a. 15 students are members of both clubs
b. 50 students only are members of the mathematics club
c. 25 students only are members of the physics club
Step-by-step explanation:
The given is:
- There are 100 students
- 65 are members of a mathematics club
- 40 are members of a physics club
- 10 are members of neither club
We need to find how many students are members of
a. both clubs?
b. only mathematics club
c. only physics club
∵ The total number of students = 100
∵ 10 are members of neither club
- Subtract 10 from 100 to find the members of the mathematics
club or the physics club
∵ 100 - 10 = 90
∴ There are 90 members of the mathematics club or physics club
∴ n(mathematics or physics) = 90
∵ 65 students are members of a mathematics club
∵ n(mathematics) = 65
∵ 40 students are members of a physics club
∴ n(physics) = 40
∵ n(mathematics or physics) = n(mathematics) + n(physics) - n(both)
∴ 90 = 65 + 40 - n(both)
∴ 90 = 105 - n(both)
- Add n(both) to each side
∴ n(both) + 90 = 105
- Subtract 90 from each side
∴ n(both) = 15
∴ 15 students are members of both clubs
a. 15 students are members of both clubs
∵ 65 students are members of the mathematics club
∵ 15 of them are members of the physics club
∴ n(mathematics only) = 65 - 15 = 50
∴ 50 students only are members of the mathematics club
b. 50 students only are members of the mathematics club
∵ 40 students are members of the physics club
∵ 15 of them are members of the mathematics club
∴ n(physics only) = 40 - 15 = 25
∴ 25 students only are members of the physics club
c. 25 students only are members of the physics club
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