Preliminary Mathematics The purpose of this document is to provide you with a refresher over some topics that will be essential for what we do in this class. We will begin with fractions, decimals, and percentages. Then, we will discuss solving equations and rewriting functions. Fractions Before discussing fractions, we must clarify two things. First, never, EVER, divide anything by 0. This is one of the most important rules in all of mathematics. Lastly, any whole number (such as, 2, -, -2, etc.) can be written as a fraction of the form x where x is the whole number. This section is perhaps the most important for what we will do in this class. Much of financial mathematics revolves around percentage rates which in turn rely on decimals which rely on fractions. Now, suppose you have a pizza it can be any kind you like and a friend you have to share it with. You know, maybe without knowing the proper notation or mathematics, that you will get half the pizza and your friend will get the other half (if it is done fairly). Now, what this also means is that you have divided one piece (the whole pizza) into 2 pieces. If we expressed this in fractions, we would have 2 = One Half. This is how we can think of fractions: the top part (the numerator) represents the number of pieces that are being divided up to supply however many people there are (the denominator). Perhaps you have heard of the term improper fractions where the numerator is greater than the denominator. In previous classes you may have been asked to convert such fractions into equivalent mixed fractions an expression involving an integer times a proper fraction. To convert an improper fraction into a mixed fraction, one must first determine how many times the denominator divides into the numerator without going over. The number from this step is the integer multiplier in the mixed fraction. The remainder of dividing the denominator into the numerator becomes the numerator in the proper fraction part of a mixed fraction while the denominator is the same as in the improper fraction. For example, if we wanted to write 6 as a mixed fraction we would have 5 since divides into 6 five times before we go over with a remainder of.
Example : Suppose you have 8 pizzas that need to be distributed amongst 5 people. (a) Express the amount that each person gets as a fraction. (b) Notice that your answer from the previous part is an improper fraction. Convert this improper fraction into a mixed fraction. For most instances, we will keep improper fractions as they are since they are much easier to work with. To truly see this, let us discuss the basic operations adding, subtracting, multiplying, and dividing. Contrary to how you learned these basic operations, we will start with multiplication and division of fractions.. Multiplying and Dividing Fractions.. Multiplication If we have two fractions, a b and c d (with b and d not 0), then their product a b c d = c d a b = ac bd. As you can see, regardless of the numerator and denominator, the product of two fractions is equal to the product of the numerators divided by the product of the denominators. Example : Find the product of and 5 6...2 Division Dividing two fractions is just as easy, it only requires one extra step. If we wish to divide the fraction a by the fraction c, all we must do is invert make the numerator the denominator, b d and make the denominator the numerator c and multiply it by a. That is, d b a b c d = a b d c = ad bc. It is important to remember that, unlike multiplication, unless both numbers are the same and not zero, a b b a. One important use of multiplying and dividing fractions is cross-multiplication. Say we have a fraction like 5 and want to know if it is equal to another fraction, such as. 7 5 The easiest way to do this is to set them equal to each other, then multiply the numerator of one with the denominator of the other. For our example here, we would get 7 5 = 05 2
on one side and 5 = 05 on the other. Thus, the two fractions are equal. In class we will see why this works. In the solving equations section we will see why this is useful. Now, we are prepared to move on to adding and subtracting two fractions..2 Adding and Subtracting Fractions.2. Like Denominators If the denominators of the two fractions are the same, all we need do is add or subtract the numerators, while keeping the denominator the the same. For example: 5 + 6 5 = 9 5. We will see in the next section that when the denominators are different, the procedure is more involved..2.2 Unlike Denominators If we have two fractions (proper or improper) with different denominators and want to add or subtract them, we run into a bit of difficulty. Continuing with the people and pizzas example, one fraction might represent how much one person can have if there are 8 pizzas split amongst friends, while another may represent the same allowance in the case of pizzas and 6 friends one fraction is thirds and one is sixths. In order to determine how much a person would have if he had 8 and we must find a common denominator 6 for the two