Figuring out the average of a group of numbers is easy. We have learned since middle school to add up the numbers and divide by however many numbers we have. This gives us the average. SAT and ACT math tests often try to be tricky and ask us to use this average formula in reverse. Knowing how to solve these types of questions will come in handy when you take the test. Here is an example:

Ex 1: The average of 4 different numbers is 808. What is the sum of the numbers?

This is an example of a simple average problem in reverse. If they gave us the four numbers and wanted the average, this problem would be easy. Instead, they gave us the average and want us to find the sum. Let's take a look at how the average formula works:

THE SUM OF THE NUMBERS
4

  = 808

 

If we want to solve for "the sum", then we just want to get "the sum" all by itself on the left side of the equation. We can do this by simply multiplying both sides by four:

4 x   THE SUM OF THE NUMBERS
4

  = 808 x 4

 

We are left with:                   THE SUM OF THE NUMBERS = 3,232.


So the answer to Example 1 is 3,232. This method will work for any average problem where you need to find the sum.

 

Let's look at a typical problem that is related but is a few steps harder:

Ex 2: The average of four different integers is 808. One of the integers is 107 and one of the integers is 800. If all of the integers are positive, what is the largest that one of the other two numbers could be?

On any problem that involves figuring out a group of four or five numbers, it is helpful to draw lines on your test booklet like this:

___ ___ ___ ___

Then fill in the numbers that you already know:

107 800 ___ ___

Doing this allows us to see what we have left to figure out. We want to figure out the greatest possible value of one of the numbers. We know that the numbers must add up to 3,232 because we already did this problem in the above example. If we know that a group of numbers has a certain sum and we want one of the numbers to be as large as possible, then we want the other unknown number to be as small as possible. Since the problem tells us that all of the numbers have to be positive, the smallest that a number could be is 1. We can now set up a problem like this:

107 + 800 +  1  +  x  = 3,232

Solving for x will give us our answer, which is 2,324.