SOLUTION: From 11 positive integer scores on a 10-point quiz, the mean is 8, the median is 8, and the mode is 7. Find the maximum number of perfect scores possible on this test.

Question 220634: From 11 positive integer scores on a 10-point quiz, the mean is 8, the median is 8, and the mode is 7. Find the maximum number of perfect scores possible on this test.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
perfect score is 10
11 scores with:
mean of 8
mode of 7
median of 8
-----
total of 11 scores with a median of 8 means that a maximum of 5 scores are below 8 and 5 scores are above 8. this will occur when there is only 1 number 8 in the distribution. we will assume only one number 8 in the distribution because that would allow for more number 10's if possible. we need at least one though, because we have 11 numbers and that means that one of the numbers has to be the median in order to get 5 on each side plus the 1 in the middle equal to 11 total.
-----
mode of 7 means that 7 is the score that has occurred with the most frequency.
-----
integer scores means that there are no fractions to worry about.
-----
we need:
1 number 8
a minimum of 2 number 7's
5 numbers above 8
5 numbers below 8
-----
let's start off with 5 number 10's.
that would be the maximum number of 10's without any other constraints except the fact that the median is 8 meaning the number of units above 8 had to be 5 at most.
-----
5 number 10's means that we would have to have 6 number 7's because 7 is the mode which is the number that occurs most frequently. we can't have 6 number 7's because the total of numbers below 8 has to be 5 and 6 is greater than 5.
-----
so 5 number 10's is no good.
-----
can we have 4 number 10's?
-----
this means we must have at least 5 number 7's and number 7 would then be the most frequent number.
if we have 4 number 10's, we would need 1 more number above 8 to make 5 numbers above 8. we'll pick the lowest number which is 9.
so far we have:
4 number 10's equals a total of 40
1 number 9 equals a total of 9
1 number 8 equals a total of 8
5 number 7's equals a total of 35
-----
we have a total of 11 numbers with a grand total of 92
-----
we have exceeded the grand total of 88 so 4 number 10's is not possible given all the constraints.
-----
can we have 3 number 10's?
-----
3 number 10's means we have to have a minimum of 4 number 7's to keep number 7 the most frequently used number in the set.
we also need 2 more numbers above 8 which means we have to choose 9 because 9 is the smallest number above 8.
so far we have:
3 number 10's equals a total of 30
2 number 9's equals a total of 18
1 number 8 equals a total of 8
4 number 7's equals a total of 28
-----
this give us a total of 10 numbers with a grand total of 84
-----
we need 1 more number that equals 4.
adding 1 number 4 give us a total of 11 numbers with a grand total of 88.
-----
the maximum number of 10's we can have is 3.
since 10 is a perfect score, then:
the maximum number of perfect scores we can have is 3.
-----

RELATED QUESTIONS
A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8.... (answered by greenestamps)

What is the mean for the following scores on a 10-point quiz: 10 10 9 9 8 8 (answered by jim_thompson5910)

I need help, I'm stuck...what is the correct answer? The table below shows the... (answered by Boreal)

Make up a data set of 10 entries for which the mean is 7, the median is 8, and the mode... (answered by partha_ban)

The table below shows the scores of a group of students on a 10-point quiz. Test Score (answered by Fombitz)

The table below shows the scores of a group of students on a 10-point quiz. Test... (answered by MathLover1,math_tutor2020)

For a 10 point quiz, the professor recorded the following scores for 8 students: 7, 8,... (answered by tylersyourtutor,89fiveohgt)

Find the median; 1,4,9,15,25,36 Find the mode; 41,43,56,67,69,72 Find the mode;... (answered by jim_thompson5910)

You have a series of twelve numbers whose mean is 7 median is 6 and mode is 8 What are... (answered by jim_thompson5910)