Chapter 10: Financial Mathematics Percentages

Transcription

1 Chapter 10: Financial Mathematics 10.1 Percentages

2 Bell Work Take your best guess at the percent for each statement. 1. % of Americans do not understand how compound interest works. (2008) 2. % of American college students with credit cards don t pay their balance in full every month and pay finance charges. (2009) 3. % of Americans have no retirement savings. (2010) 4. % of Americans have no personal savings. (2010) Excursions in Modern Mathematics, e: 1.1-2

3 Personal Finance Video Excursions in Modern Mathematics, e: 1.1-3

4 10.1 Reading Day/Quiz Read pages Take notes Quiz Excursions in Modern Mathematics, e: 1.1-4

5 Bell Work 1. Convert 34% to a decimal. 2. Convert 0.04 to a percent. 3. Convert 100% to a decimal. 4. Convert to a percent. Excursions in Modern Mathematics, e: 1.1-5

6 Percentages A fraction with denominator 100 can be interpreted as a percentage, and the percentage symbol (%) is used to indicate the presence of the hidden denominator 100. Thus, x% = x 100 Excursions in Modern Mathematics, e: 1.1-6

7 Percentages Percentages are useful for many reasons. 1. They give us a common yardstick to compare different ratios and proportions 2. They provide a useful way of dealing with fees, taxes, and tips 3. They help us better understand how things increase or decrease relative to some given baseline. Excursions in Modern Mathematics, e: 1.1-7

8 Example: Comparing Test Scores Suppose that in your English Lit class you scored 19 out of 25 on the quiz, 49.2 out of 60 on the midterm, and out of 150 on the final exam. Without reading further, can you guess which one was your best score? Not easy, right? The numbers 19, 49.2, and are called raw scores. Quiz: Midterm: Final: Excursions in Modern Mathematics, e: 1.1-8

9 Convert Decimals to Percents (#1) 0.76 = % 0.54 = % = % = % Excursions in Modern Mathematics, e: 1.1-9

10 Convert Percents to Decimals (#2) 32% = 10% = 83.2% = 7.5% = 5% = Excursions in Modern Mathematics, e:

11 Example: Is 3/20th a Reasonable Restaurant Tip? The final bill comes to $ Your friend suggests that since the service was good, you should tip 3/20th of the bill. What kind of tip is that? Ways to compute a 15% tip w/ a calculator in your head Excursions in Modern Mathematics, e:

12 Example (#3) Find a fraction that is equivalent to 30%. Find a percent that is equivalent to 3/5. Excursions in Modern Mathematics, e:

13 Example: Shopping for an ipod Option 1: You can buy the ipod at Optimal Buy, a local electronics store. The price is $399. There is an additional 6.75% sales tax. Your total cost out the door is. Excursions in Modern Mathematics, e:

14 Example: Shopping for an ipod Option 2: At Hamiltonian Circuits, another local electronic store, the sales price is $415, but you happen to have a 5% off coupon good for all electronic products. Your total cost out the door is. Excursions in Modern Mathematics, e:

15 Example: Shopping for an ipod Option 3: You found an online merchant in Portland, Oregon, that will sell you the ipod for $441. This price includes a 5% shipping/processing charge that you wouldn t have to pay if you picked up the ipod at the store in Portland (there is no sales tax in Oregon). The $441 is much higher than the price at either local store, but you are in luck: your best friend from Portland is coming to visit and can pick up the ipod for you and save you the 5% shipping/processing charge. What would your cost be then? Excursions in Modern Mathematics, e:

16 Example (#4) F = (1 + p)b Marian High School has approximately 700 students. If 16% of the student body is absent one day, how many students are absent? The population of a small town is If there was a 5% increase, what is the new population? The population of a midsize city is 80,000. If there was an 8% decrease, what is the new population? There was a 6% increase in a small town and the population is now 6,400. What was the original population? Excursions in Modern Mathematics, e:

17 Example: The Dow Jones Industrial Average The Dow Jones Industrial Average (DJIA) is one of the most commonly used indicators of the overall state of the stock market in the United States. (As of the writing of this material the DJIA hovered around 13,000.) Day 1: On a particular day, the DJIA closed at 12,875. Day 2: The stock market has a good day and the DJIA closes at 13, Excursions in Modern Mathematics, e:

18 Misleading Use of Percent Changes Percentage decreases are often used incorrectly, mostly intentionally and in an effort to exaggerate or mislead. The misuse is usually framed by the claim that if an x% increase changes A to B, then an x% decrease changes B to A. Not true! Excursions in Modern Mathematics, e:

19 Example: The Bogus 200% Decrease With great fanfare, the police chief of Happyville reports that crime decreased by 200% in one year. He came up with this number based on reported crimes in Happyville going down from 450 one year to 150 the next year. Since an increase from 150 to 450 is a 200% increase (true), a decrease from 450 to 150 must surely be a 200% decrease, right? Wrong. Excursions in Modern Mathematics, e:

20 Example: The Bogus 200% Decrease The critical thing to keep in mind when computing a decrease (or for that matter an increase) between two quantities is that these quantities are not interchangeable. In this particular example the baseline is 450 and not 150, so the correct computation of the decrease in reported crimes is 300/450 = %. Excursions in Modern Mathematics, e:

21 Example: The Bogus 200% Decrease The moral of this story? Be wary of any extravagant claims about the percentage decrease of something (be it reported crimes, traffic accidents, pollution, or any other nonnegative quantity). Always keep in mind that a percentage decrease can never exceed 100%, once you reduce something by 100%, you have reduced it to zero. An important part of being a smart shopper is understanding how markups (profit margins) and markdowns (sales) affect the price of consumer goods. Excursions in Modern Mathematics, e:

22 Example: Combining Markups and Markdowns A toy store buys a certain toy from the distributor to sell during the Christmas season. The store marks up the price of the toy by 80% (the intended profit margin). Unfortunately for the toy store, the toy is a bust and doesn t sell well. After Christmas, it goes on sale for 40% off the marked price. After a while, an additional 25% markdown is taken off the sale price and the toy is put on the clearance table. Excursions in Modern Mathematics, e:

23 Example: Combining Markups and Markdowns With all the markups and markdowns, what is the percentage profit/loss to the toy store? The answer to this question is independent of the original cost of the toy to the store. Let s just call this cost C. After adding an 80% markup to their cost C, the toy store retails the toy for a price of (1.8)C. Excursions in Modern Mathematics, e:

24 Example: Combining Markups and Markdowns After Christmas, the toy is marked down and put on sale with a 40% off tag. The sale price is 60% of the retail price. This gives (0.6)(1.8)C = (1.08)C, (which represents a net markup of 8% on the original cost to the store). Excursions in Modern Mathematics, e:

25 Example: Combining Markups and Markdowns Finally, the toy is put on clearance with an additional 25% off tag. The clearance price is (0.75)(1.08)C = 0.81C. (The clearance price is now 81% of the original cost to the store a net loss of 19%! That s what happens when toys don t sell.) Excursions in Modern Mathematics, e:

26 Example (#5) Over a period of one week, the Dow Jones Industrial Average did the following: On Monday the DJIA went up by 2.5%, on Tuesday it went up by 12.1%, on Wednesday it went down by 4.7%, on Thursday it went up by 0.8%, and on Friday it went down by 5.4%. What was the percentage increase/decrease of the DJIA over the week? Round your answer to the nearest tenth of a percentage point. Excursions in Modern Mathematics, e: