## Average (arithmetic mean)

Each of four people were given a blank piece of paper on which they wrote a positive integer. If the average (arithmetic mean) of these integers is 21, then what is the greatest possible integer that could be written on one of the pieces of paper?

## Knowsys Method

*Note: In the math section of the SAT, you will encounter questions that do not have answer choices. Instead of bubbling in a letter, you will bubble in your answer. These questions are called grid in questions, and you should always guess an answer for them because there is no penalty for getting the question wrong!*

**Read the problem carefully**. Make sure you understand the situation described in the question and take note of important details. The integers written on the pieces of paper are positive, and they are NOT distinct (different), so there could be repeats of the same number.

**Identify the bottom line.** Greatest possible integer (out of these 4) = ?

**Assess your options. ** There is only one possible method for solving this problem, and it is demonstrated below.

**Attack the problem**. Whenever a problem asks about average (arithmetic mean), the first thing you should do is write out the average formula.

Now, plug in the information you know from the problem. You were given the average and the number of people.

Multiply both sides by 4 to find the sum, which is 84. Now, think about it. The four integers written on the pieces of paper all add up to 84. What is the greatest possible number that could be included among those four? Figure out how to make the other three integers as small as possible to leave the greatest possible number left over. The integers must be positive, so each one must be at least 1. Three of them can be 1 because they do not have to be distinct. So, subtract 3 from 84 and get 81.

**Loop back.** Check to verify that you have solved for the bottom line.

The correct answer is 81.

This is a medium level problem.

**Want some help reviewing the math concepts you need to master? **Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.