A library has an average number of 510 visitors on Sunday and 240 on other days. What is the average number of visitors per day in a month of 30 days beginning with Saturday?
285
Calculating Average Library Visitors Per Day
This problem asks us to find the average number of visitors per day in a library over a specific month, given the average number of visitors on Sundays and other days, and the total number of days in the month, including the starting day.
Understanding the Month Structure
We are given a month of 30 days that begins with a Saturday. To find the average number of visitors per day, we first need to determine how many Sundays and how many 'other days' fall within this 30-day period.
If the month starts on a Saturday, the days of the week will sequence as follows:
- Day 1: Saturday
- Day 2: Sunday
- Day 3: Monday
- Day 4: Tuesday
- Day 5: Wednesday
- Day 6: Thursday
- Day 7: Friday
- Day 8: Saturday
- Day 9: Sunday
- ... and so on.
Sundays will fall on days 2, 9, 16, 23, and 30 of the month.
Determining the Number of Sundays and Other Days
In a 30-day month starting on a Saturday, the Sundays occur on days:
- 2nd day
- 9th day ($\small 2 + 7$)
- 16th day ($\small 9 + 7$)
- 23rd day ($\small 16 + 7$)
- 30th day ($\small 23 + 7$)
So, there are 5 Sundays in this 30-day month.
The number of 'other days' is the total number of days minus the number of Sundays.
Number of other days = Total days - Number of Sundays
Number of other days = $\small 30 - 5 = 25$ days
Calculating Total Visitors
We know the average number of visitors for Sundays and other days:
- Average visitors on Sunday: 510
- Average visitors on other days: 240
Now, we calculate the total visitors for the month by summing the visitors on all Sundays and all other days.
Total visitors on Sundays = Number of Sundays $\times$ Average visitors on Sunday
Total visitors on Sundays = $\small 5 \times 510 = 2550$ visitors
Total visitors on other days = Number of other days $\times$ Average visitors on other days
Total visitors on other days = $\small 25 \times 240$
$\small 25 \times 240 = 25 \times (200 + 40) = 25 \times 200 + 25 \times 40 = 5000 + 1000 = 6000$ visitors
Total visitors in the month = Total visitors on Sundays + Total visitors on other days
Total visitors in the month = $\small 2550 + 6000 = 8550$ visitors
Calculating the Average Visitors Per Day
The average number of visitors per day is the total number of visitors divided by the total number of days in the month.
Average visitors per day = $\frac{\text{Total visitors in the month}}{\text{Total number of days}}$
Average visitors per day = $\frac{8550}{30}$
To calculate $\frac{8550}{30}$:
$\frac{8550}{30} = \frac{855}{3}$
Now, perform the division:
$\frac{855}{3} = \frac{600 + 255}{3} = \frac{600}{3} + \frac{255}{3} = 200 + \frac{240 + 15}{3} = 200 + \frac{240}{3} + \frac{15}{3} = 200 + 80 + 5 = 285$
So, the average number of visitors per day is 285.
| Day Type | Number of Days | Average Visitors Per Day | Total Visitors |
|---|---|---|---|
| Sunday | 5 | 510 | $\small 5 \times 510 = 2550$ |
| Other Days | 25 | 240 | $\small 25 \times 240 = 6000$ |
| Total | 30 | $\small 2550 + 6000 = 8550$ |
Average per day = $\frac{8550}{30} = 285$
Conclusion on Average Library Visitors
Based on the calculation, the average number of visitors per day in a 30-day month beginning with Saturday, with 510 visitors on Sundays and 240 on other days, is 285.
Revision Table: Average Calculation for Library Visitors
| Concept | Description | Application in this problem |
|---|---|---|
| Average | Sum of values divided by the number of values. | Total visitors divided by total days. |
| Calculating Days | Determining specific days of the week within a given period. | Counting Sundays in a 30-day month starting Saturday. |
| Total Sum | Summing values from different categories. | Summing visitors on Sundays and visitors on other days. |
Additional Information: Average and Weighted Average
The problem we solved involves a type of calculation that is related to a weighted average. A simple average assumes each value contributes equally. However, in this case, Sunday visitors and other day visitors have different average values and occur on a different number of days. The overall average is influenced more by the days with more visitors (Sundays) and by the number of days in each category.
If all days had the same average number of visitors, we would just use that number as the average. But since Sunday is different, we calculate the total visitors over the period and then divide by the total number of days.
This is similar to calculating a weighted average where the 'weights' are the number of days each category occurs:
Weighted Average = $\frac{(\text{Average}_1 \times \text{Weight}_1) + (\text{Average}_2 \times \text{Weight}_2)}{\text{Weight}_1 + \text{Weight}_2}$
In our case:
Weighted Average = $\frac{(\text{Average Sunday Visitors} \times \text{Number of Sundays}) + (\text{Average Other Day Visitors} \times \text{Number of Other Days})}{\text{Number of Sundays} + \text{Number of Other Days}}$
Weighted Average = $\frac{(510 \times 5) + (240 \times 25)}{5 + 25}$
Weighted Average = $\frac{2550 + 6000}{30} = \frac{8550}{30} = 285$
This confirms our previous calculation method and highlights the concept of averaging values across different groups with varying frequencies.
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