M43-29

Question

There are two teams in a chess tournament, Team India and Team USA, and no player represents both teams. Each player from Team India plays with each player from Team USA exactly once, resulting in a total of 60 matches between the teams. How many players are on Team USA?

(1) For every 3 players from Team India, there are 5 players from Team USA.
 
(2) There are a total of 16 players on both teams combined.

Bunuel · Answer

Official Solution:

There are two teams in a chess tournament, Team India and Team USA, and no player represents both teams. Each player from Team India plays with each player from Team USA exactly once, resulting in a total of 60 matches between the teams. How many players are on Team USA? 
 
Assuming there are  x  players in Team India and  y  players in Team USA, a total of 60 matches implies that  xy = 60 . Hence, we can have the following possible pairs of  (x, y) :
   
(2, 30)
 
(3, 20)
 
(4, 15)
 
(5, 12)
 
(6, 10)
 
(10, 6)
 
(12, 5)
 
(15, 4)
 
(20, 3)
 
(30, 2)
   
(1) For every 3 players from Team India, there are 5 players from Team USA.
 
This implies that  x : y = 3 : 5 . From the possible pairs, only  x = 6  and  y = 10  satisfy this. Therefore, there are 10 players on Team USA. Sufficient.
 
(2) There are a total of 16 players on both teams combined. 
 
This implies that  x + y = 16 . From the possible pairs, both  x = 6 ,  y = 10  and  x = 10 ,  y = 6  satisfy this. Therefore, there can be 6 or 10 players on Team USA. Not sufficient.

Answer: A

Rebaz · Answer

Bunuel

Is the number of matches the same as the number of games played?

To calculate the number of games: (X)(Y) / 2 = Number of games, coz a game is played between 2 players

May i ask how you arrive at XY=60. It it the case that by multiplying the number of players on one team by the number of player on the other team, we get total number of matches?

Thanks in Advance!

Bunuel · Answer

Each player from the USA plays against each player from India exactly once, resulting in xy = 60 TOTAL games/matches. This concept becomes easier to understand if you imagine 6 total games and manually count the possibilities.

For example:

Team USA has two players (a and b), and Team India has three players (x, y, z). The games played are: a-x, a-y, a-z, b-x, b-y, and b-z.

Msk666 · Answer

my way of doing this is:
each indian player plays a USA player 3 times, so if a group has 3 indians and 5 USA players it means we have 3*5= 15 games (can't be more otherwise a player will be playing the other player more than once)

we have 60 games so we need four such groups, giving us 20 USA players. hence S1 sufficient. 
S2: we dont have a breakdown so cant tell anything

bumpbot · Answer

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M43-29

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