Percentages 376 REAL-LIFE MATH their assistance to a diner during the course of a meal. The money spent on the tip, which is in addition to the cost of the food and the taxes that may apply to the pur- chase of a meal, is therefore an important factor in the measurement of the total cost of food spending outside the home. From the perspective of a server working in a restau- rant, the correct calculation of the tip is important because it has a direct impact upon their personal income, as typically the tips earned by a server for their work will constitute an important part of their earnings. The calculation of a tip involves a percentage-based application, usually related to the total amount of the bill, not including sales tax. It is generally accepted that a 15% tip recognizes good service, while a 20% tip tells the server that the service was outstanding. Tips of less than 10% are treated as an expression of the diner’s dissatis- faction with the server and the establishment about the meal. Assume a 15% tip in the following examples: 15% ϭ 15/100 ϭ 0.15. Where a restaurant bill totals $28.56, without tax being added to the total, to calculate the tip: .15 ϫ 28.55 ϭ 4.28. Thus, the 15% tip on this bill is $4.28. It is unusual to leave a precise amount such as this for the tip, especially if the bill is being paid by cash. Custom may dictate that if the patron is paying by cash, a rounded figure that approximates the 15% will be left for the server, perhaps $4.25 or $4.50 in this example. Note that when using the 1% method, this tip could be also calculated as follows: 10% of 28.55 ϭ 2.85; 5% is 1 ⁄2 of 10% ϭ 1.43. Thus, the total is 4.28. COMPOUND INTEREST Bank interest is expressed as a percentage. If funds are left in a bank account as a savings, they will attract what is referred to as compound interest, which is inter- est calculated both on the principal amount as well as the accumulated interest over time. For example, in Year 1, $10,000 is deposited to a bank account that will pay the depositor 4% per year. The interest earned in Year 1 will equal $400. The interest to be calculated in Year 2 will be calculated on the original $10,000 as well as the Year 1 interest of $400, for a total of $10,400. 4% ϭ 0.04, so Year 2 interest ϭ 10,400 ϫ 0.04 ϭ $416. Total monies in the account at the end of Year 2 will be $10,816. The 4% rate will apply indefinitely until the money is withdrawn in this example. RETAIL SALES: PRICE DISCOUNTS AND MARKUPS AND SALES TAX Many aspects of retail sales advertising are expressed in percentage terms. Sale prices, discounts, markups on merchandise, and all sales tax calculations depend on per- centages. The various methods set out below assist in determining the various ways that retail sales are depen- dant upon percentages. Discounts and markups: a discount is any sale where the seller claims that the goods are being sold at less than the regular or listed price. In some cases, the original price of the item is known, as is the percentage discount. The sale price is not known and it must be calculated, as follows: A refrigerator was said to have a list or regular price of $625. In the appliance showroom, there is a tag placed on the refrigerator advertising the item as on sale at 40% off its regular price. To find the sale price, 40% ϭ 0.40; 40% of 625 ϭ 0.40 ϫ 625 ϭ 250; 625 Ϫ 250 ϭ 375. In this example, the discount of 40% is $250, and the sale price is therefore $375. As an alternative method for calculating the sale price, 100% Ϫ 40% ϭ 60%; 60% ϭ 0.60; 0.60 ϫ 625 ϭ 375. The next type of discount application commonly required in retail sales is the computing of the percentage discount advertised in any given situation. A used motor vehicle is advertised by its owner as being for sale at a price of $8,500. The advertisement states that the vehicle is worth $12,000 and that it is being sacrificed at the $8,500 price because the owner is relocating to another country to take a new job. The percentage by which the vehicle price is being discounted is calculated as follows: Percentage discount ϭ original price Ϫ sale price / original price ϫ 100%; percentage discount ϭ 12,000 Ϫ 8,500 / 12,000 ϫ 100% ϭ 3,500 / 12,000 ϫ 100% ϭ 29.17%. The opposite concept in retail sales is the notion of the markup. While discounts are typically a part of the retail process that is advertised to the public, the markup is pri- marily an internal mechanism within a particular retailer. Items that are sold in retail stores are often manufac- tured or assembled elsewhere, and they are purchased by the retailer on what is known as a wholesale basis. The ultimate sale price offered by the retail store to a pur- chaser will be the price paid by the retailer to obtain the item, plus an amount reflecting the relationship between what the retailer paid for an item themselves and what it will be sold for to the public. This amount is the markup. It is also referred to in some businesses as a margin, as in a business operating on a small margin, or the markup may also be described as the gross profit (the profit before costs and overhead is deducted). The relationship Percentages REAL-LIFE MATH 377 between cost, markup, and the retail or selling price for any item may be expressed in this simple equation: cost ϩ markup ϭ selling price. Markups will be quoted as either a percentage of the cost price or of the selling price of an item, depending upon what is customary in that particular business. To compute selling price, the following example sets out the process: A hardware store buys drills a cost of $145 per drill. The store marks up the cost 65% based on its cost. The selling price is determined by 65% ϭ 0.65; markup ϭ 0.65 ϫ 145 ϭ $94.25; selling price ϭ 94.25 ϩ 145 ϭ $239.25. Alternatively, the known markup