Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack

 It is currently 24 May 2017, 10:24

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# A certain league has four divisions. The respective

Author Message
TAGS:

### Hide Tags

GMAT Instructor
Joined: 07 Jul 2003
Posts: 770
Location: New York NY 10024
Schools: Haas, MFE; Anderson, MBA; USC, MSEE
Followers: 26

Kudos [?]: 212 [4] , given: 0

A certain league has four divisions. The respective [#permalink]

### Show Tags

16 Jul 2003, 04:59
4
KUDOS
51
This post was
BOOKMARKED
00:00

Difficulty:

95% (hard)

Question Stats:

44% (03:12) correct 56% (06:56) wrong based on 441 sessions

### HideShow timer Statistics

A certain league has four divisions. The respective divisions had 9, 10, 11, and 12 teams qualify for the playoffs. Each division held its own double-elimination tournament -- where a team is eliminated from the tournament upon losing two games -- in order to determine its champion. The four division champions then played in a single-elimination tournament -- where a team is eliminated upon losing one game -- in order to determine the overall league champion. Assuming that there were no ties and no forfeits, what is the maximum number of games that could have been played in order to determine the overall league champion?

(A) 79
(B) 83
(C) 85
(D) 87
(E) 88
[Reveal] Spoiler: OA

_________________

Best,

AkamaiBrah
Former Senior Instructor, Manhattan GMAT and VeritasPrep
Vice President, Midtown NYC Investment Bank, Structured Finance IT
MFE, Haas School of Business, UC Berkeley, Class of 2005
MBA, Anderson School of Management, UCLA, Class of 1993

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7372
Location: Pune, India
Followers: 2287

Kudos [?]: 15097 [23] , given: 224

Re: A certain league has four divisions. The respective [#permalink]

### Show Tags

03 Mar 2014, 22:18
23
KUDOS
Expert's post
10
This post was
BOOKMARKED
PareshGmat wrote:
manpreetsingh86 wrote:
AkamaiBrah wrote:
A certain league has four divisions. The respective divisions had 9, 10, 11, and 12 teams qualify for the playoffs. Each division held its own double-elimination tournament -- where a team is eliminated from the tournament upon losing two games -- in order to determine its champion. The four division champions then played in a single-elimination tournament -- where a team is eliminated upon losing one game -- in order to determine the overall league champion. Assuming that there were no ties and no forfeits, what is the maximum number of games that could have been played in order to determine the overall league champion?

(A) 79
(B) 83
(C) 85
(D) 87
(E) 88

Let's name the teams in group 1 as 1,2,3,4,5,6,7,8,9.

Case 1; team1 played with every other team and won all of its matches.
so total number of matchs =8
case 2: team2 , played with team 3,4,5,6,7,8,9 and won all of its matches.
total number of matches =7
after case 1 and case 2 we have only two teams remaining in the group1 which are team 1 and team 2. Now since question asks us for the maximum no. of matches. Therefore we must include the extra case in which team 2 defeated team 1. Now both team 2 and team 1 have 1 loss each. Now in the final match, we will found out about the eventual winner in group 1.
maximum no. of matches in group 1 are 8+7+1(in which team2 defeated team1) + 1 ( final) =17

Similarly in group 2 we have 9 + 8 +1 +1 =19
group 3 = 10+9+1+1 =21
group 4 = 11+10+1+1=23

After this we will have 4 winner from each group. lets name them as w1,w2,w3,w4
Let's assume w1 won all of its matches from the remaining three teams and eventually emerged as a winner. Therefore total matches among four winners=3

Therefore maximum total no. of matches played are 17+19+21+23+3=83

How long it took to solve this? Its taking a lot of time for me

Actually solving the problem doesn't take very long. Think of it this way:

We need to keep track of losses. Let's focus on those and forget about the wins.
Every time a game is played, someone loses. You can give at most 2 losses to a team since after that it is out of the tournament.
Consider the division which has 9 teams. What happens when 18 games are played? There are 18 losses and each team gets 2 losses (you cant give more than 2 to a team since it gets kicked out after 2 losses) so all are out of the tournament. But we need a winner so we play only 17 games so that the winning team get only 1 loss.

Similarly, the division with 10 teams can have at most 2*10 - 1 = 19 games.
The division with 11 teams can have at most 2*11 - 1 = 21 games.
The division with 12 teams can have at most 2*12 - 1 = 23 games.
This totals up to 80 games (note that the average of 17, 19, 21 and 23 will be 20 so the sum will be 4*20 = 80).

