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

Introduction / Context:
This problem involves calculating the overall average number of visitors to a library in a month, given different daily averages for Sundays and other days. You are told that the month has 30 days and begins on a Sunday. Using this information, you must find the average number of visitors per day for the entire month. It is a simple but important example of weighted averages over different types of days.

Given Data / Assumptions:
- Average visitors on Sundays = 510 per day.
- Average visitors on other days (non Sundays) = 240 per day.
- The month has 30 days and starts on a Sunday.
- We need to determine how many Sundays occur in such a month and then compute the overall daily average number of visitors.

Concept / Approach:
If a month of 30 days starts on a Sunday, the days of the month cover four complete weeks (28 days) plus 2 extra days. In every full week there is exactly one Sunday, so there are 4 Sundays in the first 28 days. The extra two days are Sunday and Monday, giving one more Sunday. Hence the total number of Sundays is 5 and the remaining 25 days are non Sundays. After finding the total visitors on Sundays and on other days, we divide the combined total by 30 to get the required average.

Step-by-Step Solution:
Step 1: Count Sundays in the month. In 4 complete weeks there are 4 Sundays. The remaining 2 days start with Sunday, so the extra days are Sunday and Monday, giving 1 more Sunday.Step 2: Total number of Sundays = 4 + 1 = 5.Step 3: Total number of non Sundays = 30 - 5 = 25.Step 4: Total visitors on Sundays = 5 * 510 = 2550.Step 5: Total visitors on other days = 25 * 240 = 6000.Step 6: Combined total visitors in the month = 2550 + 6000 = 8550.Step 7: Average visitors per day for the 30 day month = 8550 / 30 = 285.

Verification / Alternative Check:
To check, note that if every day had 240 visitors, total visitors would be 240 * 30 = 7200. On Sundays, the library actually receives 510 visitors, which is 270 more than 240. There are 5 Sundays, so the extra visitors contribute 5 * 270 = 1350 additional visitors. Hence total visitors = 7200 + 1350 = 8550. Dividing by 30 again gives 285, confirming the result.

Why Other Options Are Wrong:
Values like 290, 297 or 305 correspond to totals that are either too high or too low compared to the sum of 2550 Sunday visitors and 6000 other day visitors. Only an average of 285 visitors per day reproduces the correct total of 8550 visitors for the month.

Common Pitfalls:
Some learners miscount the number of Sundays in a 30 day month that begins on a Sunday, mistakenly taking 4 or 6 Sundays. Others forget to multiply the averages by the number of days of each type before taking the overall average. Carefully counting Sundays and using total visitors for each category avoids these mistakes.

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