can be added to 100%, creating a total percentage figure, to perform the calculation: 100% ϩ 65% ϭ 165% ϭ 1.65; selling price ϭ 1.65 ϫ 145 ϭ $239.25. SALES TAX CALCULATIONS In most jurisdictions in the world, anyone purchas- ing consumer goods, ranging from bubble gum to motor vehicles, will be faced with the imposition of a sales tax. Such taxes, depending upon the location, may be imposed by city, state or province, or national govern- ments. Tax rates vary from place to place; it is common to find 5% sales taxes. In some countries what are referred to as goods and services taxes, when combined with existing local taxes, can have a combined impact of 15% or more on a consumer purchase. When assessing the price of an item offered for sale by a retailer, the total cost of the item must be assessed with the applicable taxes taken into account. For example, a new vehicle dealer is selling a pickup truck for $21,595, plus taxes. If the applicable tax rate is 4.5%, the total cost of the item is 4.5% ϭ .045; tax ϭ .045 ϫ 21,595 ϭ $971.76; total cost ϭ 971.76 ϩ 21,595 ϭ $22,566.78. Another factor in relation to the calculation of costs is the fact that the retailer may also have paid taxes on their purchase, which are being passed along. For this rea- son, the actual savings on a discounted item that is pur- chased has two components: the available discount on the price of the goods in question, and a reduction in the sales tax otherwise applicable to the price. For example, a television is listed at a regular price of $649. It is then the subject of a “one third discount.” The total savings available to the consumer are as follows: Price discount is 1 ⁄3 discount ϭ 33.3%; discount ϭ 0.333; discount ϭ 0.333 ϫ 649 ϭ 214.17; discount price ϭ 649 Ϫ 214.17 ϭ $432.88. If the applicable sales tax was 5% the sales tax payable on the discounted price would be tax rate 5% ϭ 0.05; tax on discount price ϭ 0.05 ϫ 432.88 ϭ 21.64; total cost of discounted item ϭ tax ϩ discount price ϭ 432.88 ϩ 21.64 ϭ $454.52. Had the television been purchased at the regular price, the sales tax would have been taxed at regular price ϭ 0.05 ϫ 649 ϭ 32.45. The total cost of the television at its regular price is 649 ϩ 32.45 ϭ $681.45; total savings on the dis- counted television purchase is regular price total cost Ϫ discounted price total cost: 681.45 Ϫ 454.52 ϭ $226.93. REBATES A variation on the notion of discounts is that of the rebate. A rebate occurs where a retail business sets a par- ticular advertised or published sale price, and then will offer to refund or discount to the customer a fixed amount or percentage of the sale price. Rebates are fre- quently advertised in retail sales, and they are most com- mon in the automotive sector, and they are also employed in the sale of various kinds of electronic devices and com- puter hardware. For most circumstances, a rebate will have the same effect on a transaction as does a discount: a price that is the subject of a 10% rebate will have the same effect on a transaction as a 10% discount. However, there is one dis- tinction between the impact of a discount and that of a rebate when the rebate is not offered at the retailer, but by way of the format known as a mail-in rebate. For example, at a computer store that offers various types and brands of computers for sale, a particular com- puter manufacturer is offering a new computer monitor for sale at a price of $399, less a $50 mail-in rebate. The computer is purchased in accordance with the following transaction: sale price ϭ 399; sales tax rate ϭ 5% ϭ 0.05; sales tax ϭ 0.05 ϫ 399 ϭ $19.95; total cost ϭ 399 ϩ 19.95 ϭ $418.95. The purchaser is provided with a mail-in rebate card, which sets out the terms of the rebate, namely that upon receipt of the card, the manufacturer will send the sum of $50 payable to the purchaser within 60 days. Therefore, after 60 days, plus the time it takes to deliver the rebate to the manufacturer, the net cost to the purchaser shall be $368.95. Two percentage-based calculations come into play in this mail-in rebate example. First, the difference is sales tax payable between the mail-in rebate and an identical discount; second, the 60 days or greater that the cus- tomer’s $50 is out of the customer’s control. SALES TAX CALCULATION: IN-STORE DISCOUNT VERSUS MAIL-IN REBATE If a $50 discount had been applied to the computer monitor purchase at the time of the transaction, the sale Percentages 378 REAL-LIFE MATH price would have been reduced to $349, resulting in a total cost to the purchaser of sales tax ϭ 0.05 ϫ 349 ϭ $17.45; total cost ϭ 349 ϩ 17.45 ϭ $366.45. The difference between the rebate being obtained by the mail-in method and the discount being calculated at the time of purchase at the store is $2.50. To calculate the percentage difference between the total cost of the in store discount purchase and that of the 60-day rebate purchase: rebate cost / discount cost ϫ 100% ϭ percent- age difference, or 368.95 / 366.45 ϫ 100% ϭ 1.006%. To express the cost difference between the in-store discount and the mail-in rebate, the mail-in rebate process is 1.006% more expensive. This calculation as set out here does not place a value on other likely costs, including the time the purchaser would take to complete the rebate form, mail the rebate, and other associated steps required to have the rebate processed. IMPACT OF THE 60-DAY REBATE PERIOD ON THE COST OF THE PRODUCT As was noted in the examples dealing with the calcula- tion of percentages, an interest rate measures the value of money over a period of time. Interest rate calculations are useful not only to calculate an increase in the value of money (such as the rate on interest being compounded on money being held in a bank account), but as is illustrated by the 60-day