Now you have 4 teams. 1 loss gets a team kicked out. If you have 3 games, there are 3 losses and 3 teams are kicked out. You have a final winner!
Hence the total number of games = 80 + 3 = 83
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews SVP Joined: 03 Feb 2003 Posts: 1604 Followers: 9 Kudos [?]: 268 [11] , given: 0 ### Show Tags 16 Jul 2003, 06:36 11 This post received KUDOS 6 This post was BOOKMARKED A hard nut... As for divisional games: a minus is given for losing a game. the 9-team division) lets count minuses; 8 teams can have 2 minuses and one (a champion) can have 1 minus. Total 17 minuses. the 10-team division) lets count minuses; 9 teams can have 2 minuses and one (a champion) can have 1 minus. Total 19 minuses. the 11-team division) lets count minuses; 10 teams can have 2 minuses and one (a champion) can have 1 minus. Total 21 minuses. the 12-team division) lets count minuses; 11 teams can have 2 minuses and one (a champion) can have 1 minus. Total 23 minuses. After that, four teams remained. I assume that when 3 teams can have 1 minus and the overall champion has none. Total 3 minuses. Overall, there can be 83 minuses. Tus, it is B. Senior Manager Joined: 13 Jun 2013 Posts: 279 Followers: 13 Kudos [?]: 390 [3] , given: 13 Re: A certain league has four divisions. The respective [#permalink] ### Show Tags 27 Feb 2014, 14:30 3 This post received KUDOS 2 This post was BOOKMARKED AkamaiBrah wrote: A certain league has four divisions. The respective divisions had 9, 10, 11, and 12 teams qualify for the playoffs. Each division held its own double-elimination tournament -- where a team is eliminated from the tournament upon losing two games -- in order to determine its champion. The four division champions then played in a single-elimination tournament -- where a team is eliminated upon losing one game -- in order to determine the overall league champion. Assuming that there were no ties and no forfeits, what is the maximum number of games that could have been played in order to determine the overall league champion? (A) 79 (B) 83 (C) 85 (D) 87 (E) 88 Let's name the teams in group 1 as 1,2,3,4,5,6,7,8,9. Case 1; team1 played with every other team and won all of its matches. so total number of matchs =8 case 2: team2 , played with team 3,4,5,6,7,8,9 and won all of its matches. total number of matches =7 after case 1 and case 2 we have only two teams remaining in the group1 which are team 1 and team 2. Now since question asks us for the maximum no. of matches. Therefore we must include the extra case in which team 2 defeated team 1. Now both team 2 and team 1 have 1 loss each. Now in the final match, we will found out about the eventual winner in group 1. maximum no. of matches in group 1 are 8+7+1(in which team2 defeated team1) + 1 ( final) =17 Similarly in group 2 we have 9 + 8 +1 +1 =19 group 3 = 10+9+1+1 =21 group 4 = 11+10+1+1=23 After this we will have 4 winner from each group. lets name them as w1,w2,w3,w4 Let's assume w1 won all of its matches from the remaining three teams and eventually emerged as a winner. Therefore total matches among four winners=3 Therefore maximum total no. of matches played are 17+19+21+23+3=83 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7372 Location: Pune, India Followers: 2287 Kudos [?]: 15097 [1] , given: 224 Re: A certain league has four divisions. The respective [#permalink] ### Show Tags 22 Jul 2014, 22:14 1 This post received KUDOS Expert's post SachinWordsmith wrote: Thanks for the Reply Karishma. I follow your posts and your explanations are top notch. If I went with the same logic, the answer would be 4 lesser (eliminating the number of games by the group winners lost in the Prelims) = 79 Games. Can I generalize and use the above presented logic for say, a single elimination ( 1 Loss only) or triple elimination (3 Losses) format too? Yes, you are correct. 4 fewer games is all you can afford. Whether you can generalize will depend on the question. If the question remains the same: In case of a single loss elimination, there will be fixed number of games which will be played so there will be no maximum - minimum. Every game will have a loss and will eliminate exactly one team. In case of 3 loss elimination format, the qualifying team will not suffer any losses if we want to minimize the number of games and it will suffer 2 losses in case we want to maximize the number of games. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Manager
Joined: 08 Apr 2003
Posts: 149
Followers: 1

Kudos [?]: 45 [0], given: 0

### Show Tags

16 Jul 2003, 08:14

I generalised the solution again...

