In a hotel, 60% of the people had vegetarian lunch, 30% had non vegetarian lunch and 15% had both types of lunch.\nIf a total of 96 people were present, how many people did not eat either type of lunch?

Introduction / Context:
This problem is a classic example of the use of set theory and percentages in practical situations. It involves two overlapping groups, those who had vegetarian lunch and those who had non vegetarian lunch, with some people counted in both groups. We are asked to find the number of people who belong to neither group, using inclusion exclusion principles.

Given Data / Assumptions:

• Total number of people in the hotel = 96.
• 60% had vegetarian lunch.
• 30% had non vegetarian lunch.
• 15% had both vegetarian and non vegetarian lunches.
• We assume everyone is included in the total of 96 and that percentages are of this total.
• We must find how many did not eat either type of lunch.

Concept / Approach:
To work with overlapping categories, we use the principle of inclusion exclusion. The percentage of people who had at least one type of lunch equals the sum of the individual percentages minus the percentage of people counted twice (those who had both types). Once we find the percentage of people who had at least one type of lunch, we subtract this from 100% to find the percentage who had neither type. Then we apply this percentage to the total number of people.

Step-by-Step Solution:
Step 1: Let V be the set of people who had vegetarian lunch, N the set who had non vegetarian lunch. Step 2: Percentage in V = 60%, percentage in N = 30%. Step 3: Percentage in both V and N (intersection) = 15%. Step 4: Percentage in at least one of the two sets = 60% + 30% - 15% = 75%. Step 5: Therefore, percentage who had neither type of lunch = 100% - 75% = 25%. Step 6: Number of people who had neither type = 25% of 96 = (25 / 100) * 96 = 24.

Verification / Alternative check:
Compute actual numbers for each category. Vegetarian: 60% of 96 = 57.6 (conceptually, but the percentage is applied to the total distribution and is assumed exact for this problem). Non vegetarian: 30% of 96 = 28.8. Both: 15% of 96 = 14.4. While these numbers are fractional, the net effect using inclusion exclusion on percentages is exact. Using percentages only, we already found 25% of 96 is 24, which is an integer and matches typical exam expectations.

Why Other Options Are Wrong:
30, 36, 42, 18: Each of these corresponds to different percentages of 96 (about 31.25%, 37.5%, 43.75%, 18.75%), none matching the required 25% remaining after accounting for the overlapping lunch groups. Substituting any of these into the total leads to incorrect percentages for those who had at least one type of lunch.

Common Pitfalls:
A frequent error is to simply add 60% and 30% to get 90% and then subtract from 100% to infer that 10% had neither, ignoring the overlap. This double counts those who had both types of lunch. Always remember to subtract the intersection percentage when using inclusion exclusion. Another pitfall is being confused by fractional counts; working entirely in percentages avoids this.

Final Answer:
24 people did not eat either type of lunch.