rebate, the interest rate percentage calculation can be used to confirm a loss of value over a period of time. The calculation of the difference in the total cost of the refrigerator confirmed that the in-store discount total price of $366.45 was $2.50, or 1.006% less than the mail- in rebate total price of $368.95. The next calculation will illustrate what happens to the $50 rebate during the 60-day rebate period. Assume that if the $50 were placed in a bank account, it would earn interest at a rate of 4% per year. Had the customer purchased the refrigerator by way of an in-store discount, the $50 discount would have been an immediate benefit to the purchaser, deducted at that point from the price paid to the retailer. By waiting 60 days to receive the rebate (the mini- mum period, given that as a mail-in rebate there are addi- tional days of mail and processing by the manufacturer), the purchaser lost an opportunity to use that $50 sum. The percentage interest calculation will place a value on that loss of opportunity: loss ϭ value of rebate ϫ number of days rebate not available / length of the year ϫ interest rate; value of rebate ϭ $50; mail-in period ϭ 60 days; year ϭ 365 days; interest rate ϭ 4% ϭ 0.04; loss ϭ $50 ϫ 60 / 365 ϫ 0.04; loss ϭ 50 ϫ 0.164 ϫ 0.04; loss ϭ 0.205. In this example, the loss of opportunity for the pur- chaser on the $50 rebate paid to the purchaser after 60 days is a small figure, 20.5 cents. The total difference in cost between the in-store discount purchase and the rebated purchase is the difference in total cost, $2.50, and the loss of opportunity on the $50 rebate, $0.205, for a total of $2.705. However, as with most retail sales examples using rel- atively small numbers, it is easy to understand the impor- tance of these percentage calculations where the retail price is 10 or 100 times greater. The percentages do not change, but where the percentages are applied to larger numbers, the potential impact on a purchaser is considerable. UNDERSTANDING PERCENTAGES IN THE MEDIA It is virtually impossible to read a news article, whether in paper format or by way of Internet service, that does not make at least one reference to a statistic that is described by way of a percentage. Sports, television ratings, employment, stock prices: all are commonly described in terms a percentage. In the media, it is common for per- centage figures to be stated as a conclusion. For example, the income tax rate will be increased by 2.5% next year, for all persons earning more than $75,000 per year. To properly understand how things such as the con- sumer price index, the inflation rate, the unemployment rate, and similar issues are reported in the media, it is important to keep in mind the mathematical rules con- cerning percentages and how they are calculated. The Consumer Price Indexes (CPI) program pro- duces monthly data on changes in the prices paid by urban consumers for a representative basket of goods and services. Comparisons between prices on a month-by- month basis are useful in determining whether living costs are going up or down. To put it another way, the CPI tells how much money must be spent each month to main- tain the same standard of living month to month, as the CPI values the same items to be purchased each period. The CPI is based upon a sample of actual prices of goods that are grouped together under a number of cate- gories such as food and beverages, clothing, transporta- tion, and housing. Each individual item is priced, and the entire costs of the categories are compared with a selected base period. There are a number of adjustments that are also factored into the calculations, to take into account seasonal buying patterns at holidays and well-known sale periods. The CPI calculations are made as follows: the base period, representing the time against which the current comparison will be made, is equal to 100, based upon 1990 Percentages REAL-LIFE MATH 379 reference data. Assume that the period to be compared is in November 2005: 1990 base price ϭ $100.00; November 2005 price ϭ $189.50. The increase in the CPI index from 1990 to November 2005 is 89.5% or (189.50 Ϫ 100.00)/100. If the December cost of the consumer basket is 191.10, the increase from the base period of 1990 is 91.10% or (191.10 Ϫ 100.00)/100. To calculate the percentage increase between November and December, the following process must be carried out: the November value of 189.50 must be subtracted from the December value of 191.10, for an increase of 1.60% when compared to the 1990 rate. To calculate the percentage change between November and December: 1.6%/ 189.5% ϫ 100% ϭ 0.0084 ϫ 100 ϭ 0.84%. Therefore, there was a 0.84% increase in the consumer price index in this example between November and December. PUBLIC OPINION POLLS From time to time, specialist organizations, known as polling companies, will be commissioned to gather data from a segment of the population concerning par- ticular issues. The question asked of the people polled may involve a large national issue, such as whether capi- tal punishment ought to be permitted, or whether the maximum speed limits on national highways should be increased or decreased. In some instances, the polling organization may be hired to obtain the opinions of the public in relation to issues that pertain to a local concern, such as whether a town should permit a casino to be con- structed within its boundaries. The manner in which public opinion polls are car- ried out is a branch of social science. The methods used by the pollsters in the asking of the questions, the num- ber of people who form the sample upon which calcula- tions are made, and the age and the background of the responders are all factors that may impact upon the answers given to the polling company. From the