A library has on average 510 visitors on Sundays and 240 visitors on each of the other days. In a 30-day month that begins on a Sunday, what is the average number of visitors per day during the month?

Introduction / Context:
This problem combines averages with a calendar structure. The number of Sundays in the month affects how many days have the higher visitor count. This is a typical exam question that checks whether you can count the number of specific weekdays in a month and then use total visitors to compute an overall average.

Given Data / Assumptions:

• The average number of visitors on Sundays is 510.
• The average number of visitors on each non-Sunday is 240.
• The month has 30 days and starts on a Sunday.
• We need to find the average number of visitors per day for the entire month.

Concept / Approach:
Since the month starts on a Sunday and has 30 days, we must determine how many Sundays occur in that period. Once we know how many Sundays and non-Sundays there are, we can compute the total visitors in the month by multiplying the appropriate average for each type of day. Finally, we divide the total visitors by 30 to get the average per day.

Step-by-Step Solution:
A 30-day month that begins on Sunday has Sundays on days 1, 8, 15, 22 and 29.So there are 5 Sundays in the month.Number of non-Sunday days = 30 - 5 = 25.Average visitors on a Sunday = 510, so total Sunday visitors = 5 * 510 = 2550.Average visitors on a non-Sunday = 240, so total non-Sunday visitors = 25 * 240.Compute 25 * 240 = 6000.Total visitors in the month = 2550 + 6000 = 8550.Average visitors per day for the month = total visitors / total days = 8550 / 30 = 285.

Verification / Alternative check:
We can sanity check by noticing that the overall average should lie between 240 and 510. Since Sundays (with 510 visitors) are fewer than non-Sundays (with 240 visitors), the overall average should be much closer to 240 than to 510. The value 285 fits this expectation and matches the exact calculation.

Why Other Options Are Wrong:
Values like 290, 297, 305 or 275 are not equal to 8550 / 30. If you recompute total visitors using those averages, the implied total would not match 8550. Only 285 gives the correct total number of visitors and respects the tallies for both Sundays and other days.

Common Pitfalls:
Students sometimes miscount the number of Sundays, especially if they forget that the pattern repeats every 7 days. Others mistakenly assume 4 Sundays in every month regardless of the starting day. Errors can also occur when multiplying or adding the totals. To avoid mistakes, always list out the Sundays by date and then confirm that the total days add up correctly.

Final Answer:
The average number of visitors per day during the month is 285.