How Many Games Were Tied?

Date: 06/07/99 at 18:21:34
From: laura
Subject: How many games were tied?

Hi!
I'm a seventh grade student in Canada and I just finished writing the 
Gauss Math Competition in the Canadian Mathematics Competition series. 
There was one problem I skipped, which is extremely frustrating. Can 
you please give me some help?

Here is the problem:
In a softball league, after each team has played every other team 4 
times, the total accumulated points are: LIONS 22, TIGERS 19, MOUNTIES 
14, ROYALS 12. If each team received 3 points for a win, 1 point for a 
tie and no points for a loss, what is the total number of tied games?

My dad and I have tried to solve it using linear and quadratic 
equations, but it really doesn't offer an adequate answer.

Thanks a lot,
Laura

Date: 06/07/99 at 18:56:17
From: Doctor Anthony
Subject: Re: How many games were tied?

Total points  22 + 19 + 14 + 12 =  67

There are 4 teams, so the number of games is 4 x 3 x 4/2 = 24 games 
(There are 3 games by each team in each round and there are 4 rounds. 
But each game is played by 2 teams so we divide by 2.)

So if a = number of games which end in a win or a loss, b = number of 
games which are drawn, then

           a + b = 24
and so
               b = 24 - a


Also
         3a + 2b = 67     (remember that both teams earn one point
                            in a draw)
    3a + 48 - 2a = 67

               a = 19

So there are 19 games in which one or other team wins and then 5 games 
which are drawn.

   Therefore 5 games are tied.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/