L.A. Math: Romance, Crime, and Mathematics in the City of Angels

by James D. Stein

  0.07 P = $11.20

  P = $11.20/0.07 = $160

  You can check that $160 is the right price simply by taking 7% of $160 and observing that it equals $11.20.

  Example 2: Rutabaga Preferred stock rose 15% last year. If the stock went up 13½ points, at what price did it start the year? (Note: A “point” is actually investor-speak for a dollar. A stock selling for 50 points is selling for $50 for one share of stock.)

  Solution: If S is the starting price, 15% of S is 0.15 S, so

  0.15 S = 13.5

  S = 13.5/0.15 = 90

  The stock started at 90. Checking, 15% of 90 is 13.5. ■

  From here we move on to markups and markdowns. Markups generally don’t cause problems mathematically, but since many of them come from various governmental institutions imposing taxes, they’re a major annoyance. In real life, that is—but the major mathematical annoyance comes from confusion with discounts. Check out this common mistake, which occurs because many people erroneously compute the original price, from which a discount is taken, by adding that percentage to the discounted price

  Example 3: After a 20% discount, a TV sells for $120. What was the original price?

  Solution: The mistake is to take 20% of the discounted price of $120, which is $24, and add that to the discounted price of $120, arriving at an erroneous original price of $120 + $24 = $144.

  You, of course, now know better. Let S denote the original price of the TV. Then 20% of S is 0.20 S, and so

  S − 0.20 S = $120

  0.8 S = $120

  S = $120/0.8 = $150

  Checking, 20% of $150 is 0.20 × $150 = $30, and when $30 is subtracted from $150, the result is $120. ■

  Now let’s take a look at the trap into which the city of Linda Vista fell. Why can’t we reduce an individual’s taxes by 20% if the tax base increases by 20%? If there are originally T taxpayers, and each taxpayer is assessed D dollars, then the total revenue is clearly TD dollars. A 20% increase in the number of taxpayers will add 0.20 T taxpayers to the original T taxpayers, so there will now be T + 0.20 T = 1.2 T taxpayers. A 20% reduction in the taxes assessed each taxpayer will be 0.20 D dollars, so each taxpayer will now pay D − 0.20 D = 0.80 D dollars. Therefore, the total revenue will be the number of taxpayers multiplied by the tax per taxpayer. This comes to 1.2 T × .8 D = 0.96 TD dollars. This is only 96% of the original revenue.

  (Percentage calculation continued from p. 18)

  Incidentally, did you catch on to the fact that Pete was able to compute the number of taxpayers simply from the information that taxes had been reduced from $100 to $80 and that the city of Linda Vista was $396,000 short? Once he found out that everybody assessed had paid up, Pete hypothesized that the city council had fallen into the classic innumeracy trap of thinking that a 20% gain in the number of taxpayers compensated for a 20% loss in revenue per taxpayer. In that case, as we discovered above, the shortfall would have been 4%, since the revenue was only 96% of the original revenue. If the total revenue is denoted by R, then 4% of R would be $396,000. Therefore,

  0.04 R = $396,000

  R = $396,000/0.04 = $9,900,000

  At $100 per taxpayer, the number of taxpayers in the previous census would have been $9,900,000/100 = 99,000 taxpayers.

  Innumeracy in this respect can have potentially catastrophic consequences. A doctor may tell a nurse to reduce the dosage of a drug by 50%. When the patient relapses, the doctor tells the nurse to raise the dosage by 50%. Disaster! The doctor may think the patient is receiving the same amount of medication as he or she did originally, but the patient is only receiving three-quarters of the original amount. One shudders at the thought of a similar error being made with an airplane whose fuel has been depleted by 50%.

