GMAT Club Olympics 2024 (Day 7): The French and German teams are

Question

The French team and the German team are playing a series of games against each other, where a win results in +3 points, a draw results in 0 points, and a loss results in -2 points. If each team played fewer than 10 games in the series and each team experienced all three outcomes, how many games did each team play?

(1) The French team scored 14 points in the series.
(2) The German team scored -6 points in the series.

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:

The French and German teams are playing a series of games against each other, where a win results in +3 points, a draw results in 0 points, and a loss results in -2 points. If each team played fewer than 10 games in the series and each team experienced all three outcomes, how many games did each team play?

Assuming the French team won x games, lost y games, and drew z games, the question asks for the value of x + y + z, given that x + y + z is less than 10. Note that the French team winning x games and losing y games implies that the German team won y games and lost x games. Also, since each team experienced all three outcomes, x, y, and z must each be greater than 0.

(1) The French team scored 14 points in the series.

This implies that 3x - 2y = 14. By trial and error, we can find that only one set of (x, y) such that x + y < 10 satisfies this equation: (6, 2). For (6, 2), x + y = 8, and since z > 0, z must be 1 for x + y + z to be less than 10. Therefore, x = 6, y = 2, and z = 1. Total games played by each team = 6 + 2 + 1 = 9. Sufficient.

(2) The German team scored -6 points in the series.

This implies that 3y - 2x = -6. By trial and error, we can find that only one set of (x, y) such that x + y < 10 satisfies this equation: (6, 2). As with statement (1), for (6, 2), we can conclude that z must be 1. Therefore, x = 6, y = 2, and z = 1. Total games played by each team = 6 + 2 + 1 = 9. Sufficient.

Answer: D.

jksiingh · Answer

Win gives +3, Draw gives 0 and Loss gives -2
Both teams have atleast 1 win, 1 draw and 1 loss
Both teams have played less than 10 games

Statement 1:
French scored 14 points, which means they have had mostly wins
But let's calculate loss and draws
They have had atleast 1 loss and 1 draw, which gives them -2 points
To bring the score to 14, we need 16 points, but 16 is not a multiple of 3, that means they must have more than 1 loss
Let's try 2 losses and 1 draw, gives them -4 points
They need 18 points to have a score of 14, which means 6 wins
6 wins, 2 losses and 1 draw gives 14 points. Total is 9 games, and they can't have any more games because they've had less than 10 games
Sufficient

Statement 2:
Similarily here they have -6 points
Having atleast 1 win and 1 draw gives 3 points
But to get -6, we need -9 points which is not a divisible by 2
So they need to have atleast 2 wins, giving them 6 points
They need 6 losses to get -12 points and final score of -6
Total is 9 games, They can't play any more because theyve had less than 10 games
Sufficient
Answer is D

jack5397 · Answer

IMO - D

The French team and the German team are playing a series of games against each other, where a win results in +3 points, a draw results in 0 points, and a loss results in -2 points. If each team played fewer than 10 games in the series and each team experienced all three outcomes, how many games did each team play?

(1) The French team scored 14 points in the series. // Sufficient as per the given context they are playing against each other and experiencing all three outcomes.

Games --G1 G2 G3 G4 G5 G6 G7 G8 G9
Points-- 3. 3. 3. 3. 3. 3. 0. -2. -2

It will be opposite in the case of German as they are playing against each other.

(2) The German team scored -6 points in the series. // Sufficient as per the given context they are playing against each other and experiencing all three outcomes.

Games --G1 G2 G3 G4 G5 G6 G7 G8 G9
Points-- -2. -2. -2. -2. -2. -2. 0. 3. 3

meet459 · Answer

The French and German teams are playing a series of games against each other, where a win results in +3 points, a draw results in 0 points, and a loss results in -2 points. If each team played fewer than 10 games in the series and each team experienced all three outcomes, how many games did each team play?

(1) The French team scored 14 points in the series.
(2) The German team scored -6 points in the series.

Let us assume that x be the number of matches a team won and y be the number of matches they lost

It is given that x + y < 9 (Since there was atleast one draw) and x > 1 and y > 1

The total points by a team will be 3x - 2y

(1) The French team scored 14 points in the series. -> 3x - 2y = 14 => x = (14+2y)/3 => 4 + (2+2y)/3

We know that y starts from 2 onwards, if we keepy y = 2, we get x = 6 (Valid)
If we keep y = 3 and 4, it is not possible since x will be not be a natural number
Let's try with y = 5, we get x = 4 (Invalid because x + y < 9) and if we increase y to 8, x will be even more, hence no need to proceed further.
Hence only one possible way x = 6, y = 2, SUFF

(2) The German team scored -6 points in the series. -> 3x - 2y = -6 => x = (2y-6)/3 => x = 2y/3 - 2

We know that y starts from 3 onwards (Since it must be divisible by 3), if we keepy y = 3, we get x = 0 (Invalid, since x > 1)
Let's try with y = 6, we get x = 2 (Valid)
If we keep y = 9, we get x = 4 (Invalid since x + y < 9), if we increase y, x will be even more, hence no need to proceed further.
Hence only one possible way y = 6, x = 2, SUFF

Hence answer will be D

Lizaza · Answer

Let's try to obtain the points presented in the conditions. The only general concern is that both teams played the same number of games (obviously), they had 1 of each outcomes and the number was strictly below 10.

