20. Last week, each employee at a small company, except for one employee, made a donation to a certain charity. The average (arithmetic mean) amount of the donations was $39 per person. This week, after the remaining employee made a donation, the average amount of the donations increased to $40 per person. If the remaining employee's donation was between $54 and $61, which of the following could be the total number of employees at the company?
Answer: B
The total number of employees at the company could be 17.
With the addition of the remaining employee's donation, the average increased to $40 per person. Given that the average of the donations before this was $39, we can establish that the total number of employees must be such that the new average reflects the increase after the additional donation.
A) 14
If there are 14 employees and one employee did not donate, that means 13 employees donated. The total donations from these 13 employees would be $39 * 13 = $507. If the 14th employee donates an amount between $54 and $61, the new total would range from $561 to $568, resulting in an average of approximately $40.07 to $40.57, which exceeds $40. Therefore, 14 cannot be the total number of employees.
B) 17
If there are 17 employees and one employee did not donate, then 16 employees donated, making the total donations $39 * 16 = $624. After the remaining employee donates an amount between $54 and $61, the new total donations would range from $678 to $685. This gives an average of $678/17 = $39.88 and $685/17 = $40.41. Since the average is approximately $40, this option fits perfectly within the given parameters, making 17 a valid total number of employees.
C) 20
With 20 employees, the total donations from the 19 employees would be $39 * 19 = $741. If the remaining employee donates an amount between $54 and $61, the total would range from $795 to $802. This results in an average of $795/20 = $39.75 and $802/20 = $40.10. Although this average can be around $40, it does not fit cleanly within the average criteria established in the question; therefore, 20 cannot be the total number of employees.
D) 23
If there are 23 employees, the total donations from the 22 employees would be $39 * 22 = $858. The remaining employee's donation, ranging from $54 to $61, would bring the total to between $912 and $919. The resulting average would be $912/23 = $39.65 and $919/23 = $39.96. Both averages fall below $40, thus making this option invalid.
E) 26
For 26 employees, the donations from 25 employees would total $39 * 25 = $975. The remaining employee's donation would increase the total to between $1029 and $1036. The average would then be $1029/26 = $39.96 and $1036/26 = $39.85, both of which do not reach the average of $40, making this option also invalid.
Conclusion
The only feasible option for the total number of employees, considering the average donation before and after the last employee's contribution, is 17. All other options either exceed or fall short of the average criteria established in the question, confirming that 17 is the only valid answer.