SAT TIP: Magic With Averages, Part II - Q&A

Question

Dear Mitch

(Continued from previous Q&A):

Joanne bowled 5 games and scored: 88, 90, 98, 100, and 110. She has one game left to bowl, and it is about to begin. If she absolutely must make her life's dream come true, and end up with an average for all the games of 120, what must she score in the game about to begin?

Answer

SAT TIP: Averages, Part II:

Okay, as stated, depending upon the way they are being used, averages can be considered exactly accurate numbers or completely unreliable rough estimates for future predictions. We covered the way they are unreliable, and now for the way they are exact:

Mathematically, whenever you have a set of numbers (scores, rates, prices, etc.), and you find their average, you can determine with exactitude the total of all the scores added together. (Or the total of the rates, prices, or whatever is at issue.) Working backwards, the formula is this:

(average) x (number of items) = exact sum of all the 'items' in that group.

Example: There are 5 people on an elevator. You are not given the individual weight of any of them, but you are told that the passengers average one hundred pounds each. Now, even if none of them weighs exactly a hundred pounds (or even close), you can easily determine the total number of pounds aboard the elevator. How?

By using the formula given just above the example:

(average) x (# of members) = the total weight.

So, the five people who average 100 pounds per person have a total weight of:

5 x 100, or 500 pounds.

Why?

Because if you retrace your steps in the method you used to find the average to begin with you will notice that the second-to-last step had you adding the separate items together to derive the total of their weight (or price or whatever). Then you divided that total by the number of individuals; well, the opposite of division is multiplication, so you can sort of undo the last step and find the total, and after looking at a few you will probably see why average x # of guys = total.

Now, in a typical question from a standardized test, here comes the 'trick':

LISTEN: Averaging is about sharing evenly. If two people go into a cave, and one has 2 dollars and the other has 4, and they decide they wish to have the same amount, they can put all their money (6 dollars) in a pile between them and dole them out saying "One for you, one for me, one for you, one for me..."

They will each end up with three dollars, which means the 'rich' guy will get a little poorer and the 'poor' guy will get a little richer.

So, if the people were to know in advance that this was going to be the procedure, and the 'rich' one wants the others to end up with a certain amount, he must bring in a bunch of extra dollars so that even after evening out he/she will have that new amount left for himself.

The bowler so far got:

88, 90, 98, 100, and 110.

With one game left, and a desire to end the season with an average of 120 per game, the bowler must give enough extra points to each earlier score to bring them up to 120 each.

The 88 needs a gift of 32 points.

The 90 needs a gift of 30 points.

The 98 needs a gift of 22 points.

The 110 needs a gift of 10 points.

Total # of points given to these early scores as 'gifts': 94.

And the last game, after giving out the 94 points in gifts, must have 120 left over for himself. So that score needs to be 120 + 94.

Which equals 214 points.

ANSWER: The bowler needs to get a score on game 6 of 214 points to make the season average of her dreams.

Hope this helps,

Mitch