(1) 
How to score 14 points?
 +3 - 2 + 0 = 1  point  
 1 + 3 + 3 + 3 + 3 + 3 = 16  points  
And we only have 1 game left until 10, which is a loss with 16 - 2 = 14 points total.
The answer is 9 games, and  this is sufficient.

(2) 
How to score -6 points?
 +3 - 2 + 0 = 1  point  
 1 - 2 - 2 - 2 - 2 - 2 = -9  points  
And again we only have 1 game left until 10, which is a victory with -9 + 3 = -6 points total.
The answer is 9 games, and  this is sufficient.

The right answer is therefore D.

PK1 · Answer

French and German team are playing a series of games against each other.
Win = +3 points
Draw= 0 points
Lose = -2 points
each team played fewer than 10 games and each team experienced all three outcomes,
how many games did each team play

Game is between two teams, so ones win = ones loss.. draw will be shared
n<10..so could be minimum 3 to get all three results.
so 3<n<10
n could be 4, 5, 6, 7, 8, 9

Option 1: French team scored 14 points in the series
3x (win) - 2y(loss) =14 and total games = x+y+1(for draw)
3*6- 2*2= 14 so total games = 9
or 3*8 - 2*5 but here total games become more than 10

so this gives unique number of games i.e 9

Option 2: German scored -6 points in series
3*x(win) - 2y(loss) = -6 and total games = x+y+1
3*2 - 2*6= -6 and total games = 9 
or 3*4 - 2*9 but total games here become more than 9
so even this gives unique number of games

So D is the answer

Milkyway_28 · Answer

The key pieces of information we are given here is that each team played fewer than 10 games, meaning at the most 9 games, and that each team experienced all three outcomes. The three outcomes are +3 for a win, 0 for a draw and -2 for a loss. The last piece of obvious information worth stating is that the French team and the German team are playing against each other so they will have the same number of games played.

At first glance, it seems we do not have enough information from the statements provided but let's double check by going through them.

(1) The first statement says that "the French team have scored 14 points". Well let's see how they could have gotten there

Game 1 (win, +3). Total 3 points
Game 2 (win, +3). Total 6 points
Game 3 (win, +3). Total 9 points
Game 4 (win, +3). Total 12 points
Game 5 (win, +3). Total 15 points - We have to continue because we need exactly 14 points. There are three variations from here to get to 14 points. 1. Win, loss, loss 2. Loss, win, loss 3. Loss, loss, win. Either way doesn't matter, what is clear is that we need an additional three more steps. 
Game 6 (loss, -2). Total 13 points
Game 7 (loss, -2). Total 11 points
Game 8 (win, +3). Total 14 points - In the question it clearly states that each team must experience all three outcomes therefore, one more game needs to be played for a draw. 
Game 9 (draw, 0). Total 14 points.

We said that 9 was the maximum number of games each team played and therefore we know that they each played 9 games. Statement 1 is sufficient.

(2) "The German team scored -6 points in the series."

Game 1 (loss, -2). Total -2 points
Game 2 (loss, -2). Total -4 points
Game 3 (loss, -2). Total -6 points - we got the right number of points however the team still needs to experience a draw and a win, so we can make those the next two games.
Game 4 (win, +3). Total -3 points
Game 5 (draw, 0). Total -3 points. - From here we can go 1. win, loss, loss, loss 2. loss, win, loss, loss 3. loss, loss, win, loss 4. loss, loss, loss, win. It doesn't matter because the number of games are the same. There must be another 4 more games played.
Game 6 (loss, -2). Total -5 points
Game 7 (loss, -2). Total -7 points
Game 8 (loss, -2). Total -9 points  
Game 9 (win, +3). Total -6 points.

Once again, all 9 games had to be used up and therefore we know how many games both the french team and german team played. Statement 2 is sufficient.

So each statement on its own is sufficient. The answer is D

andreagonzalez2k · Answer

(1)
French team must have won at least 6 games (because 5*3=15 and there is no way of obtain 14 from 15 substracting 2).
And, as each team experienced all three outcomes, there are at least 8 games. And the maximum number of games is 9.

As the only case of 8 games is (win=6, draw=1, loss=1) and its score is 16, not 14, we can deduce that the number of games must be 9.

