In a class of 80 students, 53 study Art, 70 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 in a Venn diagram. Determimr the number og students who study ;
i) Art only ii) Chemistry
To determine the number of students who study Art only and those who study Chemistry, we can use a Venn diagram.
Based on the given information, we know:
- There are 80 students in total.
- 53 study Art (A).
- 70 study Biology (B).
- 36 study both Art and Biology (A ∩ B).
- 34 study both Art and Chemistry (A ∩ C).
- 6 study Biology only (B).
- 18 study Biology but not Chemistry (B - C).
For the Venn diagram, we have three sets: A (Art), B (Biology), and C (Chemistry). They should overlap since there are students who study in multiple subjects:
```
[A ∩ B ∩ C] - the region where all three circles overlap represents the students that study all three subjects.
[A ∩ B] - the region where A and B overlap, but not C represents the students that study both Art and Biology, but not Chemistry.
[A ∩ C] - the region where A and C overlap, but not B, represents the students that study both Art and Chemistry, but not Biology.
[B ∩ C] - the region where B and C overlap, but not A, represents the students that study both Biology and Chemistry, but not Art.
A, B, C respectively - the non-overlapping parts of each circle represent the students that study only Art, only Biology, or only Chemistry.
```
Let's use this information to fill out the Venn diagram. Since 34 students study Art and Chemistry, but we don't know how many of them also study Biology, we will leave that as an unknown for now (let's call this number x). Similarly, we don't know how many students study Biology and Chemistry but not Art (let's call this number y), and we also don't know the number of students who study only Chemistry (let's call this z).
From the information that 36 study both Art and Biology, we can fill in the part of the Art and Biology intersection that does not include Chemistry, which will be 36 - x.
Now, six study Biology only, which is the part of the Biology circle that doesn't overlap with any other circle.
We also know that 18 study Biology but not Chemistry. This includes those who study only Biology and those who study Art and Biology but not Chemistry, so:
18 = (Biology only) + (Art and Biology, but not Chemistry)
18 = 6 + (36 - x)
12 = 36 - x
x = 36 - 12
x = 24
This tells us that there are 24 students who study Art, Biology, and Chemistry.
Now, we can calculate the number of students who study Art only:
(Students studying Art) = (Art only) + (Art and Biology but not Chemistry) + (Art and Chemistry but not Biology) + (Art, Biology, and Chemistry)
53 = (Art only) + (36 - 24) + (34 - 24) + 24
53 = (Art only) + 12 + 10 + 24
53 = (Art only) + 46
Art only = 53 - 46
Art only = 7
There are 7 students that study Art only.
The total number of students studying Chemistry can be determined by summing those who study Chemistry and Art, Chemistry and Biology, all three subjects, and only Chemistry.
(Students studying Chemistry) = (Art and Chemistry but not Biology) + (Biology and Chemistry but not Art) + (Art, Biology, and Chemistry) + (Chemistry only)
We still don't know y (Biology and Chemistry but not Art) and z (Chemistry only), but we can proceed with the information we have:
70 students study Biology. This includes those who study Biology and Chemistry but not Art, Biology and Art, only Biology, and all three subjects:
70 = y + (36 - 24) + 6 + 24
70 = y + 12 + 6 + 24
70 = y + 42
y = 70 - 42
y = 28
So, we have 28 students who study Biology and Chemistry but not Art.
The total number of students studying Chemistry is the sum of all the groups involving Chemistry:
(Students studying Chemistry) = z + (34 - 24) + 28 + 24
But we don't have the value of z yet. To find the total number of students, we also need to consider that some students may not study any of these three subjects. Let's call the number of students who don't study any of the subjects n.
We have the total number of students in the class:
Total students = 80 = (Art only) + (Art and Biology but not Chemistry) + (Art and Chemistry but not Biology) + (Biology only, no other subjects) + (Biology and Chemistry but not Art) + (Art, Biology, and Chemistry) + (Chemistry only) + (no subjects)
80 = 7 + (36 - 24) + (34 - 24) + 6 + 28 + 24 + z + n
80 = 7 + 12 + 10 + 6 + 28 + 24 + z + n
80 = 87 + z + n
Since there can't be a negative number of students, and we've already accounted for students in some level of subjects, we can deduce that n must be 0 (all students are part of at least one of the subjects), and so z must be negative in this equation, which isn't possible.
