When Are We Ever Going To Use This Stuff?

Dear Student,

A few years ago, a class was presented with the following problem:

A farmer looks across his field and sees pigs and chickens. If he counts 42 heads and 106 feet, how many pigs are in the field?

cartoon of a chicken and a pigOne student, instead of answering the question, sent an e-mail to the instructor stating, “I feel this question is unfair and misleading because pigs do not have feet. They have hooves.”

In this class, and in all math classes for that much, we are not trying to trick you. Although you will, undoubtedly, be presented with some challenging questions, be assured they are there to make you apply and extend your thought processes. Don't waste time and effort looking for technicalities that might invalidate a question. Instead, take the questions at face value, and spend your time trying to solve them.

Next, one of the goals in assembling this book was to keep the cost very low. As a result, we have created a consumable-style workbook, which you will not be able to sell back to the bookstore at the end of the term. Take advantage of this and write in your book. After all, it is yours to keep.

Finally, we completely understand students taking this course are often Liberal Arts majors and/or need only this one math class in order to graduate. Not only that, but, in many cases, students often wonder why they need to take this class in the first place. All of us – yes, including your instructor – have sat in a math class at one time or another and thought, “When are we ever going to use this stuff?” Well, this is the class where we tell you.

- J.M. & R.Y.

Chapter 1: Consumer Math

1.1: On the Shoulders of Giants (Biographies and Historical References)

Money. It controls nearly everything in the world. Those who have it are often seen as more powerful and influential than those who do not. So, having a basic understanding of how money grows and is (or should be) spent is essential. Understanding percentages is key to understanding how money works. Budgets are based on different percents of available money being allocated in specific ways. Sale discounts and taxes are based on the percent of cost of the items at hand. Credit cards and loans operate on interest rates, which – you guessed it – are given in percents. The single largest purchase most people will make in a lifetime is a home. A basic understanding of mortgages and finance charges will make you a much wiser homebuyer and could easily save you tens of thousands of dollars over a 20-30 year period. We could go on, but you get the picture.


Image of a column chart with an arrow heading upwards with a dollar sign at the endThe Dow Jones Industrial Average (DJIA)

Charles Dow compiled his DJIA index to gauge the performance of the industrial sector of the American stock market. It is the second-oldest U.S. market index, after the Dow Jones Transportation Average, which Dow also created. When it was first published on May 26, 1896, the DJIA index stood at 40.94. It was computed as a direct average, by first adding up stock prices of its components and dividing by the number of stocks in the index. The DJIA averaged a gain of 5.3% compounded annually for the 20th century; a record Warren Buffett called "a wonderful century" when he calculated that, to achieve that return again, the index would need to reach nearly 2,000,000 by 2100. Many of the biggest percentage price moves in the DJIA occurred early in its history, as the nascent industrial economy matured. The index hit its all-time low of 28.48 during the summer of 1896.

The original DJIA was just that - the average of the prices of the stocks in the index. The current average is computed from the stock prices of 30 of the largest and most widely held public companies in the United States, but the divisor in the average is no longer the number of components in the index. Instead, the divisor, called the DJIA divisor, gets adjusted in case of splits, spinoffs or similar structural changes, to ensure that such events do not in themselves alter the numerical value of the DJIA. The initial divisor was the number of component companies, so that the DJIA was at first a simple arithmetic average. The present divisor, after many adjustments, is much less than 1, which means the DJIA, itself, is actually larger than the sum of the prices of the components. At the end of 2008, the value of the DJIA Divisor was 0.1255527090, and the updated value is regularly published in the Wall Street Journal.

Historical References in this Book

For Consumer Math… As a signer of both the Declaration of Independence and the Constitution, Benjamin Franklin is considered one of the Founding Fathers of the United States of America. His pervasive influence in the early history of the U.S. has led to his being jocularly called "the only President of the United States who was never President of the United States." Franklin's likeness is ubiquitous, and, primarily due to his keen understanding of the power of compound interest and constant advocacy for paper currency, has adorned all the variations of American $100 bills since 1928.

Image of American Currency

The Dow Jones Industrial Average (DJIA) is one of several stock market indices, created by nineteenth-century Wall Street Journal editor Charles Dow. It is an index that shows how certain stocks have traded. Dow compiled the index to gauge the performance of the industrial sector of the American stock market and, thus, the nation’s economy.

On December, 10, 2008, Bernard “Bernie” Madoff, a former chairman of the NASDAQ Stock Market, allegedly told his sons the asset management arm of his firm was a massive Ponzi scheme - as he put it, "one big lie." The following day he was arrested and charged with a single count of securities fraud, but one that accused him of milking his investors out of $50 billion. The 71-year-old Madoff was eventually sentenced to 150 years in prison.

