At Small Town High there are 100 students, each of whom takes at least one of biology, chemistry, or physics. There are 50 students in biology, 60 in chemistry, and 70 in physics. If 24 students are enrolled in exactly two of these classes, how many students are in all three?
To solve this problem, we can use the principle of Inclusion-Exclusion.
Step 1: Add the number of students in each subject:
Biology: 50
Chemistry: 60
Physics: 70
Step 2: Subtract the number of students who are enrolled in exactly two subjects:
Biology and Chemistry: 24
Biology and Physics: X (unknown)
Chemistry and Physics: X (unknown)
Step 3: Calculate the total number of students in all three subjects using the formula:
Total = Biology + Chemistry + Physics - (Biology and Chemistry) - (Biology and Physics) - (Chemistry and Physics)
Since we are looking for the number of students in all three subjects, we need to find the value of X for Biology and Physics, as well as Chemistry and Physics.
Step 4: Calculate the value of X for Biology and Physics:
X = (Biology + Chemistry + Physics - Total - Biology and Chemistry - Chemistry and Physics) / 2
= (50 + 60 + 70 - 100 - 24 - X) / 2
Simplifying this equation gives us:
2X = 256 - X
3X = 256
X = 256/3
Step 5: Substitute the value of X into the equation for Chemistry and Physics:
X = (Biology + Chemistry + Physics - Total - Biology and Chemistry - Biology and Physics) / 2
256/3 = (50 + 60 + 70 - Total - 24 - (256/3)) / 2
Simplifying this equation gives us:
256/3 = 140 - Total
Total = 140 - 256/3
Total = (420 - 256)/3
Total = 164/3
Total ≈ 54.67
Since the total number of students cannot be fractional, we round down to the nearest whole number.
Therefore, the number of students enrolled in all three subjects is 54.
To find the number of students enrolled in all three classes, we need to use the principle of inclusion-exclusion.
Let's break down the information given:
- There are 50 students in biology.
- There are 60 students in chemistry.
- There are 70 students in physics.
- 24 students are enrolled in exactly two of these classes.
To find the number of students in all three classes, we need to subtract the number of students who are enrolled in exactly two classes from the total number of students in each subject.
Let's calculate:
- The total number of students = 50 + 60 + 70 = 180.
- The number of students enrolled in exactly two classes = 24.
To find the number of students enrolled in all three classes, we'll subtract the number of students enrolled in exactly two classes from the total number of students:
Number of students in all three classes = Total number of students - Number of students enrolled in exactly two classes
Number of students in all three classes = 180 - 24
Number of students in all three classes = 156.
Therefore, there are 156 students enrolled in all three classes at Small Town High.