Question

food selections
pizza
taco
65
corn dog
a total of 320 teenagers selected a taco.
a total of 140 teenagers selected both a taco and a corn dog.
a total of 190 teenagers selected both a taco and a pizza.
how many teenagers selected a pizza, a taco, and a corn dog?
a 65
b 125
c 55
d 75

Explanation:

Step1: Use the formula for three - set Venn diagram (for the taco set)

Let \(x\) be the number of teenagers who selected all three foods (pizza, taco, and corn dog).
The formula for the number of elements in a set in a Venn diagram: \(n(T)=n(T\text{ only})+n(T\cap P)+n(T\cap C)-n(T\cap P\cap C)\)
We know \(n(T) = 320\), \(n(T\cap C)=140\), \(n(T\cap P) = 190\), and \(n(T\text{ only})=65\)
Substitute into the formula: \(320=65 + 190+140 - x\)

Step2: Solve the equation for \(x\)

First, simplify the right - hand side of the equation: \(65 + 190+140=395\)
The equation becomes \(320=395 - x\)
Add \(x\) to both sides: \(320 + x=395\)
Subtract 320 from both sides: \(x=395 - 320\)
\(x = 75\)

Answer:

D. 75