Rational expressions and equations 91     RatDivid     This program requires division and multiplication of rational expressions. It is a test of one's knowledge of factoring and of the use of parentheses to control the order of operations. 92     RatExprs     The problems in this program require simplifying rational expressions. Students should be alerted to the fact that one of the four problem types involves determining the domain of  the variable. 93     RatEqua1     These problems provide practice in solving rational equations by multiplying to cancel denominators; essentially cross-multiplying in most cases. 94     RatEqua2         Because there is more than a single ratio on at least one side of the equations, these problems make clear why the concept of "cross-multiplying" is only a special purpose short-cut, not a basic principle. 95   RatProb    This program invites the notions of factorials, permutations, and combinations to do the counting required for the calculation of probabilities. The solution to the third problem can be expressed as   unknownCpresent  divided by  populationCpresent .  If the teacher wants to emphasize the construction of the probability, it can also be expressed as: (unknown / population) * (unkn-1 /  pop-1) * (unkn-2 / pop-2) ... (unkn-present+1 /  pop-present+1) The size of the numbers rules out simplifying with factorials because of processor overflow problems, but  the tedium of inputting the answer can be avoided by approximating with: (average numerator / average denominator) ^ number present The error introduced is much less than one hundredth of 1% for the numerical ranges involved. It can make an interesting study for very advanced students. The fourth problem emphasizes the fact that the probability that  (A or B or C or ...  did occur)  is much more easily calculated as  (one minus the probability that A and B and C and ... did not occur). <<>>