excursions in modern mathematics, 7e: 10.1 - 2copyright © 2010 pearson education, inc. 10 the...

Excursions in Modern Mathematics, 7e: 10.1 - 2Copyright © 2010 Pearson Education, Inc.

10 The Mathematics of Money

10.1 Percentages

10.2 Simple Interest

10.3 Compound Interest

10.4 Geometric Sequences

10.5 Deferred Annuities: Planned Savings for the Future

10.6 Installment Loans: The Cost of Financing the Present


As a consumer, you make decisions about money every day. Some are minor –“Should I get gas at the station on the right or make a U-turn and go to the station across the highway where gas is 5¢ a gallon cheaper?”–, but others are much more significant – “If I buy that new red Mustang, should I take the $2000 dealer’s rebate or the 0% financing for 60 months option?”.

Money Matters


Decisions of the first type usually involve just a little arithmetic and some common sense (on a 20 gallon fill-up you are saving $1 to make that U-turn–is it worth it?); decisions of the second type involve a more sophisticated understanding of the time value of money (is $2000 up front worth more or less than saving the interest on payments over the next five years?). This latter type of question and others similar to it are the focus of this chapter.

Money Matters


A general truism is that people don’t like dealing with fractions. There are exceptions, of course, but most people would rather avoid fractions whenever possible. The most likely culprit for “fraction phobia” is the difficulty of dealing with fractions with different denominators. One way to get around this difficulty is to express fractions using a common, standard denominator, and in modern life the commonly used standard is the denominator 100.

Fractions


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,

Percentages

x% x100


Percentages are useful for many reasons. They give us a common yardstick to compare different ratios and proportions; they provide a useful way of dealing with fees, taxes, and tips; and they help us better understand how things increase or decrease relative to some given baseline. The next few examples explore these ideas.

Percentages


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 124.8 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 124.8 are called raw scores. Since each raw score is based on a different total, it is hard to compare them directly, but we can do it easily once we express each score as a percentage of the total number of points possible.

Example 10.1 Comparing Test Scores


■ Quiz score = 19/25: Here we can do the arithmetic in our heads. If we just multiply both numerator and denominator by 4, we get 19/25 = 76/100 = 76%.

■ Midterm score = 49.2/60: Here the arithmetic is a little harder, so one might want to use a calculator:49.2 ÷ 60 = 0.82 = 82%. This score is a definite improvement over the quiz score.



■ Final Exam = 124.8/150: Once again, we use a calculator and get:124.8 ÷ 150 = 0.832 = 83.2%.This score is the best one.



Example 10.1 illustrates the simple but important relation between decimals and percentages: decimals can be converted to percentages through multiplication by 100 (as in 0.76 = 76%, 1.325 = 132.5%, and 0.005 = 0.5%), and conversely, percentages can be converted to decimals through division by 100 (as in 100% = 1.0, 83.2% = 0.832, and 7 1/2 % = 0.075).

Convert Decimals to Percents


Imagine you take an old friend out to dinner at a nice restaurant for her birthday. The final bill comes to $56.80. Your friend suggests that since the service was good, you should tip 3/20th of the bill. What kind of tip is that? After a moment’s thought, you realize that your friend, who can be a bit annoying at times, is simply suggesting you should tip the standard 15%. After all, 3/20 =15/100 = 15%.

Example 10.2 Is 3/20th a Reasonable Restaurant Tip?


Although 3/20 and 15% are mathematically equivalent, the latter is a much more convenient and familiar way to express the amount of the tip. To compute the actual tip, you simply multiply the amount of the bill by 0.15.

In this case we get 0.15 $56.80 = $8.52.

Example 10.2 Is 3/20th a Reasonable Restaurant Tip?


Imagine you have a little discretionary money saved up and you decide to buy yourself the latest iPod. After a little research you find the following options:

■ 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

$399 + (0.0675)$399 = $399 + $26.9325 = $399 + $26.94 = $425.94

Example 10.3 Shopping for an iPod


The above calculation can be shortened by observing that the original price (100%) plus the sales tax (6.75%) can be combined for a total of 106.75% of the original price.

Thus, the entire calculation can be carried out by a single multiplication:

(1.0675)$399 = $425.94

(rounded up to the nearest penny)



■ 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. Taking the 5% off from the coupon gives the sale price, which is 95% of the original price.

Sale price: (0.95)$415 = $394.25



We still have to add the 6.75% sales tax on top of that, and as we saw in Option 1, the quick way to do so is to multiply by 1.0675.Final price including taxes:

(1.0675)$394.25 = $420.87

For efficiency we can combine the two separate calculations (take the discount and add the sales tax) into one:

(1.0675)(0.95)$415=$420.87



■ 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?



Unlike option 2, in this situation we do not take a 5% discount on the $441. Here the 5% was added to the iPod’s base price to come up with the final cost of $441, that is, 105% of the base price equals $441. Using P for the unknown base price, we have


Although option 3 is the cheapest, it is hardly worth the few pennies you save to inconven-ience your friend. Your best bet is to head to Hamiltonian Circuits with your 5% off coupon.

1.05 P $441 or P $441

1.05$420


If you start with a quantity Q and increase that quantity by x%, you end up with the quantity

PERCENT INCREASE

I 1 x100

Q.


If you start with a quantity Q and decrease that quantity by x%, you end up with the quantity

PERCENT DECREASE

D 1 x100

Q.


If I is the quantity you get when you increase an unknown quantity Q and by x%, then

PERCENT INCREASE

Q I1 x 100 .

(Notice that this last formula is equivalent to the formula given in the first bullet.)


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.) We are going to illustrate the ups and downs of the DJIA with fictitious numbers.

Example 10.4 The Dow Jones Industrial Average

■ 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 13,029.50. This is an increase of 154.50 from the previous day. To express the increase as a percentage, we ask, 154.50 is what percent of 12,875 (the day 1 value that serves as our baseline)? The answer is obtained by simply dividing 154.50 into 12,875 (and then rewriting it as a percentage).



Thus, the percentage increase from day 1 to


Here is a little shortcut for the same computation, particularly convenient when you use a calculator (all it takes is one division): 13,029.50 ÷ 12,875 = 1.02All we have to do now is to mentally subtract 1 from the above number. This gives us once again 0.012=1.2%.

154.50

12,8750.012 1.2%day 2 is


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!

Misleading Use of Percent Changes


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.

Example 10.5 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 = 0.666 . . . ≈ 66.67%.

Example 10.5 The Bogus 200% Decrease


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.

Moral to Example 10.5


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.

Example 10.6 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.


■ 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).



■ 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, 7e: 10.1 - 2copyright © 2010 pearson education, inc. 10 the...

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