The Peytonville Peacocks and the Adenville Aardvarks have each played the same number of games so far this season.
The Peacocks have a .664 average, and the Aardvarks have won 70 games. Which team is ahead?
Note that each team plays 162 games in a season, and that the team's average is the number of games won divided by the number of games played rounded off to three decimal places.
The Peacocks are ahead.
This might be a "hack" but this is how I solved the problem.
I set up a table. The number of games, x, has to be greater than or equal to 70 (otherwise the Aardvarks couldn’t win 70 games), and also less than or equal to 162 (but that doesn’t really matter as you’ll see). So my first column will be x from 70 to 162.
In the next column I have the Aardvark’s average, which is always 70/x. Notice that since the 70 is fixed, and x is increasing, the average will always be decreasing as we go down the column. At x = 106, the Aardvark’s average is .660. Beyond that their average will always be lower than the Peacocks.
Ok, so now we need to find out if it is possible for the Peacocks to have an average of .664 before x=106. What I mean is, let’s try to find the smallest x for which integer/x = 0.664 (rounded). If that x is less than 106, then we know it is possible for the Aardvarks to have a better score than the Peacocks. But if there is no such x less than 106, then we know it is impossible for the Aardvarks to have a better score than the Peacocks.
So what I did was I made a column, called P, which was the exact "number" of games the Peacocks need to win to get 0.664 average. So P = x*0.664. Notice P doesn’t have to be an integer. This is just to give me a ballpark (pun not intended). Next I will have P1, which is P rounded down to the next integer, and P2, which is P rounded up to the next integer. Then I will calculate the average of each P1 and P2, rounded to 3 decimal places.
Now I look for 0.664 in either the P1 average column, or the P2 average column. The first time I see one is at x=107. At x=107, the Aardvarks won 70 games with an average of 0.654, and the Peacocks won 71 games with an average of 0.664. There are more cases where the Peacocks have an average of 0.664, but all of those are for a greater x, so the Aardvarks average will always be decreasing.
Therefore, the Peacocks are ahead.
Posted by nikki
on 2004-11-03 13:56:54