Respuesta :
Answer:
18 cars had air conditioning and automatic transmission but not power steering.
Step-by-step explanation:
To solve this problem, we must build the Venn's Diagram of these sets.
I am going to say that:
-The set A represents the cars that had air conditioning.
-The set B represents the cars that had automatic transmission.
-The set C represents the cars that had power steering.
-The value of d represents the cars that had none of those extras.
We have that:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
In which a is the number of cars that only had air conditioning, [tex]A \cap B[/tex] is the number of cars that have both air conditioning and automatic transmission, [tex]A \cap C[/tex] is the number of cars that have both air conditioning and power steering. And [tex]A \cap B \cap C[/tex] is the number of cars that had all those three extras.
By the same logic, we have:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
How many cars had air conditioning and automatic transmission but not power steering?
This is [tex]A \cap B[/tex].
We start finding the values from the intersection of three sets
6 cars had all three of these extras. This means that
[tex]A \cap B \cap C = 6[/tex]
11 cars had both automatic transmission and power steering.
This is
[tex](B \cap C) + (A \cap B \cap C) = 11[/tex]
[tex](B \cap C) = 5[/tex]
25 cars had only power steering.
This means that [tex]c = 25[/tex]
62 cars had only automatic transmissions.
This means that [tex]b = 62[/tex].
21 cars had only air conditioning
This means that [tex]a = 21[/tex]
80 had power steering
This means that [tex]C = 80[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]80 = 25 + (A \cap C) + 5 + 6[/tex]
[tex]A \cap C = 44[/tex]
Now, we can use the equation of A or B to find [tex]A \cap B[/tex]
96 had automatic transmission. This means that [tex]B = 96[/tex]
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]96 = 62 + 5 + (A \cap B) + 11[/tex]
[tex](A \cap B) = 18[/tex]
18 cars had air conditioning and automatic transmission but not power steering.
