Question

cricket question 2

Answer 1

In a cricket tournament organised by the ICC , a total of 15 teams participated ."Australia" , as usual won the tournament by scoring the maximum number of points .The tournament is organised as a single round robin tournament - where each team plays with every other team exactly once.3 points are awarded for a win , 2 points are awarded for a tie/washed out match and 1 point is awarded for a loss ."Zimbabwe" had the lowest score ( in terms of points ) at the end of the tournament ."Zimbabwe" scored a total of 21 points .All the 15 national teams got known that at least one match played by the "Australian" team was tied/washed out

Answer 2

Which of the following is always true for the "Australia" team.

a it had at least two ties/washouts

b it had a maximum of 3 losses

c it had a maximum of 9 wins

d all the above

Answer 3

@aum can u help

Answer 4

There is a total of 15 teams including Australia.
"each team plays with every other team exactly once"
So Australia plays 14 other teams in the tournament.
Assume Australia wins W games, ties T games and loses L games. Then
W + T + L = 14 ------ (1)

Zimbabwe got the lowest score of 21 points which implies Australia's score must be greater than 21.
"3 points are awarded for a win , 2 points are awarded for a tie/washed out match and 1 point is awarded for a loss" Therefore, Australia's score is:
3W + 2T + L > 21 ------- (2) or
(2W + T) + (W + T + L) > 21
2W + T + 14 > 21 (from (1))
2W + T > 7 ----- (3)

"at least one match played by the "Australian" team was tied/washed out"
So T >= 1 ---- (4)

What can we conclude from (1), (3) and (4)?

Answer 5

at least 2 or more matches washed out..?

Answer 6

L=10/3 so option B is iradicated

Answer 7

wait option b is correct ? as we can consider integer value

Answer 8

text book gives option B.)

Answer 9

What if W = 9, T = 1, L = 4 ? This possibility is not consistent with option B.
They add up to 14 which satisfies (1)
The score is 9*3 + 1*2 + 4*1 = 27+2+4 = 33 which is higher than the Zimbabwe's score which satisfies (2). And T = 1 which satisfies 3.

I think there is one more inequality we are missing.

Answer 10

yes this satisfies it

Answer 11

book might have a typo, though i have the latest edition

Answer 12

No, I think there is one more condition we are overlooking.

Answer 13

ok @aum wolfram gave this

Answer 14

http://www.wolframalpha.com/input/?i=W+%2B+T+%2B+L+%3D+14%2C2W+%2B+T+%3E+7%2CT+%3E%3D+1%2C+solve+integers

Answer 15

\(\large\tt \color{black}{L=0,T=12,W=2}\)

\(\large\tt \color{black}{L=1,T=10,W=3}\)

\(\large\tt \color{black}{L=2,T=8,W=4}\)

\(\large\tt \color{black}{L=3,T=6,W=5}\)

\(\large\tt \color{black}{L=3,T=5,W=6}\)

Answer 16

But it says "examples of integer solutions". Not ALL integer solutions.

Answer 17

oh i was overjoyed

Answer 18

If you plug in the numbers I gave earlier it satisfies all three conditions. So it is a solution but the solution is not consistent with the answer.

Answer 19

but all of them given by wolfram gives w+l+t=14

Answer 20

That is the very first condition. All solutions will satisfy that condition including the one I gave earlier with W = 9, T = 1, L = 4

Answer 21

There has to be one more condition on the MINIMUM score that Australia must have to win the tournament. If no two countries has the same final score we can say:
Zimbabwe: 21 (minimum score. Last in the tournament)
country 1: 22 (last but one)
country 2: 23 (last but two)
.....
country 13: 34
Australia: 35

See if that will work.

Answer 22

But it is based on the assumption that no two countries can have the same score which may not be a correct assumption.

Answer 23

for L>3

wolfram gave this
http://www.wolframalpha.com/input/?i=solve+%2CW+%2B+T+%2B+L+%3D+14%2C2W+%2B+T+%3E+7%2CT+%3E%3D+1+%2CL%3E3+++over+integers

Answer 24

Wolfram said it did not understand your above input.

Answer 25

http://www.wolframalpha.com/input/?i=W+%2B+T+%2B+L+%3D+14%2C+2W+%2B+T+%3E+7%2C+T+%3E%3D+1+%2C+L%3E3+++

Answer 26

this might work
http://www.wolframalpha.com/input/?i=solve+%2CW+%2B+T+%2B+L+%3D+14%2C2W+%2B+T+%3E+7%2CT+%3E%3D+1+%2CL%3E3+++over+integers

Answer 27

The link I provided above does the same.

Answer 28

ok,so what ur intuition says which option is correct

Answer 29

A couple of replies ago I said if we make the assumption that no two countries have the same score, then the minimum score for Australia is 35. We are also given T >= 1 for Australia and we know that W + T + L = 14. From this we can create a table as follows:
T W L Score = 2T + 3W + L
1 13 0 41
1 12 1 39
1 11 2 37
1 10 3 35
1 9 4 33 (falls below the minimum score of 35 for Australia based on our assumption).

So the maximum loses is 3.


But we had to make an assumption not given in the problem to solve this problem.

Answer 30

In essence what we have done is replaced (2) in the original set of equation/inequalities with 3W + 2T + L >= 35 which changes (3) to 2W + T >= 21

Answer 31

thanks very much for ur time and easy guidance

u rock , u r great ,Bravo

Answer 32

You are welcome.

Answer 33

Click on "more solutions" in the link and you will see max L is 3:
http://www.wolframalpha.com/input/?i=W+%2B+T+%2B+L+%3D+14%2C+++2W+%2B+T+%3E%3D+21%2C++T+%3E%3D+1%2C++++L%3E%3D+0%2C+W+%3E%3D+0%2C+solve+integers

Answer 34

You can also see from the solutions, the minimum number of Wins is 7, the maximum number of Ties is 7 and the maximum number of loses is 3 in order for Australia to win the tournament where no two countries has the same final score and Zimbabwe has the lowest score at 21 in a 15 country tournament.