Consider 4 teams.

Max possible games if a team is eliminated after 2 loses = 3 + 2 = 5

Similarly, if 5 teams are there,

Max possible games = 4 + 3 = 7.

So going forth we can play 15,17,19,21 max possible games for 9,10,11,12 teams respectively.

Adding them up gives 72. Now there are two teams left in each group still. So four more games to give a winner in each team.

So 72+ 4 = 76.

Now of the four teams we can have max 3 games if its a knock out round.

So 76 + 3 = 79.

GMAT Instructor
Joined: 07 Jul 2003
Posts: 770
Location: New York NY 10024
Schools: Haas, MFE; Anderson, MBA; USC, MSEE
Followers: 26

Kudos [?]: 212 [0], given: 0

### Show Tags

16 Jul 2003, 10:30
stolyar wrote:
A hard nut...

As for divisional games: a minus is given for losing a game.

the 9-team division) lets count minuses; 8 teams can have 2 minuses and one (a champion) can have 1 minus. Total 17 minuses.

the 10-team division) lets count minuses; 9 teams can have 2 minuses and one (a champion) can have 1 minus. Total 19 minuses.

the 11-team division) lets count minuses; 10 teams can have 2 minuses and one (a champion) can have 1 minus. Total 21 minuses.

the 12-team division) lets count minuses; 11 teams can have 2 minuses and one (a champion) can have 1 minus. Total 23 minuses.

After that, four teams remained. I assume that when 3 teams can have 1 minus and the overall champion has none. Total 3 minuses.

Overall, there can be 83 minuses. Tus, it is B.

Very nicely done. This is correct and a fine approach to the problem.
_________________

Best,

AkamaiBrah
Former Senior Instructor, Manhattan GMAT and VeritasPrep
Vice President, Midtown NYC Investment Bank, Structured Finance IT
MFE, Haas School of Business, UC Berkeley, Class of 2005
MBA, Anderson School of Management, UCLA, Class of 1993

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15428
Followers: 649

Kudos [?]: 207 [0], given: 0

Re: A certain league has four divisions. The respective [#permalink]

### Show Tags

26 Feb 2014, 04:04
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1857
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Followers: 51

Kudos [?]: 2167 [0], given: 193

Re: A certain league has four divisions. The respective [#permalink]

### Show Tags

03 Mar 2014, 21:48
manpreetsingh86 wrote:
AkamaiBrah wrote:
A certain league has four divisions. The respective divisions had 9, 10, 11, and 12 teams qualify for the playoffs. Each division held its own double-elimination tournament -- where a team is eliminated from the tournament upon losing two games -- in order to determine its champion. The four division champions then played in a single-elimination tournament -- where a team is eliminated upon losing one game -- in order to determine the overall league champion. Assuming that there were no ties and no forfeits, what is the maximum number of games that could have been played in order to determine the overall league champion?

(A) 79
(B) 83
(C) 85
(D) 87
(E) 88

Let's name the teams in group 1 as 1,2,3,4,5,6,7,8,9.

Case 1; team1 played with every other team and won all of its matches.
so total number of matchs =8
case 2: team2 , played with team 3,4,5,6,7,8,9 and won all of its matches.
total number of matches =7
after case 1 and case 2 we have only two teams remaining in the group1 which are team 1 and team 2. Now since question asks us for the maximum no. of matches. Therefore we must include the extra case in which team 2 defeated team 1. Now both team 2 and team 1 have 1 loss each. Now in the final match, we will found out about the eventual winner in group 1.
maximum no. of matches in group 1 are 8+7+1(in which team2 defeated team1) + 1 ( final) =17

Similarly in group 2 we have 9 + 8 +1 +1 =19
group 3 = 10+9+1+1 =21
group 4 = 11+10+1+1=23

After this we will have 4 winner from each group. lets name them as w1,w2,w3,w4
Let's assume w1 won all of its matches from the remaining three teams and eventually emerged as a winner. Therefore total matches among four winners=3

Therefore maximum total no. of matches played are 17+19+21+23+3=83

How long it took to solve this? Its taking a lot of time for me
_________________

Kindly press "+1 Kudos" to appreciate

SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1857
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Followers: 51

Kudos [?]: 2167 [0], given: 193

Re: A certain league has four divisions. The respective [#permalink]

### Show Tags

04 Mar 2014, 18:57
Thank you for the explanation
_________________

Kindly press "+1 Kudos" to appreciate

Intern
Joined: 18 Sep 2013
Posts: 7
Followers: 0

Kudos [?]: 0 [0], given: 18

Re: A certain league has four divisions. The respective [#permalink]

### Show Tags

10 Mar 2014, 11:09
Dear Manpreet

I was applying the same trick but got confused in between.

