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Mean/Average

When thinking about averages we like to think about cookies. Okay, we always like to think about cookies...averages are only a convenient excuse.

Suppose Paul got 3 cookies and Mary got 5. That distribution doesn't seem quite fair. Let's redistribute so Paul and Mary each receive the same number of cookies. We have 3 + 5 = 8 cookies total, which means to divide the cookies fairly between Paul and Mary we should give each person 4 cookies. We should do it quickly though, because Mary just caught our eye and we're fairly certain she's onto us.

Sample Problem

Anita got 2 cookies, Jonas got 3, and Ella got 2. If we were to redistribute the cookies fairly, how many would each person get?

We have 2 + 3 + 2 = 7 cookies total. If we divide these cookies evenly between the 3 people, we see that each person should receive

Hope they're chewy rather than crunchy cookies, or cutting these things will be a nightmare.

The mean, or average, of a set of values is the size of a "fair" portion. To find the average of a set of values, we add up all the values and divide by the number of portions. "Average'' and ''mean'' refer to the same thing, and you may be asked to find either one. Their synonymousness doesn't extend outside the world of algebra though. Your mother won't care if you tell her a bully at school was being "average" to you.

Sample Problem

Louisa got 5 cookies, Danielle got 8 cookies, and Marcus got 2 cookies. What is the average number of cookies each person got?

The average number of cookies is the number each person would get if we divided the cookies fairly. We add up all the different numbers of cookies:

5 + 8 + 2 = 15.

Then we divide by the number of portions, which is 3, since there are 3 people:

15 รท 3 = 5.

The average number of cookies received by each person is 5. Again, we had better take care of this situation with haste. Mary from our earlier problem caught wind of the cookie surplus and is on her way over.

Another phrase you might hear is the phrase "on average." This means roughly the same thing as "find the average of..." In the example above we could say that, on average, each person got 5 cookies. It's only another way of saying "5 is the average of the number of cookies each person got."

Word problems involving averages can do some interesting things. They can't bend their legs behind their heads... nothing that interesting, but still. Interesting.

Sample Problem

Linda bought four cookies that cost $0.25 each and two cookies that cost $0.50 each. On average, how much did she spend per cookie?

We want to make a list of all the cookie prices, and find the average of the numbers in that list. Since Linda bought four cookies for $0.25 each, the number $0.25 will show up four times in the list. Similarly, the number $0.50 will show up twice. The prices of the cookies were

0.25, 0.25, 0.25, 0.25, 0.50, 0.50.

If we add these numbers and divide by 6 (the total number of cookies purchased), we get

or about $0.33 per cookie. Maybe next time Linda will consider buying cookies by the box. Preferably in bulk from Costco.

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