  Many of the misunderstandings concerning percentages occur because of a failure to realize that the computation of a percentage requires a base number upon which one computes the percentage. Suppose that a stock is selling for 100, and the price rises 30% and then declines 30%. The base number for figuring the price rise percentage is 100; 30% of 100 is 30, so the price after the rise is 130. This number, 130, is the base number for figuring the price decline of 30%. And 30% of 130 is 39, so the stock price falls to 130 − 39 = 91. Notice that, even if the stock had fallen 30% first and then risen 30%, the final price would again be 91. That’s because we are performing two successive multiplications, and it doesn’t matter in which order we perform them.

  Failure to understand percentages has repercussions in other areas of mathematics. Percentages are often used to convey probabilistic notions. For instance, a meteorologist might say that there is a 50% chance of rain on Saturday and a 50% chance of rain on Sunday. A recent poll showed that many people were under the impression that the above forecast was equivalent to saying that it was certain that it would rain during the weekend! Well, it’s not, and we shall have more to say about this in chapter 9.

  APPENDIX 3

  AVERAGES AND RATES IN “A MATTER OF TIME”

  If mathematicians were to vote on the most useful notion in mathematics, the concept of averages would be up close to the top. In fact, many would probably award it the title. Averages occur throughout all of mathematics. They represent one of the best ways of summarizing past information, and in the absence of more pertinent data, the best way to predict the future. Averages play critical roles in such widely diverse topics as percentages, probability and statistics, algebra, and calculus.

  An average is a quotient, and one model for division is sharing a quantity of items in a fair fashion. If four people are to share twelve slices of pizza fairly, how many slices should each person receive?

  Simple as this example may seem, it can be used to provide an easy introduction to the concept of an average. If twelve slices of pizza are shared among four people, the average number of slices each person receives is three. Of course, this does not mean that each person actually receives three slices. An average in this instance represents a way of summarizing data by looking at what would have happened if fair sharing had actually taken place.

  When we say that the average is three slices, there is a very important, but often unspoken phrase: “per person.” An average is a quotient, and a quotient consists of a numerator and a denominator. When we are dealing with real-world quantities, the numerators and denominators are measured in units. The numerator units in the above example are slices of pizza, the denominator units are persons. To fully understand an average, one must know what is being shared (the numerator units—pizza slices) and among what the shared quantity is being shared (the denominator units—people). The units of measurement for averages are “numerator units per denominator unit”—in this instance, slices per person. All the concepts in this chapter involve quotients.

  THE IMPORTANCE OF UNITS

  When computing an average, a number by itself is meaningless—both the numerator and denominator units must be specified. To see how important this is, ask yourself if you would take a job if the salary was 5.

  Assuming that the job isn’t distasteful or dangerous, you almost certainly would take the job if the salary was $5 per second. You almost certainly wouldn’t take the job if the salary was 5 cents per year.

  Averages: Summarizing the Past, Predicting the Future

  An average is a quotient. Baseball, the national pastime (although rapidly being eclipsed by football), provides an excellent source for the computation of averages. A player’s batting average is the quotient of the number of hits the player has achieved divided by the total number of official at-bats (an official at-bat occurs any time a player is not automatically awarded first base via a base on balls or being hit by a pitch). If a player has 500 official at-bats and gets hits in 150 of these, his batting average is 150/500 = 0.3 = 0.300 (pronounced “three hundred”). (A player whose batting average rounded to the nearest thousandth is .273 is said to be hitting “two seventy-three.”) A player’s batting average is the average number of hits per official at-b
at.

  Computing averages is not difficult, but it is surprisingly easy to be misled by the form in which the information is presented.

  Example 1: Evan buys $6.00 worth of hamburger at $1.50 a pound and then goes to another store where he buys another $6.00 worth of hamburger at $1.00 a pound. What is the average price of the hamburger?

  Solution: If you computed the answer by saying that, since the same amount was purchased, the average price of the hamburger must be $1.25, the average of the prices $1.50 and $1.00, you have fallen into exactly the same trap as Freddy did during the story!

