When asked what her math average was, a student coyly responded: "I've taken four tests to date and when I add the highest score with my lowest score and divide the result by the sum of my other two scores, I get a ratio of 1.4"
Assuming all scores are equally weighted and each grade is a real number between 0 and 100 inclusive, find four scores that produce the highest possible average (arithmetic mean), and show that a higher average is not possible.
(In reply to
Solution by tomarken)
I came up with more or less the same results as the one given in tomarken's previous post. However, we had better answer the question posed to us! An average of 66.66 is NOT the highest possible!! (I'm nitpicking here, just to let you all know ahead of time.)
I wasn't able to come up with an elegant proof, but it seemed to me that the way to maximise the average would be to let all 3 lower scores be equal, and the highest score be 100. In this case, and using the notation of tomarken's post, we come up with A=100 and B=C=D=500/9=55.55555...
This gives our coy math student an average grade of
200/3=66.66666...
Sorry to be so pedantic, but I like to see exact answers! Kudos to tomarken for such an elegant explanation of the solution method - although looking carefully, I think we need to include the possibility of equality in your stated inequality (ie ¡Ü instead of just <). And thanks to Dennis for a fun problem - it was quite a poser and kept me thinking for a while!
-John
re: Solution
