In a certain game, each of 5 players received a score between 0 and 100, inclusive. If their average (arithmetic mean) score was 80, what is the greatest possible number of the 5 players who could have received a score of 50? Explain.
(A)None
(B)One
(C)Two
The average is defined this way:
average=1/5 (score1 + score2 + ...)
Here. We want to find the number n of folks that can get a 50 with a class average of 80.
80=1/5(n*50 + sum of other scores)
Now, to maximize n, we need to maximize the sum of the other scores, or
80=1/5(n*50 + 100*(5-n))
solve for n
To solve for n, we can simplify the equation as follows:
80 = (n*50 + 100(5-n))/5
Simplifying further:
400 = n*50 + 500 - 100n
Rearranging the terms:
400 = -50n + 500
Subtracting 500 from both sides:
-100 = -50n
Dividing both sides by -50:
2 = n
Therefore, n equals 2.
This means that the greatest possible number of players who could have received a score of 50 is 2, which corresponds to option (C) Two.
To solve for n, let's simplify the equation:
80 = 1/5(n*50 + 100*(5-n))
First, distribute the 100:
80 = 1/5(n*50 + 500 - 100n)
Simplify further:
80 = 1/5(500 - 50n + 100n)
80 = 1/5(500 + 50n)
To get rid of the fraction, multiply both sides of the equation by 5:
5 * 80 = 500 + 50n
400 = 500 + 50n
Subtract 500 from both sides:
400 - 500 = 50n
-100 = 50n
Now we can solve for n:
n = -100/50
n = -2
Since you can't have a negative number of players, it means there cannot be any players who received a score of 50 in order to achieve an average score of 80.
Therefore, the answer is (A) None.