Arithmetic

Mean, Median & Mode

Easy

The following table shows the number of employees and their median age in eight companies located in a district.

Company Number of employees Median age
A 32 24
B 28 30
C 43 39
D 39 45
E 35 49
F 29 54
G 23 59
H 16 63

It is known that the age of all employees are integers. 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, ..., the age of every employee in G is strictly less than the age of every employee in H.

The median age of employees across the eight companies is:

Entered answer:

✅ Correct Answer: 45

Add up all the employees across the eight companies:

$32 + 28 + 43 + 39 + 35 + 29 + 23 + 16 = 245$

Since there are $245$ employees (an odd number), the median is at position:

$\dfrac{245 + 1}{2} = 123\text{rd value}$

When you have an odd number of values arranged in order, the exact middle one is at position $\dfrac{n+1}{2}$ .

Since every employee in $A$ is younger than every employee in $B$ , who are all younger than every employee in $C$ , and so on — all $245$ employees can be thought of as standing in a single line from youngest to oldest. We just need to find which company the $123$ rd person belongs to.

Using cumulative count:

Company	Number of Employees	Cumulative Count	Positions Covered
$A$	$32$	$32$	$1$ st to $32$ nd
$B$	$28$	$60$	$33$ rd to $60$ th
$C$	$43$	$103$	$61$ st to $103$ rd
$D$	$39$	$142$	$104$ th to $142$ nd
$E$	$35$	$177$	$143$ rd to $177$ th
$F$	$29$	$206$	$178$ th to $206$ th
$G$	$23$	$229$	$207$ th to $229$ th
$H$	$16$	$245$	$230$ th to $245$ th

The $123$ rd value falls in Company $D$ (since it lies between the $104$ th and $142$ nd positions).

Company $D$ has $39$ employees, so its median is the:

$\dfrac{39+1}{2} = 20\text{th employee within } D$

The $123$ rd employee overall is the $123 - 103 = 20$ th employee within $D$ — which is exactly $D$ 's median.

This means the $123$ rd employee's age is exactly the median age of Company $D$ , which is $45$ .

$\boxed{\text{Median age} = 45}$

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Company	Number of employees	Median age
A	32	24
B	28	30
C	43	39
D	39	45
E	35	49
F	29	54
G	23	59
H	16	63

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