In a clasd 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 mot Chemistry.
Let's define the different groups of students according to the subjects they study:
- Let A be the number of students studying Art.
- Let B be the number of students studying Biology.
- Let C be the number of students studying Chemistry.
- Let A ∩ B be the number of students studying both Art and Biology.
- Let A ∩ C be the number of students studying both Art and Chemistry.
- Let B ∩ C be the number of students studying both Biology and Chemistry.
- Let B_only be the number of students studying Biology only.
- Let B_not_C be the number of students studying Biology but not Chemistry.
Based on the information provided:
- A = 53 students study Art.
- B = 70 students study Biology.
- A ∩ B = 36 students study both Art and Biology.
- A ∩ C = 34 students study both Art and Chemistry.
- B_only = 6 students study Biology only.
- B_not_C = 18 students study Biology but not Chemistry.
Taking B_not_C into account, we know that:
B_not_C = B_only + (B ∩ C)
18 = 6 + (B ∩ C)
This means that B ∩ C = 18 - 6 = 12 students study both Biology and Chemistry.
Now we want to find a few more values, such as the total number of students studying Chemistry (C), and also those who study only Art or Chemistry.
The total number of students studying both Biology and another subject (either Art or Chemistry) is the sum of those who study Biology and Art (A ∩ B), those who study Biology and Chemistry (B ∩ C), and those who study Biology only (B_only). Since B = 70 and we know B_only, A ∩ B, and B ∩ C, we can write:
B = B_only + A ∩ B + B ∩ C
70 = 6 + 36 + 12
70 = 54
Please note that in the last statement, something is incorrect because the addition result of 6 + 36 + 12 is not equal to 70; it equals 54. There seems to be a contradiction in the information provided.
If 70 students study Biology, and only 6 study Biology only, it should mean that there are 70 - 6 = 64 students who study Biology and another subject (either Art or Chemistry). But with the given overlaps of A ∩ B and B ∩ C, we only account for 36 + 12 = 48 students, which is fewer than the 64 we'd expect from subtracting the Biology only from the total studying Biology.
Since we've found a discrepancy in the provided information, please review the data provided to ensure its accuracy. If additional context or corrections are available, feel free to elaborate, and we can attempt to solve the problem again.