Question

After a fundraiser, the event organisers surveyed all the participants who attended about how many different types of sweets they had during the party. The attendants could have taken any number of sweets available of any kind. The chart below shows the percentage of people surveyed who consumed a particular type of sweet. The participants could have taken one piece or more of the sweets they consumed.
Chart Data (picked by percentage): Ice-cream (80%), Chocolate (55%), Custard (75%), Tiramisu (60%), Jello (40%), Halwa (55%), Donuts (48%).
According to the data, the maximum percentage of people who would have taken at least one of every sweet would be \underline{\hspace{1cm}, while the minimum percentage of people who would have taken at least one sweet would be \underline{\hspace{1cm}}.}

Hint:

For set theory questions without overlap data:

The \textbf{maximum} possible intersection (AND) is the minimum of the individual set sizes.
The \textbf{minimum} possible union (OR) is the maximum of the individual set sizes.

Answer 1

Step 1: Understanding the Concept:
This problem involves interpreting data from a chart to determine the maximum possible intersection and the minimum possible union of several sets (the groups of people who ate each type of sweet).
Part 1: Maximum percentage who took at least one of every sweet.
Step 2: Detailed Explanation:
This asks for the maximum possible size of the intersection of all seven sets. Let \(S_1, S_2, \dots, S_7\) be the sets of people who ate each of the seven sweets. We want to find the maximum possible value of \(|S_1 \cap S_2 \cap \dots \cap S_7|\).
The intersection of several sets can be no larger than the smallest of those sets. For a person to be in the intersection, they must have eaten every sweet, so they must be counted in every percentage. The maximum possible overlap is therefore limited by the sweet that was eaten by the fewest people.
Let's list the percentages:

Ice-cream: 80%
Chocolate: 55%
Custard: 75%
Tiramisu: 60%
Jello: 40%
Halwa: 55%
Donuts: 48%
The minimum percentage is 40% for Jello. Therefore, the maximum percentage of people who could have eaten every single sweet is 40%.
Part 2: Minimum percentage who took at least one sweet.
Step 3: Detailed Explanation:
This asks for the minimum possible size of the union of all seven sets, i.e., \(|S_1 \cup S_2 \cup \dots \cup S_7|\).
The union of several sets must be at least as large as the largest of those sets. The minimum possible value for the union occurs if all the smaller sets are subsets of the largest set (maximum overlap). For example, if everyone who ate chocolate also ate ice cream, the union of those two sets would simply be the set of people who ate ice cream.
The largest percentage is 80% for Ice-cream. At a minimum, 80% of the people surveyed ate at least one sweet (the ice cream). It is possible that all the people who ate the other sweets also ate ice cream, in which case the total percentage of people eating at least one sweet would be exactly 80%. Therefore, the minimum percentage of people who would have taken at least one sweet is 80%.
Step 4: Final Answer:
The maximum percentage for "at least one of every sweet" is 40.
The minimum percentage for "at least one sweet" is 80.