Round-robin soccer tournament

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16206; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sat Jan 31, 2009 12:27 pm

There are 12 teams in a round-robin soccer tournament, where each team plays every other team once. The winning team of each game receives 3 points and the losing team receives 0 points. If there is a tie, each team receives 1 point. If the total number of points given to all teams is 185, how many games ended in a tie?
A) 13
B) 14
C) 15
D) 16
E) 17

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Source: Beat The GMAT — Problem Solving | Sat Jan 31, 2009 12:27 pm

ontopofit: Senior | Next Rank: 100 Posts; Posts: 68; Joined: Sun Dec 21, 2008 12:00 am; Thanked: 3 times

by ontopofit » Sat Jan 31, 2009 12:41 pm

Is that A?
there will be a total 66 matches played.(11*12/2)

for 66 matches there wil be a total 198 pts.

for a draw game 2 pts are given (to both teams)
let x be no. of games drawn

hence, 198 -3x +2x = 185,
x=13

shulapa: Senior | Next Rank: 100 Posts; Posts: 35; Joined: Thu Nov 13, 2008 8:05 am; Thanked: 6 times

by shulapa » Sat Jan 31, 2009 12:43 pm

I like this question, as it combines two subjects.

First we should determine how many games were played. This is similar to the famous question of how many people shaked hands in a class reunion. So, the number of games are 12C2 = 66.

Now we can create two equations with X as the number of games ended in tie and Y the number of games ended in one team's victory.

x + Y = 66 -> Y = 66 - X
3Y + 2X = 185 -> 3(66-X) + 2X = 185

Solving the equation gives X = 19.

Thus, A

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16206; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sat Jan 31, 2009 12:52 pm

Awesome - the answer is A.
I love it when there's more than one aproach (so often the case with GMAT questions)

Here's my solution as well:

If there are 12 teams then there are 12C2 (66) games played. If all of the games had ended with one team winning (no ties), the total number of points given to all teams would have been 66x3=198. So, we can conclude that some games ended in a tie.
Notice that if a game ends in a tie, a total of only 2 points are given to the teams (1 point less than the points given when a game is won). So, for every winning game that we replace with a tie game, the total number of points given to the teams decreases by 1.
If no games end in a tie, a total of 198 points are given. So, a total of 185 points means that there were 13 ties (198-13=185).
The answer is A

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gaggleofgirls: Master | Next Rank: 500 Posts; Posts: 138; Joined: Thu Jan 15, 2009 7:52 am; Location: Steamboat Springs, CO; Thanked: 15 times

by gaggleofgirls » Sat Jan 31, 2009 8:32 pm

The 123 teams play a total of 12C2 games = 66 games.

There are 3 points given in games that do not end in ties and only 2 points give in games that do end if ties.

If all games were W/L 9no ties), there there would be 198 points given.

But since there were only 185 points given, the 'missing' points each represent 1 tie game.

So 198-185 = 13

Answer = A.

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