Since, we allowed 3 matches between team 1 and team 2 in the first group, are you assuming that more than one match can be played between two teams as long as the team does not lose 2 matches in total?

Thanks & regards
Senior Manager
Joined: 13 Jun 2013
Posts: 279
Followers: 13

Kudos [?]: 390 [0], given: 13

Re: A certain league has four divisions. The respective [#permalink]

### Show Tags

10 Mar 2014, 13:34
ShantnuMathuria wrote:
Dear Manpreet

I was applying the same trick but got confused in between.

Since, we allowed 3 matches between team 1 and team 2 in the first group, are you assuming that more than one match can be played between two teams as long as the team does not lose 2 matches in total?

Thanks & regards

yes, you are right. see the question asks us to find the maximum number of games that could have been played to determine the overall league champion. hence we go for the additional matches.

i hope it is clear now.
Senior Manager
Joined: 17 Dec 2012
Posts: 460
Location: India
Followers: 27

Kudos [?]: 426 [0], given: 14

Re: A certain league has four divisions. The respective [#permalink]

### Show Tags

10 Mar 2014, 22:32
1
This post was
BOOKMARKED
The intuition is each team is eliminated after 2 losses. It has to lose 2 games and it can lose only 2 games. The winner loses only once or may not lose at all. But since maximum number of games is asked we will assume that the winner loses once.

Each loss corresponds to 1 game played.

For the 4 divisions the number of games played will be 2*8 + 1, 2*9 +1 , 2*10 +1 and 2*11+1= 17,19,21,23 resp= 80 total

Applying the same logic to the finals, the maximum number of games played will be 1*3 + 0=3

Total = 80+3=83
_________________

Srinivasan Vaidyaraman
Sravna
http://www.sravnatestprep.com

Classroom and Online Coaching

Intern
Joined: 03 Oct 2012
Posts: 26
Concentration: Strategy, General Management
GMAT 1: 750 Q49 V42
Followers: 0

Kudos [?]: 9 [0], given: 142

Re: A certain league has four divisions. The respective [#permalink]

### Show Tags

22 Jul 2014, 04:17
Hi,

Could someone comment on how they would solve/approach it if the question were : What's the minimum number of games that could be played before a winner is determined?
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7372
Location: Pune, India
Followers: 2287

Kudos [?]: 15097 [0], given: 224

Re: A certain league has four divisions. The respective [#permalink]

### Show Tags

22 Jul 2014, 06:25
SachinWordsmith wrote:
Hi,

Could someone comment on how they would solve/approach it if the question were : What's the minimum number of games that could be played before a winner is determined?

Use the same logic as that used for maximum number of games. You know that each team must lose in at least 2 games (and at most 2 games). The only difference will be that the qualifying team will not lose any match to minimize the number of matches.

