Question

A library has an average of 510 visitors on Sundays and 240 on other days. The average number of visitors per day in a month of 30 days beginning with a Sunday is:

• A. 285
• B. 290
• C. 297
• D. 305

Hint:

Carefully count the number of specific days (e.g., Sundays) in the given month, especially when the starting day is specified. Avoid making assumptions about the number of weekend days if not explicitly stated.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem asks to calculate the average number of visitors per day for a month of 30 days, given different average visitor numbers for Sundays and other days, and knowing that the month starts on a Sunday.

Step 2: Key Formula or Approach:
Average = $\frac{\text{Total number of visitors}}{\text{Total number of days}}$.
1. Determine the number of Sundays in the month.
2. Determine the number of "other days" in the month.
3. Calculate the total visitors for Sundays.
4. Calculate the total visitors for "other days".
5. Sum these to get the total visitors for the month.

Step 3: Detailed Explanation:
Given:
- Total days in the month = 30.
- Month begins with a Sunday.
- Average visitors on Sundays = 510.
- Average visitors on other days = 240.
1. Number of Sundays in the month:
Since the month starts on a Sunday, the Sundays will fall on days: 1, 8, 15, 22, 29.
Number of Sundays = 5.
2. Number of other days:
Number of other days = Total days - Number of Sundays = $30 - 5 = 25$.
3. Total visitors on Sundays:
Total Sunday visitors = $5 \text{ Sundays} \times 510 \text{ visitors/Sunday} = 2550$.
4. Total visitors on other days:
Total other day visitors = $25 \text{ days} \times 240 \text{ visitors/day} = 6000$.
5. Total visitors for the month:
Total visitors = Total Sunday visitors + Total other day visitors = $2550 + 6000 = 8550$.
6. Average number of visitors per day:
Average = $\frac{\text{Total visitors}}{\text{Total days}} = \frac{8550}{30} = \frac{855}{3} = 285$.

Step 4: Final Answer:
The average number of visitors per day is 285.

Answer 2

The month has 30 days beginning on a Sunday, so there are exactly 5 Sundays (days 1, 8, 15, 22, 29) and 25 other days. A weighted average of 510 visitors on Sundays and 240 on the other days must fall closer to 240, since the 25 "other" days heavily outweigh the 5 Sundays. We can check each option against this weighted total.

1. 285: Total visitors would be \( 285 \times 30 = 8550 \). Checking this against the actual breakdown: \( 5 \times 510 + 25 \times 240 = 2550 + 6000 = 8550 \), an exact match.
2. 290: This would require a total of \( 290 \times 30 = 8700 \) visitors, which is 150 more than the 8550 that the actual Sunday and weekday counts produce, so this overstates the true average.
3. 297: This corresponds to a total of \( 297 \times 30 = 8910 \) visitors, 360 more than the correct total of 8550. Since there is no way to add extra visitors beyond what the Sunday and weekday averages give, this value is too high.
4. 305: This implies a total of \( 305 \times 30 = 9150 \), a full 600 more than the actual 8550, making it the furthest option from the true weighted total.

Because the 25 non-Sunday days at 240 visitors each dominate the total, the average is pulled well below the midpoint of 510 and 240, and only 285 reproduces the exact visitor count of 8550.

Therefore, the correct answer is 285.