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.

Labeling the highest score A, the lowest D, and the two middle scores B and C, we have the following:

A + D = 1.4 (B + C)

A + D - 100 < (B + C) / 2

(This second is derived from the fact that D must be the lowest of the four numbers, and D can be lowest when A is 100.)

Substituting the first equation into the second and doing a little algebra shows that the limit of B + C is 111 1/9

So we can say that the limiting values of B and C are each 55 5/9, A is 100 and D is solved for appropriately (it is slightly less than B and C, I guess it depends on how far out you want to carry the decimal...)

For the sake of simplicity, since it's Sunday morning and I'm still a little groggy, let's say B and C are each 55.55, then from the first equation we could say A is 100 and D is 55.54, and this would produce an arithmetic mean of 66.66.

Posted by tomarken on 2006-12-10 11:42:30