By this time in your mathematical education, it is likely that you have encountered the concept of the arithmetic mean, A.K.A. average. Given a set of numbers, the arithmetic mean is equal to the sum of those numbers divided by the "number of numbers."
So if you were told that n was equal to the arithmetic mean of the numbers 1, 2, 4, and 9, you could solve for n. You would just add up the numbers 1, 2, 4, and 9, obtaining a sum of 16. Then you would divide by 4, obtaining the arithmetic mean of 4. So you would eagerly scribble down the answer, n=4.
But sometimes, things get a bit trickier. Sometimes you are given the arithmetic mean, but you don't know all of the original numbers. For example, maybe you know that the arithmetic mean of the set {1, 2, 4, 9, n} is 6. How would solve for n this time? Don't let this type of problem faze you. You already know the definition of the arithmetic mean (the sum of the numbers divided by the number of numbers) and you already know how to deal with unknown values (solve an algebraic equation). All you have to do is to put these two pieces of knowledge together!
Write an equation based on the definition of the arithmetic mean, using n as a variable:
(1 + 2 + 4 + 9 + n) / 5 = 6.
1 + 2 + 4 + 9 + n = 30
16 + n = 30
n = 14
Ta-da! Now, that wasn't so bad, was it?
There are all sorts of variations on this theme. For example, sometimes there is not just one unknown value, but two. Suppose you know that the arithmetic mean of the set {1, 2, 4, 9, a, b} is 6. Can you find the mean of a and b? You may notice that there is not enough information to solve for either a or b on its on. However, even without knowing what each value is, it is still possible to find their mean. Start by setting up an equation, just as you did before, when there was only one variable:
(1 + 2 + 4 + 9 + a + b) / 6 = 6.
1 + 2 + 4 + 9 + a + b = 36
16 + a + b = 36
a + b = 20
You know the sum of a and b now. What next? Remember your original goal: to find the mean of a and b. That's just the sum of a and b divided by 2! So take 20 and divide it by 2--your final answer is 10.
Sometimes, the Mathcounts test-writers turn these kinds of problems into word problems: "Bozo scored 12%, 14%, 20%, and 24%. What does he need to score on his next test in order to average a 20% on all five tests?" The only difference here is that you have to take the extra first step of translating English into Math. Like before, you write an algebraic equation--where the variable represents Bozo's fifth test score--and then solve the equation. (The answer is 30%).