In a class of 40 students it is known that 22 of them study Art, 18 of them study study Chemistry.5 students study all the 3 subjects, 9 study Art and Biology, 7 study Art but not Chemistry and 11 study exactly one subject.
sounds like pretty much everything is known.
Please help me with the answer
To solve this problem, we can use a Venn diagram. Let's start by drawing three overlapping circles representing Art, Chemistry, and Biology.
Step 1: Fill in the given information in the Venn diagram:
- The total number of students in the class is 40.
- 22 students study Art.
- 18 students study Chemistry.
- 5 students study all three subjects (Art, Chemistry, and Biology).
Step 2: Calculate the number of students who study Art and Biology:
- 9 students study Art and Biology. Fill in this number in the overlapping region between Art and Biology.
Step 3: Calculate the number of students who study Art but not Chemistry:
- 7 students study Art but not Chemistry. Fill in this number in the section of the Art circle not overlapped with Chemistry.
Step 4: Calculate the number of students who study exactly one subject:
- 11 students study exactly one subject. We can determine the number of students who study only Biology by subtracting the number of students who study all three subjects (5) and the number of students who study both Art and Biology (9) from the total number of students who study Biology.
- Biology = Total number of students studying Biology - Students studying all three subjects - Students studying Art and Biology = 11 - 5 - 9.
- Fill in the number of students who study only Biology in the section of the Biology circle not overlapped with Art and Chemistry.
Step 5: Calculate the remaining number of students for each subject:
- Now that we have filled in most of the information in the Venn diagram, we can calculate the remaining number of students for each subject.
- The number of students studying only Art = Number of students studying Art - Students studying all three subjects - Students studying Art but not Chemistry = 22 - 5 - 7.
- The number of students studying only Chemistry = Number of students studying Chemistry - Students studying all three subjects = 18 - 5.
- The number of students who don't study any of the three subjects = Total number of students - Students studying at least one subject = 40 - (students studying Art + students studying Chemistry + students studying Biology - students studying all three subjects) = 40 - (22 + 18 + 11 - 5).
- Fill in these numbers in the appropriate sections of the Venn diagram.
Step 6: Calculate the number of students who study only Chemistry:
- The number of students studying only Chemistry = Number of students studying Chemistry - Students studying all three subjects = 18 - 5.
- Fill in the number of students who study only Chemistry in the section of the Chemistry circle not overlapped with Art.
Step 7: Calculate the number of students who study only Art:
- The number of students studying only Art = Number of students studying Art - Students studying all three subjects - Students studying Art but not Chemistry = 22 - 5 - 7.
- Fill in the number of students who study only Art in the section of the Art circle not overlapped with Chemistry.
The Venn diagram is now complete, and you can read off the numbers for each part of the diagram to answer specific questions about the students' subject choices.
To find the number of students studying Biology, we need to use the principle of inclusion-exclusion. We have the following information:
Number of students studying Art (A) = 22
Number of students studying Chemistry (C) = 18
Number of students studying all 3 subjects (A∩C∩B) = 5
Number of students studying Art and Biology (A∩B) = 9
Number of students studying only Art (A - A∩C) = 7
Number of students studying exactly one subject (A∪C∪B - A∩C∩B - A∩B) = 11
Let's solve this step by step:
1. Number of students studying either Art, Chemistry, or Biology (A∪C∪B):
A∪C∪B = (A + C + B) - (A∩C + A∩B + C∩B) + (A∩C∩B)
= (22 + 18 + B) - (5 + 9 + C∩B) + 5
= 40 + B - (C∩B)
2. Number of students studying only Biology (B - A∩C∩B - A∩B - C∩B):
B - A∩C∩B - A∩B - C∩B = B - (C∩B)
3. Number of students studying Biology:
B = Number of students studying either Art, Chemistry, or Biology (A∪C∪B) - Number of students studying only Biology (B - A∩C∩B - A∩B - C∩B)
Let's substitute the known values:
B = (40 + B - (C∩B)) - (B - (C∩B))
Next, we simplify the equation:
B = 40 + B - C∩B - B + C∩B
Notice that C∩B appears both as a positive and negative term, so it cancels out:
B = 40
Therefore, the number of students studying Biology is 40.