GMAT Club Olympics 2024 (Day 5): There are two teams in a chess

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.
 
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Bunuel · Accepted Answer

GMAT Club Official Explanation:

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, resulting in a total of 60 matches. 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.

Kinshook · Answer

Given: 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, resulting in a total of 60 matches.
Asked: How many players are on Team USA?

Let the number of players in Team India & Team USA be x & y respectively.
Total matches played = x * y = 60
y = ?

(1) For every 3 players from Team India, there are 5 players from Team USA.
x/y = 3/5; x = 3y/5; 
xy = 60; (3y/5)*y = 60; y^2 = 60*5/3 = 100; y = 10; x = 6
The number of players on Team USA = y = 10
SUFFICIENT

(2) There are a total of 16 players on both teams combined.
x + y = 16; x = 16-y
xy = 60
(16-y)y = 60
y^2 - 16y + 60 = 0
(y-10)(y-6) = 0
y = 10 or y = 6
Since unique value of y can not be ascertained.
NOT SUFFICIENT

IMO A

AthulSasi · Answer

Assuming X players in Team India and Y players in Team USA.

Given that X * Y = 60  - Eq1

We need to find Y.

Statement 1  implies
 \frac{ X}{Y} = \frac{3}{5}

5X=3Y

X= \frac{3}{5} Y

Substituting in Eq1
 \frac{3}{5} Y * Y = 60

Y^2 = 100

Y = 10

Sufficient.

Statement 1  implies
 X+Y = 16

X = 16-Y

Substituting in Eq1

(16-Y)*Y = 60

Y^2-16Y+60 = 0

(Y-10)(Y-6)=0

Y can be either 6 or 10.

Since Y can have multiple values, Statement 2 is not sufficient.

Correct Asnwer A

xhym · Answer

We know that each player from Team India (I = Indian players) plays each player from Team USA (U = USA players), so:

I*U=60

Statement 1

This means that  \frac{I}{U} = \frac{3}{5}  so that means that  I = \frac{3}{5}U

I*U=60

\frac{3}{5}U*U=60

U^2=100

U = 10  (we ignore -10 as we cannot have negative players on a team)

-> Statement 1 alone is sufficient.

Statement 2

This means that  I + U = 16  so that means that  I = U-16

I*U=60

(U-16)*U=60

U^2-16U-60=0

(U-6)(U-10)=0

U=6  and  U=10

This gives two options, each of which is valid, so we cannot conclusively know which of these is the right one.

-> Statement 2 alone is not sufficient.

The answer is  A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

DG1989 · Answer

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, 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.

Solution:  Let X represent the number of players on Team India and Y represent the number of players on Team USA.
The total number of matches played between the two teams = 60
Since each player from Team India played with each player from Team USA
X * Y = 60  ---------(1)
 We need to find Y

Statement 1:    For every 3 players from Team India, there are 5 players from Team USA 
This means  \frac{X}{Y}  =  \frac{3}{5} 
X =  \frac{3Y}{5}

From (1)
X * Y = 60
 \frac{3Y}{5}  * Y = 60
 Y^2  = 100
Y = 10
  SUFFICIENT

Statement 2:    There are a total of 16 players on both teams combined. 
This means X + Y = 16
X = 16 - Y

From (1)
X * Y = 60
(16 - Y) * Y = 60
 Y^2  - 16Y + 60 = 0
(Y - 6)(Y - 10) = 0
Y = 6 or 10
Since we cannot determine a unique value of Y
  INSUFFICIENT

The correct answer is A

OmerKor · Answer

Ok, We need to fight back if we want to WIN.
Green team, it's time to wake up!
Fifth Day, Here we go:
Let's get started with our explanation for this topic:

Glance - the Question:
Question: We have here kind of combinatorics question.

Rephrase - Reading and Understanding the question:
Given:
60 matches
Each one MUST play with eachother
*We don't know if each player can play two times or third..
[?]: What is the number of players in the U.S team.
1 value = Sufficient 2 values = Insufficient
Rephrase:
India * USA = Matches
1 60 60
60 1 60
2 30 60
30 2 60
3 20 60
20 3 60
4 15 60
15 4 60
5 12 60
12 5 60
6 10 60
10 6 60

Solve:
(1) 3 India = 5 USA
The only match in our table is: 6 and 10
So we have 10 players in USA team.
Sufficient.

(2) 16 players total
can we know if the USA team is the 6 or the 10 ?
we CANNOT know that.
Insufficient.

Answer Choice A is out Answer

**Perhaps I missed the problem here. but anyway, I just want to mention that it will be better to write that each person can play Only once.

THE END
I hope you liked the explanation, I have tried my best here.
Let me know if you have any questions about this question or my explanation.

**Perhaps I missed the problem here. but anyway, I just want to mention that it will be better to write that each person can play Only once. THE END I hope you liked the explanation, I have tried my best here. Let me know if you have any questions about this question or my explanation. https://cdn.gmatclub.com/cdn/files/forum/images/smilies/1f44f-1f3fb.png

d_patel · Answer

Question stem tells us following details:

Two teams in tournament - Team India and Team USA
No player represents both teams.

If Number of players in Team India =  x  and Number of players in Team USA =  y ,
Each player from Team India plays with each player from Team USA, resulting in a total of 60 matches between the teams.
This gives, 
( xC_1 )( yC_1 ) = 60
 xy = 60 
 
We need to find y.

Statement-1 
For every 3 players from Team India, there are 5 players from Team USA.
So,  \frac{x}{y }= \frac{3}{5} 
x =  \frac{3y}{5}

And we know  xy = 60

\frac{3y^2}{5} = 60

y^2 = 100 
  y = +10  or  y = -10 
But number of players cannot be negative.

So, we get  y=10  as single solution.

Statement-1 is sufficient.

Statement-2 
There are a total of 16 players on both teams combined.
So,  x+y= 16 
x =  16 - y

And we know  xy = 60

(16-y)y = 60

16y - y^2 = 60 
  y^2 - 16y + 60  = 0  
  (y-10)(y-6)  = 0  
  y = 10  or  y = 6

So, either USA has 10 players or 6 players. We cannot be sure.

Statement-2 is not sufficient.

Final answer -  A.  Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Lizaza · Answer

How to get the number of matches? Well, if everyone plays everyone, then it's just the multiple of the number of players:
 i * u = 60,  where  i  stands for the Indian players and  u  stands for the American ones.
The question is:  u = ?

(1)
Basically this says that the ratio is  \frac{i}{u }= \frac{3}{5} 
and then   i = \frac{3u}{5} 
Therefore,  i*u = \frac{3u}{5}*u = \frac{3u^2}{5} = 60 
From here, we can easily find  u , so  this is sufficient.

(2)
 i+u = 16 
and  i*u = 60 
While it's quite clear that the necessary values are 6 and 10, we have no way of distinguishing, which of these is America and which - India.
Therefore,  this is not sufficient .

The correct answer is A.

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