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0 Review of Algebra

0.1 Sets of Real Numbers 0.2 Some Properties of Real Numbers 0.3 Exponents and Radicals 0.4 Operations with Algebraic Expressions 0.5 Factoring 0.6 Fractions 0.7 Equations, in Particular Linear Equations 0.8 Quadratic Equations

Lesley Griffith works for a yacht supply company in Antibes, France.??????????????????????

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0 Review of Algebra

0.1 Sets of Real Numbers
0.2 Some Properties of Real Numbers
0.3 Exponents and Radicals
0.4 Operations with Algebraic Expressions
0.5 Factoring
0.6 Fractions
0.7 Equations, in Particular Linear Equations
0.8 Quadratic Equations

Lesley Griffith works for a yacht supply company in Antibes, France. Often, she needs to examine receipts in which only the total paid is reported and then determine the amount of the total which was French value-added tax, known as TVA for “Taxe à la Value Ajoutée”. The French TVA rate is 19.6%. A lot of Lesley’s business comes from Italian suppliers and purchasers, so she also must deal with the similar problem of receipts containing Italian sales tax at 18%.

A problem of this kind demands a formula, but many people are able to work through a particular instance of the problem, using specified numbers, without knowing the formula. Thus if Lesley has a 200-Euro French receipt, she might reason as follows: If the item cost 100 Euros before tax, then the receipt total would be for 119.6 Euros with tax of 19.6—and then with a leap of faith—tax in a receipt total of 200 is to 200 as 19.6 is to 119.6. Stated mathematically,

\frac{\text{tax in 200}}{200} = \frac{19.6}{119.6} \approx 0.164 = 16.4%

At this point it is fairly clear that the amount of TVA in a 200-Euro receipt is about 16.4% of 200 Euros, which is 32.8 Euros. In fact, many people will now guess that

\text{tax in } R = R\left(\frac{p}{100+p}\right)

gives the tax in a receipt R, when the tax rate is p%. Thus, if Lesley feels comfortable with her deduction, she can multiply her Italian receipts by

\frac{18}{118}

Of course, most people do not remember formulas for very long and are uncomfortable basing a monetary calculation on a leap of faith. The purpose of this chapter is to review the algebra necessary for you to construct your own formulas, with confidence, as needed. In particular, we will derive Lesley’s formula, without a mysterious invocation of proportion, from principles with which everybody is familiar. This usage of algebra will appear throughout the book, in the course of making general calculations with variable quantities.

In this chapter we will review real numbers and algebraic expressions and the basic operations on them. The chapter is designed to provide a brief review of some terms and methods of symbolic calculation. No doubt you have been exposed to much of this material before. However, because these topics are important in handling the mathematics that comes later, an immediate second exposure to them may be beneficial. Devote whatever time is necessary to the sections in which you