(18) Out of 120 students in SSS 3, [tex]\frac{5}{8}[/tex] like football and [tex]\frac{2}{5}[/tex] like swimming. Every student like at least one of these sports.
(a) What fraction of the students in [tex]SSS _3[/tex] like both subjects?
(b) What fraction of the students like football but not swimming?
(c) What fraction of the students like swimming only?
Answer (1)
Calculate the number of students who like football: 8 5 × 120 = 75 .
Calculate the number of students who like swimming: 5 2 × 120 = 48 .
Find the number of students who like both sports using the inclusion-exclusion principle: 120 = 75 + 48 − n ( F ∩ S ) , which gives n ( F ∩ S ) = 3 . Thus, the fraction is 120 3 = 40 1 .
Calculate the fraction of students who like football only: 120 75 − 3 = 120 72 = 5 3 .
Calculate the fraction of students who like swimming only: 120 48 − 3 = 120 45 = 8 3 .
The fraction of students who like both subjects is 40 1 , football only is 5 3 , and swimming only is 8 3 .
Explanation
Understand the problem We are given that there are 120 students in SSS 3. We also know that 8 5 of the students like football and 5 2 of the students like swimming. Every student likes at least one of these sports. We need to find the fraction of students who like both sports, the fraction who like football but not swimming, and the fraction who like swimming only.
Define the sets Let F be the set of students who like football, and S be the set of students who like swimming. We are given:
Total number of students = 120 Fraction of students who like football = 8 5 Fraction of students who like swimming = 5 2 Every student likes at least one of the sports.
Calculate the fraction of students who like both sports (a) To find the fraction of students who like both sports, we use the principle of inclusion-exclusion:
n ( F ∪ S ) = n ( F ) + n ( S ) − n ( F ∩ S )
Where: n ( F ∪ S ) = Total number of students = 120 n ( F ) = Number of students who like football = 8 5 × 120 = 75 n ( S ) = Number of students who like swimming = 5 2 × 120 = 48 n ( F ∩ S ) = Number of students who like both sports
So, 120 = 75 + 48 − n ( F ∩ S ) n ( F ∩ S ) = 75 + 48 − 120 = 123 − 120 = 3
Therefore, the fraction of students who like both sports is 120 3 = 40 1 = 0.025 .
Calculate the fraction of students who like football only (b) To find the fraction of students who like football but not swimming, we subtract the number of students who like both sports from the number of students who like football:
Number of students who like football only = n ( F ) − n ( F ∩ S ) = 75 − 3 = 72
Therefore, the fraction of students who like football only is 120 72 = 10 6 = 5 3 = 0.6 .
Calculate the fraction of students who like swimming only (c) To find the fraction of students who like swimming only, we subtract the number of students who like both sports from the number of students who like swimming:
Number of students who like swimming only = n ( S ) − n ( F ∩ S ) = 48 − 3 = 45
Therefore, the fraction of students who like swimming only is 120 45 = 8 3 = 0.375 .
State the final answer (a) The fraction of students who like both subjects is 40 1 .
(b) The fraction of students who like football but not swimming is 5 3 .
(c) The fraction of students who like swimming only is 8 3 .
Examples
Understanding the preferences of students in a school can help in planning extracurricular activities. For example, if a school knows the fraction of students interested in different sports, they can allocate resources effectively to promote those sports. This can lead to increased participation and a healthier student body. Moreover, this type of analysis can be extended to other areas such as music, arts, and academic clubs, allowing the school to create a well-rounded program that caters to the diverse interests of its students.
