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. The median age of an employee across the eight companies is …………..

Hint:

To find the median across multiple groups, first calculate the cumulative number of employees in each group and determine which group contains the median position.

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 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 so on.
The task is to find the median age of all employees across the eight companies.
Step 2: Calculate the total number of employees.
The total number of employees across all eight companies is: \[ 32 + 28 + 43 + 39 + 35 + 29 + 23 + 16 = 245 \] Thus, there are 245 employees in total. 
Step 3: Determine the position of the median.
Since there are 245 employees, the median employee is at position:
\[ \frac{245 + 1}{2} = 123 \] So, we need to determine which company contains the 123rd employee when we list all the employees from the lowest median age to the highest.
Step 4: Identify the median employee's company.
Company A has 32 employees (positions 1 to 32, median age = 24).
Company B has 28 employees (positions 33 to 60, median age = 30).
Company C has 43 employees (positions 61 to 103, median age = 39).
Company D has 39 employees (positions 104 to 142, median age = 45).
The 123rd employee is in Company D, where the median age is 45.
Step 5: Conclude the median age.
Thus, the median age of an employee across the eight companies is: \[ \boxed{45} \]