Question

At a refreshment stand 73% of the beverages are cold and 42% of the beverages are caffeinated. Of the total beverages at this refreshment stand 37% are cold caffeinated beverages. Beverages are only hot or cold and caffeinated or decaffeinated. What percentage of the total beverages are hot decaffeinated beverages?

a) 31%
b) 27%
c) 25%
d) 23%

Answer 1

To find the percentage of hot decaffeinated beverages, we can use set theory and calculate the complement of the given information. None of the provided options are correct.

Step-by-step explanation:

To find the percentage of hot decaffeinated beverages, we can use the concept of set theory and calculate the complement of the given information. Let's define the variables:

A = Set of cold beverages

C = Set of caffeinated beverages

Given that 73% of beverages are cold (A) and 42% of beverages are caffeinated (C), we can represent this as:

P(A) = 0.73, P(C) = 0.42

Furthermore, we are given that 37% of the total beverages are cold caffeinated beverages (A ∩ C). Using this information, we can calculate:

P(A ∩ C) = 0.37

Now, to find the percentage of hot decaffeinated beverages, we need to find the complement of (A ∪ C) and subtract it from 1:

P(Hot decaffeinated) = 1 - P(A ∪ C)

To find P(A ∪ C), we can use the formula:

P(A ∪ C) = P(A) + P(C) - P(A ∩ C)

Substituting the given values, we get:

P(Hot decaffeinated) = 1 - (0.73 + 0.42 - 0.37) = 1 - 0.78 = 0.22

Therefore, the percentage of hot decaffeinated beverages is 22%, which is not among the given options. None of the provided options are correct.