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Nikki, Mikey, Akeem, Katie and Davey have all entered a mathematics competition. In each round of the competition, the entrants were put into pairs as far as possible (if there were an odd number of participants, all but one would be paired) and answered a set of questions at the same time as their opponent.
The winner was the entrant who gave correct answers to most of the questions. The winner got two points, one point was awarded for a draw and zero points for a loss. Every entrant played another entrant exactly once.
As we know, Nikki, Mikey, Akeem, Katie and Davey are all very good mathematicians,and they were delighted to find that they had all won the competition with the same score. In fact they had all dropped only 4 points. All of the other entrants only scored 4 points.
When the judges examined the results, there had been no draws.
How many points did Nikki, Mikey, Akeem, Katie and Davey all score?
When they met their friends Sophie, Stevie, Sabina, Marge and Chrissie, they discovered that they had been in a different group of entrants for the same competition. The friends admitted that they had dropped four times as many points even though they all scored the same number, but they noted that in their round the other entrants also scored exactly the same number of points as they had each dropped.
Davey thought that there were the same number of entrants in each round, so he said that he was sorry that his friends were not through to the next round, but, Sophie said that they too got through.
How many entrants were in their round?
welcome news: you wrote this
As we know, Nikki, Mikey, Akeem, Katie and Davey are all very good mathematicians,and they were delighted to find that they had all won the competition with the same score. In fact they had all dropped only 4 points. All of the other entrants only scored 4 points.
From the above the only possible solution is that each person played each other person in the group and won two matches and lost two matches. They also won all their matches against the other players.
each of the other players got 4 points: they must have therefore won two matches. This suggests that there are 5 other players each of whcih won two and lost two of their matches between themselves.
Since each of the friends beat the other 5 players (scoring 10 points) and beat two of their own friends (scoring 4 points) they must have all scored 14 points.
For the second group each of them lost 16 points: As they (presumably) lost 4 points playing each other then they must have lost 12 points playing the other competitors (a total of 60 points or 30 games) (this means there must have been at least 6 others)
suppose there are n other people. The total number of points scored between them is n^2-n so the total including the friends is n^2-n+60 and this must equal 16.n
So : n^2 -n + 60 = 16n
OR: n^2 -17n + 60 = 0
or (n-5)(n-12) = 0 giving n = 5 or n = 12.
However we know that n>5 so n= 12
So the total entrants was 17
Please could you explain fully how you did this part: how did you get 60??Please explain fully!
suppose there are n other people. The total number of points scored between them is n^2-n so the total including the friends is n^2-n+60 and this must equal 16.n
So : n^2 -n + 60 = 16n
OR: n^2 -17n + 60 = 0
or (n-5)(n-12) = 0 giving n = 5 or n = 12.
However we know that n>5 so n= 12
So the total entrants was 17
thanks
The winner was the entrant who gave correct answers to most of the questions. The winner got two points, one point was awarded for a draw and zero points for a loss. Every entrant played another entrant exactly once.
As we know, Nikki, Mikey, Akeem, Katie and Davey are all very good mathematicians,and they were delighted to find that they had all won the competition with the same score. In fact they had all dropped only 4 points. All of the other entrants only scored 4 points.
When the judges examined the results, there had been no draws.
How many points did Nikki, Mikey, Akeem, Katie and Davey all score?
When they met their friends Sophie, Stevie, Sabina, Marge and Chrissie, they discovered that they had been in a different group of entrants for the same competition. The friends admitted that they had dropped four times as many points even though they all scored the same number, but they noted that in their round the other entrants also scored exactly the same number of points as they had each dropped.
Davey thought that there were the same number of entrants in each round, so he said that he was sorry that his friends were not through to the next round, but, Sophie said that they too got through.
How many entrants were in their round?
welcome news: you wrote this
As we know, Nikki, Mikey, Akeem, Katie and Davey are all very good mathematicians,and they were delighted to find that they had all won the competition with the same score. In fact they had all dropped only 4 points. All of the other entrants only scored 4 points.
From the above the only possible solution is that each person played each other person in the group and won two matches and lost two matches. They also won all their matches against the other players.
each of the other players got 4 points: they must have therefore won two matches. This suggests that there are 5 other players each of whcih won two and lost two of their matches between themselves.
Since each of the friends beat the other 5 players (scoring 10 points) and beat two of their own friends (scoring 4 points) they must have all scored 14 points.
For the second group each of them lost 16 points: As they (presumably) lost 4 points playing each other then they must have lost 12 points playing the other competitors (a total of 60 points or 30 games) (this means there must have been at least 6 others)
suppose there are n other people. The total number of points scored between them is n^2-n so the total including the friends is n^2-n+60 and this must equal 16.n
So : n^2 -n + 60 = 16n
OR: n^2 -17n + 60 = 0
or (n-5)(n-12) = 0 giving n = 5 or n = 12.
However we know that n>5 so n= 12
So the total entrants was 17
Please could you explain fully how you did this part: how did you get 60??Please explain fully!
suppose there are n other people. The total number of points scored between them is n^2-n so the total including the friends is n^2-n+60 and this must equal 16.n
So : n^2 -n + 60 = 16n
OR: n^2 -17n + 60 = 0
or (n-5)(n-12) = 0 giving n = 5 or n = 12.
However we know that n>5 so n= 12
So the total entrants was 17
thanks