perspective of percentages, it is important to appreciate that virtually all such public opinion polls are translated, and reported in the media, as a percentage fig- ure. The meaning to be attached to the percentage quoted as the result of the poll must be examined carefully. For example, a sample of 4,000 people were asked the following questions: Should cigarette sales in their city be banned completely? Should smoking be banned in every public place in their city? For the first question, the fol- lowing results were noted: 1,900 said, “yes”; 1,800 said, “no”; 250 were “not sure”, and 50 “refused to answer.” For the second question, the following results were noted: 2,100 said, “yes”; 1,550 said, “no”; 300 were “not sure”; and 50 “refused to answer.” What are the different ways that the results of each of these questions can be expressed as a percentage? Depending upon how the percentage calculation is used in each case, what answers may be given to each of the ques- tions? The percentage calculation for each answer to ques- tion 1 on the ban of cigarette sales is “yes” ϭ 1,900/4,000 ϭ 47.5%; “no” ϭ 1,800/4,000 ϭ 40%; “not sure” ϭ 250/ 4,000 ϭ 6.25%; “refused” ϭ 50/4,000 ϭ 1.25%. If the poll was to exclude those who refused to answer the question, and only calculate the responses from people who did answer, the percentages for each answer are “yes” ϭ 1,900/3,950 ϭ 48.1%; “no” ϭ 1,800/3,950 ϭ 345.6%; “not sure” ϭ 250/3,950 ϭ 6.3%. If the poll were further defined as all respondents who had made up their minds and therefore had a posi- tive opinion on the issue, the formula is “yes” ϭ 1,900/3,700 ϭ 51.35%;“no” ϭ 1,800/3,700 ϭ 48.65%. By taking these steps, the polling company might choose to state this result as “more than 50% of respondents to the poll who had formed an opinion on the question were in favor of a ban on the sale of cigarettes in the city.” If the poll is defined by who is in favor of the ques- tion, the formula is “yes” ϭ 1,900/4,000 ϭ 47.5%; “all other responses” ϭ 2,100/4,000 ϭ 52.5%. The polling company might state this result as “less than 50% of all respondents to the poll stated that they were in favor of a ban on cigarette sales in the city.” The result to the question 2 to ban cigarette smoking in public places generates the following percentage cal- culations: “yes” ϭ 1,650/4,000 ϭ 41.25%; “no” ϭ 1,550/4,000 ϭ 38.75%; “not sure” ϭ 700/4,000 ϭ 17.5%; “refused” ϭ 100/4,000 ϭ 2.5%. Using the same analysis as carried out with question 1, if the persons who refused to answer the question are also eliminated from the sample: “yes” ϭ 1,650/3,900 ϭ 42.3%; “no” ϭ 1,550/3,900 ϭ 39.8%; “not sure” ϭ 700/3,900 ϭ 17.9%. If the persons who were not sure in their answers to the question are removed from the sample: “yes” ϭ 1,650/3,200 ϭ 51.5%; “no” ϭ 1,550/3,200 ϭ 48.5%. In the same manner as is set out in the question 1 analysis, the manner in which the percentages are calcu- lated in each case can support different conclusions. With the question 2 calculations, when the whole sample of 4,000 answers is examined, only 41.25% of those ques- tioned supported the ban on smoking in public places. By restricting the sample to those with a definitive opinion, a majority of those questioned may be said to support the proposed ban. Percentages 380 REAL-LIFE MATH USING PERCENTAGES TO MAKE COMPARISONS It is common in media reports to compare different results in related topics. For example, government spend- ing may be reported in a particular year as having increased 5% over the previous year. The population of a particular state may be stated as having increased by 3% over the past decade. These calculations are relatively straightforward, because the comparison is being made between single entities, namely a government budget, which would be calculated and measured to be reflected as a total figure, or population, which would have been measured by way of a population count, known as a census. Percentages are more difficult to put into perspective when they are employed to compare less certain items. For example, if the two public opinion questions and the various answers are compared by way of percentage cal- culations, the results are not always certain. In question 1, when only the respondents who had either a yes or a no opinion were calculated, the number of those in favor of the ban on cigarette sales was 51.35%, and those opposed to such a ban was 48.65%. In the ques- tion 2 analysis, when only the respondents with a yes or no opinion were counted, the number of those in favor of banning smoking in all public places was 51.5%, those opposed totaled 48.5%. Based upon the determination of percentage figures that are virtually identical (51.35% and 51.5%) in each question, it would be possible to state the following as a conclusion from the two sets of polling questions, namely a majority of people in the city are in favor of both a ban on cigarette sales and a ban on smoking in all public places. However, having worked through the calculation to each of the percentages that form the basis of this state- ment, it is also clear that the use of those percentages in the manner contemplated by this conclusion is not the entire picture. If other parts of the calculation are used to determine a conclusion, it could also be stated that as 47.5% of all respondents were in favor of the ban on cig- arette sales, and then a further 41.25% were in favor of the public places ban, the following conclusions are valid: less than 50% of persons polled were in favor of any restriction upon cigarette purchase or usage in the city; a little over 2 out of 5 people polled were in favor of these restrictions. Percentages and statistics of all types are often stated as a definitive answer or conclusion to an issue. As illus- trated in