Question

The following table shows the number of employees and their median age in eight companies located in a district. \[ \begin{array}{|l|l|l|} \hline \textbf{Company} & \textbf{Number of Employees} & \textbf{Median Age} \\ \hline A & 32 & 24 \\ B & 28 & 30 \\ C & 43 & 39 \\ D & 39 & 45 \\ E & 35 & 49 \\ F & 29 & 54 \\ G & 23 & 59 \\ H & 16 & 63 \\ \hline \end{array} \] It is known that the age of all employees are integers. It is also known that the age of every employee in A is strictly less than the age of every employee in B, the age of every employee in B is strictly less than the age of every employee in C, and the age of every employee in G is strictly less than the age of every employee in H. In company F, the lowest possible sum of the ages of all employees is ………….

Hint:

In problems involving minimizing the sum of ages, assign the lowest possible integer values for each employee while respecting the constraints (e.g., the median age or age restrictions from other companies).

Answer 1

Step 1: Understand the problem. We are given the following data for the eight companies: \[ \begin{array}{|l|l|l|} \hline \textbf{Company} & \textbf{Number of Employees} & \textbf{Median Age} \\ \hline A & 32 & 24 \\ B & 28 & 30 \\ C & 43 & 39 \\ D & 39 & 45 \\ E & 35 & 49 \\ F & 29 & 54 \\ G & 23 & 59 \\ H & 16 & 63 \\ \hline \end{array} \] It is known that the age of every employee in company A is strictly less than the age of every employee in company B, and so on. We are to calculate the lowest possible sum of the ages of all employees in company F, given that the median age in company F is 54.
Step 2: Consider the lowest possible age for each employee.
Since the median age in company F is 54, the middle employee's age is 54. To minimize the sum of ages, we assign the lowest possible ages to the employees based on the given constraints.
The first 14 employees can have the minimum possible age of 55, as their ages must be strictly greater than the median age of company E (49).
The 15th employee will have the median age of 54.
The remaining 14 employees can have the same age of 54.
Thus, the lowest possible sum of the ages of all employees in company F is calculated as follows: \[ \text{Sum} = 14 \times 55 + 15 \times 54 \] Step 3: Calculate the sum. \[ \text{Sum} = 14 \times 55 + 15 \times 54 = 770 + 740 = 1510 \] Thus, the lowest possible sum of the ages of all employees in company F is: \[ \boxed{1510} \]