fractions and then express each as an equivalent fraction with that common denominator as the denominator. Admittedly, this seems like a lot, so lets tackle each step separately. Finding a common denominator is exactly what it sounds like: we need to find a number that is a multiple of both denominators. This can be done in two ways. The first is the easiest, but will usually require simplification if you want the fraction to be in lowest terms. In the first method, all we do is multiply the two denominators together, thereby creating a number which is a multiple of both. The other method is more exhaustive, but useful if you need the least common denominator. In the second method one must list the multiples of each denominator and find the first entry that is common to both lists. For small numbers, method two may be easier to do mentally. So, to continue our 8 and example, the 6 least common denominator would be 6 (why?). Alternatively, you could find a common denominator of 8 by multiplying the two values. Let us now progress to finding equivalent fractions. Given a fraction, like, an equivalent fractionis a fraction created by multiplying 2 the numerator and denominator by the same value. Thus, 2 = 6 4 = n 2n
for any n. Now we can combine these procedures to add/subtract fractions with unlike denominators. Two of the most useful tools in algebra are adding 0 and multiplying by. You may scoff and wonder why these are useful, but after this section you will see why. Suppose we have found a common denominator for two fractions (suppose 6, for definiteness). Now, if we take any number except for 0 and divide it by itself we get ; that is, r r = for any r 0. This fact becomes handy when we do the following: () Change each original fraction into an equivalent fraction that has our common denominator on the bottom; (2) We now have the same denominator for each fraction (was it magic?) and can add/subtract them as previously discussed. Finally, let s finish our example with these pizzas. 8 = 6 6 We found that 8 and have a common denominator of 6. Similarly, we have that 6. Thus, we can easily see that 8 + 6 = 6 6 + 6 = 9 6. 2 Decimals and Percentages 2. Understanding Decimal Places All of us have seen numbers like 0.245..., but maybe not all of us have seen how decimal numbers relate to fractions and percentages. First, we need to understand that each place to the right of the decimal place has a specific representation just like the numbers to the left of the decimal place. Figure : A handy chart for decimal places As you can see in Figure, each place to the right has a name similar to each place to the left. By tenth, we mean, by hundredth we mean, etc. One key to decimal 0 00 numbers is that if we multiply a decimal number by a power of 0, the decimal place moves 4
to the right as many places as there are zeros. If we divide by a power of ten, the decimal moves to the left accordingly. For example, 0.856789 000 = 85.6789, and 0.856789 000 = 0.000856789. 2.2 Fractions to Decimals and Back Again If we want to express as a decimal, we would have =. since it is three-tenths. 0 0 If we have a fraction like, we no longer have something like 0, 00, 000, etc. as a 2 denominator. But, if you recall equivalent fractions, we can write one half as 2 = 5 0 =.5. Thus, we can easily write fractions as decimals by finding an equivalent fraction that has a power of 0 in the denominator. But what if we can t do this? Say, for instance, we have the fraction. We know that equivalent fractions of one-third would be = 9 = 99 = 999 = 9999 =... Notice that no matter how far out we go, we won t get a power of 0. Thus, we say that =.... =. where the bar over the indicates that it repeats forever to the right. 2.2. Terminating Decimals In the event that the decimal number terminates becomes all 0 s after a certain point the process of going from a decimal to a fraction is even easier. First, determine the number of decimal places before the number terminates, call this number k. Then, the denominator of the fraction is k zeros { }} { 0 k = 00... 00. Then, the decimal until it terminates itself (after the decimal point) becomes the numerator. For example the decimal number 8 places { }} { { }} { 2454 0. 2454 =. } 00000000 {{ } 8 zeros 5
Any numbers to the left of the decimal place take the role of the integer in a mixed fraction. Thus, 5.4 = 5 4 000. 2.2.2 Non-terminating Decimals What if the decimal number never terminates, like 0....? The key here is to see if there is a pattern repeating s, repeating 2, repeating 775, etc. that repeats continually. Notice here that the pattern must repeat forever, not just occur one or two or so times. If we have a decimal number that does have such a forever repeating pattern, say x =.878787..., we do the following: use the repeating part (87 in this case) as the numerator and the denominator is as many 9 s as there are digits in the pattern ( in this case). So, we have 0.878787... = 87 999. *This section uses some steps that will be discussed in the Solving Equations section* What if there are digits between the decimal point and the pattern, as in the number x = 0.4878787...? Well, notice that 000x = 4.878787.... If we now subtract x from 000x, we have 000x x = 999x = 4.878787... 