the questions posed above, it is important that the method employed in calculating the percentage be understood if one is to truly understand the significance of the percentage figure that is stated as a conclusion. Where the methodology behind a particular percentage is not stated in a particular media report, the percentage must be regarded with caution. SPORTS MATH Another common media report in which percentages are employed in a variety of ways is that of the sports commentary. There are a seemingly limitless number of ways that sport and athletic competition commentaries are enhanced by the use of statistics, many of which are dependent upon percentage calculations. In the media, there is a recognition that certain sta- tistics go beyond analysis of an individual performance, but are descriptors that convey a definition of enduring excellence. The “300 hitter” is a description applied to a Miami Heat’s Dwayne Wade goes up and scores against the Atlanta Hawks in the game in Miami. Players are often rated by percentages, such as their field goal percentage. AP/WIDE WORLD PHOTOS. REPRODUCED BY PERMISSION. Percentages REAL-LIFE MATH 381 solid offensive professional baseball player, while a “400 hitter” is in an ethereal world inhabited by legends like Ted Williams and Ty Cobb. A 90% free-throw shooter in basketball has a similar instantaneous public recognition. The American humorist Samuel Langhorne Clemens, better known as Mark Twain (1835–1910), once said that there are three kinds of lies: % lies, damn lies, and statistics. Whenever a percentage is referenced in a sports article, as with any other media usage of percent- ages, care must be taken to determine whether the per- centage figure being quoted is an accurate indicator of performance, or whether at best it is a lesser or insignifi- cant fact adding only color, and not necessarily insight, concerning the sporting event. Sports examples of percentage calculation usage are based on daily examples found in the media around the world. For instance, in basketball, an example would be Amanda and Claire as members of their girls’ high school basketball team. The coach of the team has been asked to select a most valuable player for the season. While the coach has a personal view of each player based on his assessment of their play through practice and games all season, he decides to do an analysis of their respective offensive statistics. Each player had the following statistics after the completion of the 20-game high school season: Amanda scored 160 total points; 108 2-point shots attempted; 62 2-point shots made; 10 3-point shots attempted; 6 3-point shots made; 21 free throws attempted; 18 free throws made; 17.5 minutes played per 32-minute game. Claire scored 322 total points; 341 2-point shots attempted; 125 2-point shots made; 22 3-point shots attempted; 5 3-point shots made; 81 free throws attempted; 57 free throws made; 28.8 minutes played per 32-minute game. The team scored 887 points on the season. How can percentages be used to help determine who is having the better season? Conversely, do percentage cal- culations distort any elements of the performance of these players? If the 2-point shooting of each player is compared, by calculating the percentage accuracy of each player through the entire season, the following comparison can be made: Amanda ϭ 62 shots made/108 shots attempted ϭ 57.4%. Claire ϭ 125 shots made/341 shots attempted ϭ 36.66%. The 3-point shooting percentage calculation is as fol- lows: Amanda ϭ 6 shots made/10 shots attempted ϭ 60%. Claire ϭ 5 shots made/22 shots attempted ϭ 22.7%. The players’ free-throw shooting percentages are cal- culated as follows: Amanda ϭ 18/21 ϭ 85.7%. Claire ϭ 57/81 ϭ 70.4%. If a newspaper report was written setting out the coach’s analysis of the respective play of Amanda and Claire, it is quite possible that such a report might describe Amanda as a better shooter than Claire because her shooting percentages in every area of comparison (2- point shooting, 3-point shooting, and free-throw shoot- ing) are better than Claire’s. Conversely, Claire has scored the most points and she has played more minutes per game than Amanda. When those statistics are assessed, the following percentage calculations can be determined: For Amanda, 160 points scored/887 team points scored ϫ 100% ϭ 18% of the team offense. For Claire, 322 points scored/887 team points scored ϫ 100% ϭ 36.3% of the team offense. Further, Amanda generated her 18% of the team offense while playing 17.5 minutes per game. Claire pro- duced her 36.3% of the team offense while playing 28.8 minutes per game. There are certain hard conclusions that the coach in this scenario may have reached based upon the percent- age calculations that pertain to Amanda and Claire. Amanda is a more accurate shooter in every aspect of the shooting game. It is likely that based upon these percent- ages, the coach will create opportunities for Amanda to shoot more often next season. However, as with many applications of the percent- age calculation in a sports context, it is important to have more information about the team and the players to give the percentage statistics more context, and to put the per- centages into a better perspective. If Amanda is a weak defensive player, her offensive percentages are placed in a different light. If Claire had performed all season known to all rivals as the team’s best player, and thus attracted extra attention from opponents, her shooting percentages would be weighed differently. Baseball statistics may be the most identifiable per- centage in sport, usually expressed as a decimal. For example, a strong hitter in the North American profes- sional leagues will be referred to as a “300 hitter,” mean- ing a batsman with an average of 