  Many problems involving averages simply require you to keep focused on the numerator and denominator of the quotient that you use to compute the average. In this case, the numerator is $12.00, the amount of money spent, and the denominator is 10 pounds, the amount of hamburger purchased. The average price is therefore $12.00/10 pounds = $1.20 per pound. ■

  Many of the mistakes made in computing averages are variations of the error that Freddy made during the story. Suppose that one is given two different data sets and computes an average for each data set, and then computes the average of all the data together. In general, it is not true that the average for all the data is the average of the averages for each data set. In example 1, the trap is to compute the average of two average prices. The way to avoid this trap is to compute the numerator and denominator for the entire data set—computing the average of two averages is very likely to lead to a wrong answer.

  (Calculating averages continued from p. 27)

  In the story, Freddy made the common mistake of assuming that the average rate is the average of the rates. It certainly sounds convincing, but it is only true when the denominators of both rates are equal. The actual average rate of Freddy’s round trip to San Diego can be computed by looking at the total distance traveled (240 miles) and dividing by the total time taken (3 hours to go the 120 miles to San Diego at 40 miles per hour; 6 hours to return at 20 miles per hour). Although the average of 40 and 20 is 30, the average speed on the trip is 240/9 = 26⅔ miles per hour. As Pete observed, the actual average speed is different from Freddy’s estimate of 30 miles per hour because time, the denominators of the two rates, differed for the two legs of the round trip.

  That’s not the only way that averages can cause confusion, as the following example shows.

  Example 2: The four executives at Mirage Financial are paid annual salaries of $100,000, and the six assistants get annual salaries of $40,000. The executives got raises of $10,000, and the staff got raises of $2,000. The company told its stockholders that the average raise was 7%, and it told the employees that the average raise was 8.125%. What’s going on here?

  Solution: Welcome to the wonderful world of creative accounting! There were four raises of 10% and six raises of 5%; the average of these ten percentages is 7%. On the other hand, the initial payroll was $640,000, and after the raises the payroll was $692,000, an increase of 8.125%. ■

  There is some truth in both of these numbers—they are both correctly figured, but they are differently defined. This example indicates why it is often difficult to find out what is really happening with the finances of a company that has a complicated balance sheet.

  Averages are often used as a basis for estimates of future performance. This reliance on averages to estimate future performance exists even where the reasons that resulted in that average are not well understood. If a hundred years of weather data show that the average rainfall in an area is thirty inches per year, planning for water use and distribution is made on the assumption that the future rainfall will continue to average thirty inches per year.

  Using past averages as future estimates is common practice. A company estimates future sales by looking at past averages. It is important to realize that averages can be used to estimate the future, and plans based on those estimates may often be extremely useful, but averages based on past data cannot predict the future. Even though past rainfall may have averaged thirty inches per year, the coming year may see either a drought or a flood.

  Rates

  Rates are quite similar to averages, as both averages and rates are expressed as quotients.

  From a mathematical standpoint, there is no difference between an average and a rate. However, there is a sort of unspoken agreement that the word “average” is generally used to summarize data, whereas the word “rate” is generally used to describe an ongoing process or to facilitate exchanges.

  If twelve slices of pizza were eaten by four people, each person ate an average of three slices; this summarizes what happened in the past. We could also say that the rate of pizza consumption is three slices per person; here the ongoing process of eating pizza is being emphasized. The numerical measures (three) of both average and rate are the same, the units of measurement (slices per person) are the same, so from the mathematical standpoint, they are indistinguishable.

  Exchanging money for goods or services is basic to business. If hamburger costs $2 per pound, a denominator unit (pound of hamburger) can be exchanged for $2. This exchange is how business commonly sees rates, but exchanges also underlie rates in other situations. For example, if a car goes at a rate of forty miles per hour, a denominator unit (one hour) can be exchanged for forty miles. This may be a somewhat strange way to view driving speed, but it shows that the rate concept has applications in different areas.