0.4878787... = 4.7. If we divide throughout by 999, we have x = 999x 999 = 4.7 999 = 47 99900. As you can see now, we have a fractional form for our decimal number. 2. Percentages and Decimals We are now prepared to discuss percent (from the Latin per centum meaning of one hundred ) values. Just like fractions and decimals, percentages provide an answer to how much of something we have. Maybe we have half of it (50%) maybe we have five of it (500%) maybe we have one-tenth of it (0%). To transition from percent to decimals all we have to do is move the decimal point to the left 2 places and drop the percent symbol: 97.85% = 0.9785. To convert from a decimal number to a percentage, all we do is move the decimal point 2 places to the right and add the percent symbol:.4056 =.4056%. 6
If we have to go from a fraction to a percentage rate, we simply need to first convert to a decimal and then to a percentage. Solving Equations When handling non-terminating decimals we performed some algebraic operations that you may have forgotten. An equation is algebraic if it involves one or more unknown values which we call variables x, y, z,... for example. When it comes to actually solving for values, we will only have one variable, while our financial formulas will be rewritten for different variables using the operations we learn here. Now, let us review the basic rules/procedures for solving an equation such as (x + 2) = 8. Now, to this equation, we can do the following: () Add or subtract the same value to BOTH sides; (2) Multiply or divide BOTH sides by the same value; () Distribute, factor, combine like terms to ONE side. Notice that some operations can be done to one side without having to be done to the other, while others must be done to both sides. To better understand this, think of the equation as a balanced scale. We want to not only keep it in balance at all times, but we also want to eventually find out what our mystery weight is (the variable x). Rule () above can be thought of as rearranging items on one of the sides. Since nothing is being added or taken away, that scale is balanced. Rules () and (2) change the weight on one side, so to keep it balanced, we have to do the same thing to the other side. Now, before you get too worried, we won t be doing much algebra at all in this class, but being able to rewrite formulas is a must in financial mathematics and that is where the algebra will be used. Well, to end the section, let s solve this equation for x. First, using R(), we can add to both sides so that we have = 0 { }} { (x + 2) + = 9 }{{} 8 + = 9 Well, using R(2) here we can divide both sides by this will cancel the out of our product on the left and obtain 9 = (x + 2) {}}{ =. We now have a pretty easy problem: we need a number that when added to 2 gives us. Well, after we use R() and subtract 2 from both sides we see that x =. 7.
There is one important thing to mention here: canceling terms in a fraction is allowed if they are multiplied. Recall what we learned with equivalen fractions if we multiply the top and bottom by the same thing, we have the same fraction. To better see this, let us do two examples. First, given the fraction (x + 2) 6 = (x + 2) 2 we can cancel the in the top and bottom since it is common in both. But, if we have the fraction x + 2 6 we cannot cancel a out since is not common to both 2 and x (it is certainly a part of x). So, if ever you are about to cancel terms in a fraction, make sure they can in fact be canceled. One easy way to check this is to see if the value can be factored out, as in x + 6 = 4 Rewriting Formulas can be multiplied by each { }} { (x + 2). In this section, we will simply do one example to see how what we just learned can be used. Conside the following formula (will be introduced in Chapter ): M = P + P RT. It is not important what the letters stand for. If I were to ask you to find a formula for P, though, all you would need to do is solve the equation for P. Well, on the right hand side, I can factor out P since it is common to both terms, and get M = P ( + RT ). Recall that a number divided by itself is equal to. Now, if I use R(2), I can divide both sides by ( + RT ) to get M ( + RT ) = P ( + RT ) ( + RT ) = P. We now have a formula for P, as desired. We will use this procedure a lot in the latter part of the class, but knowing these skills we be useful throughout the quarter. 8
5 Recommended Exercises For practice, work on the Cumulative Review on pages 22-24 from Salzman, 8th edition. This won t be collected as homework, but it will help you to see how well you understand the material, and will also assist you in knowing what you may need help on. If your struggles occur with the algebra, do not be too worried as we will only utilize that material occasionally. Being comfortable with percents, decimals, and fractions however is an absolute must in this class. 9