0.300, or a 30%, success rate. This percentage is calculated by the following formula: Number of hits/Number of at bats ϫ 100% ϭ Batting average. However, as befits a sport that has been played pro- fessionally in North America since the 1870s, statistics have grown out of the game, some clear to even the aver- age fan, and some very obscure. A key percentage used to calculate offensive contributions is that of “on base per- centage,” which measures how often a batter advances to first base by any of the means available in baseball, namely hit, walk, hit by pitched ball, etc. The percentage Percentages 382 REAL-LIFE MATH is calculated by the following formula: Total number of times on base / Total number of at bats or plate appear- ances ϫ 100% ϭ On base percentage. A very intricate set of percentages has made its way into the analytical end of baseball through the work of Bill James. His approach, which he termed sabermetics, is an attempt to use scientific data collection and interpre- tation methods that employ various types of percentages in different aspects of baseball to conclude why certain teams succeed and others fail. North American football is also riddled with statis- tics. One of those measurements is that concerning the most prominent player on the field, the quarterback. How often the quarterback may successfully throw the ball down field is an important statistic, referred to as passing completion. This percentage is calculated by: Number of passes completed/Number of passes thrown ϫ 100%. However, much like the basketball examples set out above, this percentage on its own is potentially deceiving. A quarterback who throws 80% of his passes for comple- tions, but never throws a pass for a score, is unlikely to be as successful as the 50% passer who throws for 20 touch- downs in a season. TOURNAMENTS AND CHAMPIONSHIPS With the rise in the popularity and the sophistication of college sports in the United States, coupled with the impossibility of having hundreds of teams in a given sea- son playing one another head to head, statistical tools were developed to weight the relative abilities of teams that would not necessarily meet in a season, but each of whom would seek selection to an elite end-of-season tournament or championship. In American college basketball, hockey, and football, tournament selection is made using what is known as the RPI, or ratings percentage index. This interesting and much debated tool is defined in college basketball as fol- lows: RPI ϭ Team winning percentage/25% ϩ Oppo- nents winning percentage/50% ϩ Opponents’ opponents winning percentage/25%. If a team had a record of 16 wins and 12 losses in a sea- son, they would therefore have a team winning percentage of 16 of 57.14%. The team played opponents whose total record was 400 wins and 354 losses. The opponents’ winning percentage is 53.05%. These opponents played teams whose winning percentage was 49.1%, the opponents’ opponents’ winning percentage: RPI ϭ 57.14/25 ϩ 53.05/50 ϩ 49.1/25, which is RPI ϭ 2.28 ϩ 1.06 ϩ 1.96 ϭ 5.304. A team will typically have a bigger and better RPI if the team combines its own success with an ability to beat strong opponents that have themselves played a strong schedule. Therefore, a team at the end of a particular sea- son that has a lesser record than a rival, but that has played what the RPI determines to be a demonstrably more difficult schedule, may be selected to compete over the team with the better win/loss record. The RPI has a number of nuances that are not the subject of this text, but it is important to understand that the percentage cal- culation is at the root of any RPI determination. Percentiles The percentile is a ranking and performance tool that is closely related to the concept of percentages. A per- centile represents a place on a scale or a field of data, pro- viding a rank relative to the other points on the scale. Percentiles are calculated by dividing a data set into 100 groups of values, with at most 1% of the data values in each group. Percentages can be expressed in any number from 0 to virtual infinity, with either a positive or negative value as circumstance may determine. However, it is commonly accepted that in many applications where a percentage calculation determines a grade or a score in a particular activity, the percentage is expected to be between 0% and 100%. For example, where a school assignment was graded at 17/20, the assignment has a percentage grade of 85%. In situations where there is a large class of students, it is often desirable to rank them in order of performance. Ranking provides a measure of how a particular student has performed relative to every other comparable stu- dent. For example, hundreds of thousands of potential university students in the United States, with many thou- sands more worldwide, test for the standard Scholastic Aptitude Test (SAT) every year. The SAT is tested at a multitude of test sites, at various times. Each test in a given year is similar, but the exact questions asked on each of the tests will vary. The SAT has a complicated scoring system generating scores from 0 to 1600, and the administrators of the test recognize that assessing stu- dents who have taken different versions of the SAT is very difficult. For this reason the percentile ranking becomes important, as it measures where every student stands rel- ative to every other student who took the test. Determining where an individual students stands relative to everyone else who took the test is a terrific tool with which to assess relative performance. This determi- nation is done by calculating the percentile. Percentages REAL-LIFE MATH 383 SAT SCORES OR OTHER ACADEMIC TESTING The percentile grew from the concept of percentages; for that reason, founded upon