The mean of a set of numbers is . If one number is increased by , the mean increases to . How many numbers are in the set?
Source: NCTM Mathematics Teacher, January 2006
The formula for the mean of numbers is simple
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 or while others find more like or 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 so each ant gets grains. The ants start eating their food and wish there were more to eat when suddenly a fairy dumps grains to the now empty pile. The ants happily divide the grains and found that each gets more grains.
So how many ants are there?
Suppose the ants gather more grains and the mean of this new gathering equals . There are two ways of producing a mean of : either ants gather different amounts some more than , some less than or they gather exactly each. We use the second way to find the number of ants, .