Now can you come up with the answer?
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 03 Oct 2012 Posts: 26 Concentration: Strategy, General Management GMAT 1: 750 Q49 V42 Followers: 0 Kudos [?]: 9 [0], given: 142 Re: A certain league has four divisions. The respective [#permalink] ### Show Tags 22 Jul 2014, 06:35 Thanks for the Reply Karishma. I follow your posts and your explanations are top notch. If I went with the same logic, the answer would be 4 lesser (eliminating the number of games by the group winners lost in the Prelims) = 79 Games. Can I generalize and use the above presented logic for say, a single elimination ( 1 Loss only) or triple elimination (3 Losses) format too? Manager Joined: 12 Sep 2014 Posts: 169 Concentration: Strategy, Leadership GMAT 1: 740 Q49 V41 GPA: 3.94 Followers: 0 Kudos [?]: 80 [0], given: 103 Re: A certain league has four divisions. The respective [#permalink] ### Show Tags 04 Oct 2014, 13:08 stolyar wrote: A hard nut... As for divisional games: a minus is given for losing a game. the 9-team division) lets count minuses; 8 teams can have 2 minuses and one (a champion) can have 1 minus. Total 17 minuses. the 10-team division) lets count minuses; 9 teams can have 2 minuses and one (a champion) can have 1 minus. Total 19 minuses. the 11-team division) lets count minuses; 10 teams can have 2 minuses and one (a champion) can have 1 minus. Total 21 minuses. the 12-team division) lets count minuses; 11 teams can have 2 minuses and one (a champion) can have 1 minus. Total 23 minuses. After that, four teams remained. I assume that when 3 teams can have 1 minus and the overall champion has none. Total 3 minuses. Overall, there can be 83 minuses. Tus, it is B. Deceptive question and you're right! I double counted at the end making mine 18, 20 and so far to get 87 total instead of 83! Intern Joined: 18 May 2011 Posts: 4 Followers: 0 Kudos [?]: 0 [0], given: 13 Re: A certain league has four divisions. The respective [#permalink] ### Show Tags 01 Apr 2015, 21:42 Hi, I had a crazy thought. Here our goal is to maximize the matches. In Divison A we have 9 teams let them be a,b,c,d,e,f,g,h,i I let team "a" play with all other teams twice and lose every time. No. of matches=16 Similarly for b=14,c=12 etc. So we get a total of 72 matches in Group A alone. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7372 Location: Pune, India Followers: 2287 Kudos [?]: 15097 [0], given: 224 Re: A certain league has four divisions. The respective [#permalink] ### Show Tags 01 Apr 2015, 22:23 rmselva wrote: Hi, I had a crazy thought. Here our goal is to maximize the matches. In Divison A we have 9 teams let them be a,b,c,d,e,f,g,h,i I let team "a" play with all other teams twice and lose every time. No. of matches=16 Similarly for b=14,c=12 etc. So we get a total of 72 matches in Group A alone. Here is what the question says: "Each division held its own double-elimination tournament -- where a team is eliminated from the tournament upon losing two games --" When a loses its matches against b and c, it will be eliminated. It will not play any other game. It doesn't need to lose 2 matches against the same team for every team. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Director
Joined: 07 Aug 2011
Posts: 579
GMAT 1: 630 Q49 V27
Followers: 3

Kudos [?]: 449 [0], given: 75

Re: A certain league has four divisions. The respective [#permalink]

### Show Tags

01 Apr 2015, 23:20
VeritasPrepKarishma wrote:
Actually solving the problem doesn't take very long. Think of it this way:

We need to keep track of losses. Let's focus on those and forget about the wins.
Every time a game is played, someone loses. You can give at most 2 losses to a team since after that it is out of the tournament.
Consider the division which has 9 teams. What happens when 18 games are played? There are 18 losses and each team gets 2 losses (you cant give more than 2 to a team since it gets kicked out after 2 losses) so all are out of the tournament. But we need a winner so we play only 17 games so that the winning team get only 1 loss.

Similarly, the division with 10 teams can have at most 2*10 - 1 = 19 games.
The division with 11 teams can have at most 2*11 - 1 = 21 games.
The division with 12 teams can have at most 2*12 - 1 = 23 games.
This totals up to 80 games (note that the average of 17, 19, 21 and 23 will be 20 so the sum will be 4*20 = 80).

Now you have 4 teams. 1 loss gets a team kicked out. If you have 3 games, there are 3 losses and 3 teams are kicked out. You have a final winner!
Hence the total number of games = 80 + 3 = 83

Excellent solution .
Thanks VeritasPrepKarishma.
_________________

Thanks,
Lucky

_______________________________________________________
Kindly press the to appreciate my post !!

Re: A certain league has four divisions. The respective   [#permalink] 01 Apr 2015, 23:20

Go to page    1   2    Next  [ 23 posts ]

Similar topics Replies Last post
Similar
Topics:
A certain sports league has ten teams in its Western Confere 2 01 Oct 2012, 04:19
57 There are 8 teams in a certain league and each team plays 21 16 Jun 2016, 06:00
8 There are 8 teams in a certain league and each team plays 15 08 May 2017, 07:40
8 With respect to the division whose percent of total income 8 10 Oct 2016, 07:23
A certain company has 500 employees split among three divisions, such 3 02 Dec 2012, 04:18
Display posts from previous: Sort by