the concept of 100, and if the data comprising the test results is regarded as a unit of 100, percentile ranking proceeds in bands from 0 to 99, with the 99th band being that that includes the highest score or scores in the sample. Each percentile in the sample may have more than one score within it. Further, percentiles are not sub- divided. For example, there may be as many as 20,000 test scores produced from one round of SAT testing. If eight students scored a perfect 1600 on the SAT, they would each be described as having a result in the 99th percentile even if, say, 10 students with slightly lower scores were also in the 99th percentile. Similarly, if the 55th per- centile, representing 1% of all scores from that test, was determined to be all of the scores between 1010 and 1040, all scores within that percentile band would be described as in the 55th percentile. One formula to calculate the percentile for a given data value is: Percentile ϭ (number of values below x ϩ 0.5)/number of values in the data set ϫ 100%. As an example, the following is a sample of the shoe sizes for a 12-member high school boys basketball team: Sample: 14, 12, 10, 10, 13, 11, 10, 9, 9, 10, 11, 9. How is the percentile rank of shoe size 12 determined? First, the shoes sizes must be arranged in values smallest to largest, which create this set: 9, 9, 9, 10, 10, 10, 11, 11, 12, 13, 14. The num- ber of values below 12 is eight, and the total number of values in the data set is 12. The formula to express the per- centile rank of the value 12 is (8 ϩ 0.5)/12 ϫ 100% ϭ 70.83%. The percentile ranking of the value of the size 12 can therefore be expressed as the 1st percentile. To calculate the percentile ranking of the size 10, there are three identical sizes in the data set. There are three values in the set below 10. The formula would be (3 ϩ 0.5)/12 ϫ 100% ϭ 29.1%. The percentile ranking of the value of all three of the size 10s is expressed as the 29th percentile. It is also common to express a ranking using a broader term. For example, a student may be described as being in the top 20% of their class, or in the top quarter. These expressions are a paraphrasing of the percentiles known as deciles (groups of 10 percentiles) and quartiles (groups of 25 percentiles). Deciles divide the data set into 10 equal parts, and quartiles divide the set into four equal parts. The 50th percentile, the 5th decile, the 2nd quartile, and the median are all equal to one another. Final grades in academic courses are typically expressed as a percentage. Even where alternate methods are used to express performance (as with alpha grades A through F), or as a grade point average, each alternative has an equiv- alency expressed as a percentage. The percentages are then matched to a particular letter grade that has a range of percentages within it. For example, Aϩ is the equiva- lent of 90–100%; A is the equivalent of 80–89%; B is the equivalent of 70–79%; C is the equivalent of 60–69%; D is the equivalent of 50–59%; and F is the equivalent of below 50%. Letter grades function in a similar way as percentiles, in that each grade includes a potential range of percent- age scores, and like the percentile, the percentage scores are not ranked within the assigned grade. Any area of human performance that is subject to ranking will likely employ percentiles as a measuring stick. Topics can be as diverse as the relative rate of obe- sity in children, ranking increases or decreases in funding rates for hospitals and schools, and comparing the rela- tive safety rates in relation to speed on highways. These are three of the almost limitless ways that percentiles can be used to assist in a ranking of performance. Potential Applications The better understanding of a multitude of everyday concepts and activities will be determined, directly or indirectly, by an appreciation of the ability to perform the percentage calculation. As further examples, percentages play a key role in the following areas: • Voting patterns and election results: Percentages are used to take the large numbers of persons who may vote in an election, and reduce the figures to a result that is often easier to understand. • Automobile performance: Octane is a term that is familiar to everyone who has ever used gasoline as a fuel for a vehicle. In general terms, the octane rating refers to how much the fuel can be compressed before spontaneously igniting, an important factor in optimizing the performance of the internal com- bustion engine. While the public generally associates high octane requirements as required for certain motor vehicle models with more powerful engines and vehicle performance, the octane rating repre- sents the percentage between the hydrocarbon octane (or similar composition) in relation to the hydrocarbon heptane. For example, an 87 octane rat- ing (a common minimum in the United States) rep- resents an 87 percent octane, 13 percent heptane mixture in the fuel. Percentages 384 REAL-LIFE MATH • Clothing composition and manufacture: Most clothing is sold with a tag or other indication as to its material composition. For example, it is com- mon to see a label on a shirt indicating 65% cot- ton, 35% polyester, or a sweater marked as 100% wool. • Vacancy rates: The availability of vacant apartment space in a particular city is of great importance to prospective residents and existing apartment dwellers alike. The vacancy rate is expressed as a per- centage to provide interested persons with an indica- tor as to the relative ease or difficulty to obtain particular types of rental accommodation. Vacancy rates can be viewed as of a particular period (for example, the vacancy rate in Spokane was 1.8% in April), or as a calculation increase or decrease from period to period (for example, the