## 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

Solution
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?
$300/n=15$
$n=20$

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, $\dfrac{300}{15}=20$.