The average age
Given this information about averages, find the ages of these teachers.
Age
11 to 14
Challenge level
Problem
The sum of the ages of 5 teachers is 190.
The mean age of the teachers is 10 years greater than the median age. The modal age is 5 years less than the median age. One of the teachers is 56 years old.
How old are the other teachers?
This problem is taken from the World Mathematics Championships
Student Solutions
The sum of the ages of 5 teachers is 190, so the mean age is 190$\div$5 = 38.
The mean age of the teachers is 10 years greater than the median age, so the median age must be 10 less than 38, so the median age is 28 - so the ordered list of the ages is __, __, 28, __, __.
The modal age is 5 years less than the median age, so the modal age is 23.
There must be at least 2 teachers who are 23. The third youngest teacher has the median age, so the youngest three teachers are 23, 23, 28.
Of the two older teachers, one is 56. So four of the teachers are 23, 23, 28 and 56. Subtracting these from 190 will give the age of the remaining teacher.
23 + 23 + 28 + 56 = 130
So the other teacher must be 60, since 130 + 60 = 190.