Mean of Numbers

The mean of a set of numbers is 20. If one number is increased by 300, the mean increases to 35. How many numbers are in the set?
Source: NCTM Mathematics Teacher, January 2006

The formula for the mean of n numbers a_1,a_2,a_3,\cdots,a_n is simple
mean = \dfrac{a_1+a_2+a_3+\cdots+a_n}{n}
To gain a concrete understanding of the mean, imagine a set of ants that share the food that they find. During the course of the day, some ants find a few grains of food like 10 or 11 while others find more like 20 or 30 grains. They do not keep track of who is doing more or who is doing less and put all the grains in a pile. At the end of the day, they count the grains and divide them equally among themselves.
Today, the mean equals 20 so each ant gets 20 grains. The ants start eating their food and wish there were more to eat when suddenly a fairy dumps 300 grains to the now empty pile. The ants happily divide the 300 grains and found that each gets 15 more grains.
So how many ants are there?

Answer: 20

Alternative solution
Suppose the ants gather 300 more grains and the mean of this new gathering equals 15. There are two ways of producing a mean of 15: either ants gather different amounts some more than 15, some less than 15 or they gather exactly 15 each. We use the second way to find the number of ants, \dfrac{300}{15}=20.


About mvtrinh

Retired high school math teacher.
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