18 students study biology but not chemistry (B - C).
- 36 students study art and biology (A ∩ B).
Based on this information, we can fill in the Venn diagram step by step.
First, we fill in the overlapping region between art and biology (A ∩ B) with 36 students since 36 students study both art and biology.
Next, we fill in the remaining part of the biology circle (B) that is not in the overlapping region with chemistry. Since 6 students study biology only, we can write 6 in the B - C region.
Similarly, we can fill in the overlapping region between art and chemistry (A ∩ C) with 34 students.
Now, we can determine the number of students who study art only (A only) and the number of students who study chemistry (C).
To find the number of students who study art only, we subtract the number of students who study both art and biology (A ∩ B) and the number of students who study art and chemistry (A ∩ C) from the total number of students who study art (53).
A only = 53 - (36 + 34) = 53 -70 = -17 (Since it is not possible to have negative numbers of students, we can conclude that there is an error or inconsistency in the given information or calculations.)
Similarly, to find the number of students who study chemistry, we subtract the number of students who study both art and chemistry (A ∩ C) from the total number of students who study chemistry or biology (36 + 18).
C = (36 + 18) - 34 = 54 - 34 = 20
Therefore, the number of students who study (i) art only is -17 (which is not possible with the given information), and (ii) the number of students who study chemistry is 20.
To illustrate the information on a Venn diagram, we can start by drawing three intersecting circles, one for each subject - art, biology, and chemistry.
Now, let's fill in the given information:
- 53 students study art, so we write "53" in the intersection of the art circle and the total section.
- 60 students study biology, so we write "60" in the intersection of the biology circle and the total section.
- 36 students study art and biology, so we write "36" in the intersection of the art and biology circles.
- 34 students study art and chemistry, so we write "34" in the intersection of the art and chemistry circles.
- 6 students study biology only, so we write "6" in the part of the biology circle that does not intersect with any other circle.
- 18 students study biology but not chemistry, so we write "18" in the part of the biology circle that intersects with the art circle but not the chemistry circle.
Now, we need to find the number of students who study art only and the number of students who study chemistry only.
- To find the number of students who study art only, we add up the number of students in the art circle that is not part of any other intersection. So, art only = Art circle - (Art-Biology) - (Art-Chemistry).
art only = 53 - 36 - 34 = 17
- To find the number of students who study chemistry, we subtract the number of students who study biology but not chemistry from the number of students in the total chemistry circle. So, chemistry = Chemistry circle - (Biology but not Chemistry).
chemistry = 34 - 18 = 16
Therefore, the number of students who study art only is 17, and the number of students who study chemistry is 16.