From a survey of 200 college students, a marketing research company found that 140 students owned iPhones, 60 owned cars, and 50 owned both cars and iPhones.
(a) How many students owned either a car or an iPhone (but not both)?
______ students
(b) How many students do not own either a car or an iPhone?
______students
To solve this problem, we will use a combination of set theory and basic arithmetic operations.
Let's define the following sets:
A = Set of students who own iPhones
B = Set of students who own cars
Given information:
|A| = 140 (Number of students who own iPhones)
|B| = 60 (Number of students who own cars)
|A ∩ B| = 50 (Number of students who own both cars and iPhones)
(a) To find the number of students who own either a car or an iPhone (but not both), we need to find the cardinality of the set (A ∪ B) - (A ∩ B).
Using the formula: |A ∪ B| = |A| + |B| - |A ∩ B|
Substituting the given values, we get:
|A ∪ B| = 140 + 60 - 50
= 150
Therefore, the number of students who own either a car or an iPhone (but not both) is 150 students.
(b) To find the number of students who do not own either a car or an iPhone, we need to find the cardinality of the complement set of (A ∪ B) with respect to the universal set U (which represents the total number of college students surveyed).
Using the formula: |U| - |A ∪ B|
Given that the total number of students surveyed is 200:
|U| = 200
Substituting the values, we get:
|U| - |A ∪ B| = 200 - 150
= 50
Therefore, the number of students who do not own either a car or an iPhone is 50 students
To find the number of students who owned either a car or an iPhone (but not both), we need to subtract the number of students who owned both from the sum of students who owned cars and students who owned iPhones.
(a) How many students owned either a car or an iPhone (but not both)?
1. Start by adding the number of students who owned cars and the number of students who owned iPhones:
Cars: 60 students
iPhones: 140 students
Total: 60 + 140 = 200 students
2. Subtract the number of students who owned both cars and iPhones:
Both cars and iPhones: 50 students
Total cars or iPhones (but not both): 200 - 50 = 150
Therefore, 150 students owned either a car or an iPhone (but not both).
(b) To find the number of students who do not own either a car or an iPhone, we need to subtract the total number of students who own either a car or an iPhone from the total number of students surveyed.
(b) How many students do not own either a car or an iPhone?
1. Subtract the number of students who own either a car or an iPhone (found in part a) from the total number of students surveyed:
Total students surveyed: 200 students
Total cars or iPhones (but not both): 150 students
Students who do not own either: 200 - 150 = 50
Therefore, 50 students do not own either a car or an iPhone.
140 students owned iPhones; of these 50 also owned cars; 140 - 50 = 90
90 students own only an iphone
60 owned cars; of these 50 also own a car; 60-50 = 10; 10 students own ONLY a car
The number of student who own either a car or an iphone, but not both is 90 + 10 = 100
The number of students who do not own either a car or an iphone is 200 - 50 - 10 - 90 = 50 students