Given a number of different fractions, create a new fraction whose numerator is the sum of all those fractions' numerators and whose denominator is the sum of the denominators. Call it y. Call the smallest of the original fractions x and the largest z.
Prove that for all cases, x < y < z.
As every student who properly understands the grading system most teachers use, adding the numerators and denominators of fractions is a way of averaging them. The higher the denominator, the more the fraction is weighted.
For example, if you get 49/50 on a quiz and 94/100 on a test, then the weighted average is 143/150. This is the way most every teacher (that I have had) averages grades.
Obviously, the average of a set of different fractions is between the lowest and highest fractions in the set. However, this isn't necessarily true when the denominator is negative, because then it is weighted negatively in the average.

Posted by Tristan
on 20041016 00:57:38 