In a particular week, the average number of people who visited a park per day is 40. If we exclude the holidays in that week, the average number of visitors per day increases by 16. Further, if we also exclude the day on which the maximum number of 112 people visited the park, the average becomes 42. How many holidays were there in that week?

Introduction / Context:
This question combines averages with exclusion of certain days, such as holidays and a peak day with the maximum number of visitors. It tests multi step reasoning about totals and averages when different subsets of the data are considered, making it a good higher level average problem.

Given Data / Assumptions:
- Consider a week with 7 days. - Average visitors per day for the entire week = 40. - When holidays are excluded, the average for the remaining non holiday days increases by 16, so it becomes 56. - When we further exclude the day with the maximum 112 visitors, the average for the remaining days becomes 42. - We assume one independent maximum day with 112 visitors. - We need to find the number of holidays in that week.

Concept / Approach:
The key idea is to translate each average into a total. First, we find the total visitors in the week from the average 40. Then we let h be the number of holidays. The number of non holiday days is (7 - h), and we know the average for those days is 56, so we can find their total. The total visitors for holidays is the difference between the weekly total and the non holiday total. After that, we exclude both the holidays and the single peak day with 112 visitors and use the new average 42 to create another equation. Solving these equations gives us h, the number of holidays.

Step-by-Step Solution:
Step 1: Total visitors in the full week = average * number of days = 40 * 7 = 280 visitors. Step 2: Let the number of holidays in the week be h. Step 3: Then, number of non holiday days = 7 - h. Step 4: When holidays are excluded, the average visitors per non holiday day is 56. Step 5: So total visitors on non holiday days = 56 * (7 - h). Call this Tn. Step 6: Total visitors on holidays = total weekly visitors - non holiday visitors = 280 - Tn. Step 7: Now we also exclude the peak day with 112 visitors. This peak day is one of the non holiday days. Step 8: After excluding holidays and this peak day, the remaining number of days = (7 - h) - 1 = 6 - h. Step 9: The total visitors on the remaining days = Tn - 112. Step 10: We are told that the average for these remaining days is 42, so: (Tn - 112) / (6 - h) = 42. Step 11: Substitute Tn = 56 * (7 - h) into the equation: (56 * (7 - h) - 112) / (6 - h) = 42. Step 12: Expand the numerator: 56 * (7 - h) = 392 - 56h. Step 13: So the numerator becomes 392 - 56h - 112 = 280 - 56h. Step 14: The equation is now (280 - 56h) / (6 - h) = 42. Step 15: Multiply both sides by (6 - h): 280 - 56h = 42 * (6 - h). Step 16: Expand the right side: 42 * 6 - 42 * h = 252 - 42h. Step 17: So 280 - 56h = 252 - 42h. Step 18: Rearrange: 280 - 252 = 56h - 42h. Step 19: 28 = 14h, so h = 28 / 14 = 2. Step 20: Therefore, there are 2 holidays in the week.

Verification / Alternative check:
Let us verify by constructing a possible scenario. With h = 2 holidays, non holiday days = 5. Total weekly visitors = 280. Non holiday total = 56 * 5 = 280. This means holiday visitors total 0, so holidays are days with no visitors, which is acceptable. Now non holiday days total 280 visitors across 5 days, average 56. The peak day has 112 visitors. Removing this day leaves 4 days with total visitors 280 - 112 = 168. Average for those 4 days = 168 / 4 = 42, which matches the given second adjusted average. This confirms that h = 2 is consistent.

Why Other Options Are Wrong:
Option A (1 holiday): Would lead to different totals, and the second average of 42 could not be satisfied simultaneously. Option C (3 holidays) and D (4 holidays): Changing h alters the denominator and the total visitors in a way that breaks the equation (280 - 56h) / (6 - h) = 42. Option E (5 holidays): Leaves too few non holiday days to maintain both average conditions.

Common Pitfalls:
Typical mistakes include assuming that holidays must have zero visitors without checking consistency, or mixing up the number of days remaining after removing both holidays and the peak day. Another common error is to try to solve intuitively without writing equations, which often leads to confusion. Writing clear expressions for totals and averages at each stage is the safest method.

Final Answer:
The number of holidays in the week is 2.