It is not necessary but we can study all the cases:
- win=6, draw=1, loss=1: score=16
- win=6, draw=1, loss=2: score=14
- win=6, draw=2, loss=1: score=16
- win=7, draw=1, loss=1: score=19

So the only valid case is: win=6, draw=1, loss=2
And the number of games is 9.

SUFFICIENT

(2)
German team:
3*win - 2*loss = -6
win = loss*2/3 - 2

As win must be a positive integer, loss must a multiple of 3.

Cases:
- loss=3 -> win=0; not possible as each team experienced all three outcomes
- loss=6 -> win=2: possible as draw can be 1 and total games are 9

No bigger values for loss are possible.

SUFFICIENT

IMO D

Purnank · Answer

Given - The French team and the German team are playing a series of  games against each other ,  where a win results in +3 points, a draw results in 0 points, and a loss results in -2 points. 
To find - If each team played  fewer than 10 games  in the series  and each team experienced all three outcomes , how many games did each team play?

lets assume a, b and c are the wins, draws and loss for french team and c, b and a are the wins, draws and loss for german team.
plus each team experienced all 3 outcomes so  a≥1 ,  b≥1 , &  c≥1 
Therefore,  a+b+c<10  and total points by team french = 3a + 0b - 2c =  3a - 2c 
Similarly, total points by german team =  3c - 2a 
lets check conditions now,

1st - The French team scored 14 points in the series.
Therefore, 3a - 2c = 14
possible values for (a,c) are (6,2) (8,5) infinite values possible
But given  a+b+c<10 
so only pair (6,2) satisfies this plus b≥1 means min b can be 1
6+2+1 < 10 also satisfies 
hence total games played by them are 9.
 Sufficient.

2nd - The German team scored -6 points in the series.
Therefore, 3c - 2a = -6
possible values for (a,c) again (6,2) (9,4) infinite values possible
But given  a+b+c<10 
so only pair (6,2) satisfies this plus b≥1 means min b can be 1
6+2+1 < 10 also satisfies 
hence total games played by them are 9.
 Sufficient.

Answer is D.

jairovx · Answer

Understand the given information 
Each team played fewer than 10 games
Each team experienced all three outcomes (win, draw, loss)
Win = +3 points
Draw = 0 points
Loss = -2 points

Analyze Statement (1) 
French team scored 14 points
Let x = number of wins, y = number of losses
14 = 3x - 2y
2y = 3x - 14
y = (3x - 14) / 2
For y to be an integer, 3x - 14 must be even
Possible solutions: x = 6, y = 2 (8 games)

Analyze Statement (2) 
German team scored -6 points
Let a = number of wins, b = number of losses
-6 = 3a - 2b
2b = 3a + 6
b = (3a + 6) / 2
For b to be an integer, 3a + 6 must be even
Possible solutions: a = 2, b = 6 (8 games)

Conclusion 
Each statement alone is sufficient to determine that eight games were played.

IMO. D

captain0612 · Answer

Choice D

Games played between French and German team be x.

Given: x < 10

1. For every win a team gets +3 points
2. For every draw a team gets 0 points
3. For every loss a team gets -2 points
4. Each team played fewer than 10 games
5. Each team experienced at least 1 loss, 1 draw and 1 win

Question: How many games did each team play.

Approach: Since, we know that each team played less than 10 games, total number of games played by either teams is x ( because teams only played against each other)

Statement 1:

French team scored 14 points.

Let number of wins, draws and loss be w, d and l respectively, and these are Positive Integers > 0

Score of french team = 14 => 3w + 0d -2l = 14

3w - 2l = 14
( sum or difference between 2 numbers is even only both the numbers are even or both the numbers are odd)
Therefore, 3w must be even since 2l is always even. For 3w to be even w must be even.

Now, let's try plugging possible values for w = { 2, 4, 6} it cannot 8 or more because of condition 4 (games less than 10)

For w = { 2, 4 } value of 3w is less and cannot sum to 14. Hence only value of w is 6. Now if w = 6, then l must 2

we know w = 6, l = 2. based on condition 4 total games cannot be more than 9. Hence w + d + l < 10
6 + 2 + d < 10 => d < 2. So, only  d = 1 is possible.

hence number of games played = 8,  Sufficient

Statement 2:

German team scored -6 points

Let number of wins, draws and loss be w, d and l respectively, and these are Positive Integers > 0

Score of German team = -6 => 3w + 0d -2l = -6

3w - 2l = -6
Same logic as Statement 1
3w to be even w must be even

Now, let's try plugging values for w = { 2 } cannot be greater than 4 of condition 4 (games less than 10)

Hence only value of w is 2. Now if w = 2, then l must 6

we know w = 2, l = 6. based on condition 4 total games cannot be more than 9. Hence w + d + l < 10
6 + 2 + d < 10 => d < 2. So, only  d = 1 is possible.

hence number of games played = 8,  Sufficient

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