  APPENDIX 4

  SEQUENCES AND ARITHMETIC PROGRESSIONS IN “THE WORST FORTY DAYS SINCE THE FLOOD”

  The ability to recognize and use patterns is one of the most important aspects of intelligence. It is only in the last ten thousand or so years that we have created societies and the advances that go with them. Props to us! I’m pretty sure these advances began when we first started to use the pattern of recurring seasons to develop agriculture.

  Predictions based on patterns form the foundation of science and technology, as well as everyday life. Medical researchers study the pattern of the beating heart in the hopes of isolating cues that will alert them to incipient heart attacks, but you and I also try to pay attention to the patterns of behavior of our friends, our co-workers, and our loved ones. Things go better when we figure out how people will react.

  One of the great things about numbers is that they provide an environment that makes it possible to recognize patterns, and these number patterns often reflect things that happen in the real world.

  SEQUENCES AND ARITHMETIC PROGRESSIONS

  A sequence is an unending string of numbers. Some well-known examples of sequences are

  The counting numbers: 1, 2, 3, 4, … (the dots indicate that the numbers continue);

  The odd numbers: 1, 3, 5, 7, …; and

  The prime numbers: 2, 3, 5, 7, … (a prime number has only two whole number divisors, itself and 1. So 13 is prime because its only whole number divisors are 13 and 1, and 12 is not prime because it has 2, 3, 4, and 6 as whole number divisors).

  The numbers that make up a sequence are called its terms. In the sequence 1, 3, 5, 7, … of odd numbers, 1 is called the first term of the sequence, 3, the second term, 5, the third term, etc.

  Letters can be used in algebra when we wish to talk about numbers and their properties without specifying a particular number. For example, we use the letter x in the equation 2x + 6 = 2(x + 3) to denote any number. When we wish to talk about sequences or properties of sequences without specifying a particular sequence, we use the notation a1, a2, a3, … to denote a sequence. The numbers 1, 2, 3, … in the above notation are called subscripts. For instance, a7, which is read “a sub seven,” or more casually, “a seven,” is the seventh term of the sequence. When we wish to talk about a term of the sequence without specifying a particular term, we use the notation an.

  There are several different ways to describe sequences. The sequence 1, 3, 5, 7, … of odd numbers can be described

  (1) by using words: the nth term of the sequence is the nth odd number.

  (2) by using a formula: an = 2n
− 1. Notice that using a formula enables us to compute the exact value of any term in the sequence. For instance, a100 = 2 × 100 − 1 = 199.

  (3) by using a recursive definition, which describes a sequence in roughly the same way that one would give instructions on how to use a ladder: put your foot on the first step, and whenever you are standing on a rung, put one foot on the next higher step and bring your other foot up to join it. In this instance, the recursive definition would be a1 = 1, an = an−1 + 2.

  We start with the definition that a1 = 1. When we let n = 2, the recursive formula becomes

  a2 = a2 − 1 + 2 = a1 + 2 = 1 + 2 = 3, so a2 = 3

  Now we can let n = 3, and the recursive formula becomes

  a3 = a3 − 1 + 2 = a2 + 2 = 3 + 2 = 5

  We can now use a3 to help us compute a4, etc.

  Of the three ways of describing a sequence, describing by means of a formula is the most useful because it enables us to compute directly any term in the sequence. The formula an = 2n − 1 can be used to compute the one hundredth term in the sequence as we did above. If we wanted to find the one hundredth term using words as the description, we would have to write out the first one hundred odd numbers, and if we wanted to find the one hundredth term by using the recursive definition, we would have to use it ninety-nine times (the first use gave us a2, the second use gave us a3, etc.).

  Unfortunately, sometimes a formula is not available. In the case of the sequence of prime numbers, there is no known formula or recursive definition that enables us to compute the nth prime number. As a matter of fact, mathematicians have actually been able to prove that it is impossible to find either a formula or a recursive definition to compute prime numbers! And a good thing, too—because this difficulty in computing prime numbers underlies the security that protects your passwords. And, of course, your bank account.