This inconsistency implies a mistake in the calculation process. So, let's reassess the total:
Total students = 80 = (Art only) + (Art and Biology - Art, Biology, and Chemistry) + (Art and Chemistry - Art, Biology, and Chemistry) + (Biology only) + (Biology and Chemistry - Art, Biology, and Chemistry) + (Art, Biology, and Chemistry) + (Chemistry only)
80 = 7 + (36 - x) + (34 - x) + 6 + y + x + z
80 = 7 + (36 - 24) + (34 - 24) + 6 + 28 + 24 + z
80 = 7 + 12 + 10 + 6 + 28 + 24 + z
80 = 87 + z (here we don't consider students with no subjects)
z = 80 - 87
z = -7
Again, we have a negative number, which means there must be an error in our assumptions or calculations. Given the complexity of the problem and potential for confusion, it might be best to redo the calculations step by step:
1. All the students that study both Art and Biology includes those who study all three subjects:
A ∩ B = (A ∩ B ∩ C) + (A ∩ B - C)
36 = x + (36 - x)
2. All the students that study both Art and Chemistry include those who study all three subjects:
A ∩ C = (A ∩ B ∩ C) + (A ∩ C - B)
34 = x + (34 - x)
3. All the students that study Biology but not Chemistry includes those who study only Biology and those who study both Biology and Art but not Chemistry:
(B - C) = B only + (A ∩ B - C)
18 = 6 + (36 - x)
18 = 6 + (36 - x)
12 = 36 - x
x = 36 - 12
x = 24
This tells us there are 24 students who study all three subjects.
Let's try to calculate the students who study only Chemistry:
Let's sum up variables including the groups of students we know:
80 = (Art only) + (A ∩ B - C) + (A ∩ C - B) + (B only) + (B ∩ C - A) + (A ∩ B ∩ C) + (C only)
80 = (Art only) + (36 - x) + (34 - x) + 6 + (B ∩ C - A) + x + (C only)
80 = (Art only) + 36 - 24 + 34 - 24 + 6 + (B ∩ C - A) + 24 + (C only)
80 = (Art only) + 52 + (B ∩ C - A) + (C only)
Now, to find out (B ∩ C - A), which is the variable y we defined earlier as the number of students that study Biology and Chemistry but not Art, we use the info that 18 students study Biology but not Chemistry:
B - (B ∩ C) = 18
(B only) + (A ∩ B - C) = 18
6 + (36 - x) = 18
6 + (36 - 24) = 18
6 + 12 = 18
As 18 = 18, our calculation is consistent here, so the number of students who study Biology and Chemistry only (B ∩ C - A), would be the students included in the Biology total (70), subtracting those who study only Biology, those who study Biology and Art, and the 24 who study all three subjects:
70 = (B only) + (A ∩ B - C) + (B ∩ C - A) + (A ∩ B ∩ C)
70 = 6 + (36 - 24) + (B ∩ C - A) + 24
70 = 6 + 12 + (B ∩ C - A) + 24
70 = 42 + (B ∩ C - A)
(B ∩ C - A) = 70 - 42
(B ∩ C - A) = 28
So, y = 28 students study Biology and Chemistry but not Art.
Now let's recalculate the total number of students studying Chemistry:
(Students studying Chemistry) = (A ∩ C - B) + (B ∩ C - A) + (A ∩ B ∩ C) + (C only)
Here (A ∩ C - B) is the number of students that study Art and Chemistry but not Biology, which we can calculate as:
(A ∩ C) - (A ∩ B ∩ C) = 34 - 24 = 10
So,
(Students studying Chemistry) = 10 + 28 + 24 + (C only)
Now we can finish finding how many students are in the (C only) group:
80 = (Art only) + (A ∩ B - C) + (A ∩ C - B) + (B only) + (B ∩ C - A) + (A ∩ B ∩ C) + (C only)
80 = 7 + 12 + 10 + 6 + 28 + 24 + (C only)
80 = 87 + (C only)
(C only) = 80 - 87
(C only) = -7
We've again reached a negative number, which is not possible. Unfortunately, there seems to be a discrepancy or inconsistency in the given data, which is why we're unable to reconcile the numbers with the conditions provided.
Please double-check the data given for this question. There might be a typographical error, or the information provided might contradict itself. If you have revised or additional information, please share it so we can assist you accurately.