A library has an average of 510 visitors on Sundays and 240 on other days. What is the average number of visitors per day in the month of June beginning with a Sunday?

### Concept & Strategy This problem relies on basic calendar knowledge and calculating a weighted average. To find the overall average visitors for the month, we must determine the exact number of Sundays versus regular days, calculate the total visitors for each category, and divide by the total days in the month. $$\text{Overall Average} = \frac{(\text{Sundays} \times \text{Avg}) + (\text{Other Days} \times \text{Avg})}{\text{Total Days}}$$ ### Step-by-Step Solution * **Calendar Analysis:** The month of June has exactly 30 days. We are told June begins on a Sunday (June 1st is a Sunday). Sundays will occur on the 1st, 8th, 15th, 22nd, and 29th. Therefore, there are exactly 5 Sundays in this month. The number of "other days" = $30 - 5 = 25$ days. * **Calculating Total Visitors:** Total visitors on all Sundays = $5 \text{ days} \times 510 \text{ visitors/day} = 2550$ Total visitors on other days = $25 \text{ days} \times 240 \text{ visitors/day} = 6000$ Grand total of visitors in June = $2550 + 6000 = 8550$ * **Calculating the Final Average:** $$\text{Average per day} = \frac{\text{Grand Total}}{\text{Total Days}}$$ $$\text{Average} = \frac{8550}{30}$$ $$\text{Average} = \frac{855}{3} = 285$$ ### Exam Strategy & Shortcut **Deviation / Baseline Method:** Instead of multiplying large numbers, pick a baseline average. Let's assume every day has a baseline of 240 visitors. If every day had 240 visitors, the average would be 240. However, there are 5 Sundays where the library gets an *extra* 270 visitors ($510 - 240 = 270$). Total extra visitors = $5 \times 270 = 1350$. Now, distribute these 1350 extra visitors equally across all 30 days of the month: Extra per day = $\frac{1350}{30} = 45$. Overall Average = Baseline + Extra per day = $240 + 45 = 285$. This method avoids calculating $25 \times 240$ and keeps the numbers small and manageable. ### Common Pitfall The most frequent error is incorrectly counting the number of Sundays. Students might assume a 30-day month always has 4 weeks and therefore exactly 4 Sundays, leading to a wrong calculation. Always map out the days if a specific starting day is mentioned. ### Final Answer **Therefore, the correct answer is 285.**