vacancy rate in Toronto fell 0.7% last month). Where to Learn More Books Boyer, Carl B. A History of Mathematics. New York: Wiley and Sons, 1991. Upton, Graham, and Ian Cook. Oxford Dictionary of Statistics. London: Oxford University Press, 2000. Web sites College Board. “Scholastic Aptitude Test.” (March 29, 2005.) Ͻhttp://www.collegeboard.comϾ. NCAA Tournament Selection, 2005. (March 29, 2005.) Ͻhttp://www.ncaa.comϾ. Key Terms Fraction: The quotient of two quantities, such as 1/4. Percentage: From Latin per centum, meaning per hun- dred, a special type of ratio in which the second value is 100; used to represent the amount present with respect to the whole. Expressed as a percent- age, the ratio times 100 (e.g., 78/100 ϭ .78 and so .78 ϫ 100 = 78%). Ratio: The ratio of a to b is a way to convey the idea of relative magnitude of two amounts. Thus if the num- ber a is always twice the number b, we can say that the ratio of a to b is “2 to 1.” This ratio is some- times written 2:1. Today, however, it is more com- mon to write a ratio as a fraction, in this case 2/1. [...]... their first appearance around 177 0, and became accepted and widely used around 1820 In 179 5, graphical scales were used to help convert old measurements to R E A L - L I F E M A T H Stem and leaf display 3 2 1 0 23 37 001112223889 2244456888899 69 Figure 1: A stem and leaf plot display the new, metric measurements The French mathematician Johann Heinrich Lambert ( 172 8– 177 7) used graphs extensively in... H Scatter graphs are used to plot much of the experimental data that scientists collect For example, if a scientist 4 07 Plots and Diagrams 190 185 180 Height (cm) 175 170 165 160 155 150 M F Gender Figure 2: A composite box plot 200 Height (cm) 190 180 170 160 150 140 140 140 160 170 180 Arm Span (cm) 190 200 Figure 3: A scatter plot on a scatter graph measured the distance a car, traveling at a constant... 2005) Perspective from MathWorld Ͻhttp://mathworld.wolfram.com/ Perspective.htmlϾ (April 8, 2005) Wired News “Sin City Expands Digital Frontier.” Jason Silverman April 1, 2005 Ͻhttp://www.wired.com/news/ digiwood/0,1412, 670 84,00.htmlϾ (April 8, 2005) Other The Lord of the Rings: The Fellowship of the Ring Extended Edition DVD special features New Line Home Entertainment, 2002 3 97 Overview Photography,... large numbers of people, sometimes motivates lens designers to use maximum apertures that change according to the focal length An 18 70 mm (0 .7 2.8 in) f/3.5-4.5 zoom lens would have a maximum aperture that ranges from f/3.5 at 18 mm (0 .7 in) focal length to f/4.5 at 70 mm (2.8 in) focal length DEPTH OF FIELD Depth of field refers to the range of distance from the lens, or depth, throughout which objects... 21.4 meters (about 70 .2 feet) The equation that determines a perimeter of a circle (also known as its circumference) is p ϭ 2␲r or p ϭ ␲d (where ␲ ϭ approximately equal to 3.14159, r ϭ radius of the circle, and d ϭ circle’s diameter and d ϭ 2r) As a specific example, a circle with a diameter of 7. 5 meters (about 24.6 feet) has an approximate perimeter of p ϭ ␲ (7. 5 meters) ϭ 3.14159 (7. 5 meters) ϭ 23.6... (translated from French by John Meldrum) Elements of the History of Mathematics Berlin, Germany: Springer-Verlag, 1994 Boyer, Carl B A History of Mathematics Princeton, NJ: Princeton University Press, 1985 Bunt, Lucas N.H., Phillip S Jones, and Jack D Bedient The Historical Roots of Elementary Mathematics Englewood Cliffs, NJ: Prentice-Hall, Inc., 1 976 Web sites Rores, Chris Rorres Drexel University “Archimedes.”... Entertainment, 2002 3 97 Overview Photography, literally writing with light, is full of mathematics even though modern auto-exposure and auto-focus cameras may seem to think for themselves Lens design requires an intimate knowledge of optics and applied mathematics, as does the calculation of correct exposure When mastered, the mathematics of basic photography allow artists, journals, and scientists to create... each other and the line of regression A correlation coefficient between 0.0 and Ϯ0.3 is considered a weak, a correlation coefficient between Ϯ0.3 and Ϯ0 .7 is considered a moderate, and a correlation coefficient between Ϯ0 .7 and Ϯ1.0 is considered a high Mathematically, the correlation coefficient is the sum of the squares of the individual errors, which are the vertical deviations, to measure how well... high-technology corrugated metal barriers that can withstand the blast of high-order detonations 3 87 Perimeter and anti-ram barriers that can withstand the repeated assault by enemy tanks and other motorized vehicles Potential Applications P L A N E TA R Y E X P L O R AT I O N Perimeter is such a general term within mathematics that its use will always be important for new applications For example, as mankind... mastered, the mathematics of basic photography allow artists, journals, and scientists to create more compelling and insightful images whether they are using film or digital cameras Photography Math Fundamental Mathematical Concepts and Terms THE CAMERA In its simplest form, the camera is a light-tight box containing light sensitive material, either in the form of photographic film or a digital sensor . 4.5%, the total cost of the item is 4.5% ϭ .045; tax ϭ .045 ϫ 21,595 ϭ $ 971 .76 ; total cost ϭ 971 .76 ϩ 21,595 ϭ $22,566 .78 . Another factor in relation to the calculation of costs is the fact that. example, a circle with a diameter of 7. 5 meters (about 24.6 feet) has an approximate perimeter of p ϭ␲ (7. 5 meters) ϭ 3.14159 (7. 5 meters) ϭ 23.6 meters (about 77 .4 feet). By knowing the shape of. 5 shots made/22 shots attempted ϭ 22 .7% . The players’ free-throw shooting percentages are cal- culated as follows: Amanda ϭ 18/21 ϭ 85 .7% . Claire ϭ 57/ 81 ϭ 70 .4%. If a newspaper report was written