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 crossmultiplying 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 "crossmultiplying" is only a special purpose shortcut, 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) * (unkn1 / pop1) * (unkn2 / pop2) ... (unknpresent+1 / poppresent+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).