Benjamin Franklin

Image of Benjamin FranklinThroughout his career, Benjamin Franklin was an advocate for paper money. He published A Modest Enquiry into the Nature and Necessity of a Paper Currency in 1729, and even printed money using his own press. In 1736, he printed a new currency for New Jersey based on innovative anticounterfeiting techniques, which he had devised. Franklin was also influential in the more restrained and thus successful monetary experiments in the Middle Colonies, which stopped deflation without causing excessive inflation. In 1785 a French mathematician wrote a parody of Franklin's Poor Richard's Almanack called Fortunate Richard. Mocking the unbearable spirit of American optimism represented by Franklin, the Frenchman wrote that Fortunate Richard left a small sum of money in his will to be used only after it had collected interest for 500 years. Franklin, who was 79 years old at the time, wrote to the Frenchman, thanking him for a great idea and telling him that he had decided to leave a bequest of 1,000 pounds (about $4,400 at the time) each to his native Boston and his adopted Philadelphia. By 1940, more than $2,000,000 had accumulated in Franklin's Philadelphia trust, which would eventually loan the money to local residents. In fact, from 1940 to 1990, the money was used mostly for mortgage loans. When the trust came due, Philadelphia decided to spend it on scholarships for local high school students. Franklin's Boston trust fund accumulated almost $5,000,000 during that same time, and was used to establish a trade school that became the Franklin Institute of Boston.

Charles Ponzi & His Scheme

A Ponzi scheme is a fraudulent investment operation that pays returns to investors from their own money or money paid by subsequent investors, rather than from profit. Without the benefit of precedent or objective prior information about the investment, only a few investors are initially tempted, and usually for small sums. After a short period (typically around 30 days) later, the investor receives the original capital plus a large return (usually 20% or more). At this point, the investor will have more incentive to put in additional money and, as word begins to spread, other investors grab the "opportunity" to participate, leading to a cascade effect deriving from the promise of extraordinary returns. The catch is that at some point, the promoters will likely vanish, taking all the remaining investment money with them. If the promoters wait around for too long, the whole scheme will collapse under its own weight, as the investments slow. Legal authorities also shut many of these schemes down, after a promoter fails to fulfill the original claims.

"Charles Ponzi" by Boston Library (21 December 2008/weekinreview/21rampell.html NYT); en.wikipedia.org - Boston Library (NYT); en.wikipedia.org. Licensed under Public Domain via Wikimedia Commons - http://commons.wikimedia.org/wiki/File:Charles_Ponzi.jpg#mediaviewer/File:Charles_Ponzi.jpgThe scheme is named after Charles Ponzi, who became notorious for using the technique after emigrating from Italy to the United States in 1903. Ponzi did not invent the scheme, but his operation took in so much money that it was the first to become known throughout the United States. Ponzi went from anonymity to being a well-known Boston millionaire in just six months after using such a scheme in 1920. He canvassed friends and associates to back his scheme, offering a 50% return on investment in 45 days. About 40,000 people invested about $15 million all together; in the end, only a third of that money was returned to them. People were mortgaging their homes and investing their life savings, and most chose to reinvest, rather than take their profits. Ponzi was bringing in cash at a fantastic rate, but the simplest financial analysis would have shown that the operation was running at a large loss. As long as money kept flowing in, existing investors could be paid with the new money. In fact, new money was the only source Ponzi had to pay off those investors, as he made no effort to generate legitimate profits.

Ponzi lived luxuriously: he bought a mansion in Lexington, Massachusetts with air conditioning and a heated swimming pool (remember, this was the 1920’s), and brought his mother from Italy in a first-class stateroom on an ocean liner. As newspaper stories began to cause a panic run on his Securities Exchange Company, Ponzi paid out $2 million in three days to a wild crowd that had gathered outside his office. He would even canvass the crowd, passing out coffee and donuts, all while cheerfully telling them they had nothing to worry about. Many people actually changed their minds and left their money with him.

On November 1, 1920, Ponzi pleaded guilty to a single count of mail fraud before Judge Clarence Hale, who declared before sentencing, "Here was a man with all the duties of seeking large money. He concocted a scheme, which, on his counsel's admission, did defraud men and women. It will not do to have the world understand that such a scheme as that can be carried out ... without receiving substantial punishment." Ponzi was sentenced to five years in federal prison, and after three and a half years, he was released to face 22 Massachusetts state charges of larceny. He was eventually released in 1934 following other indictments, and was deported to his homeland, Italy, as he hadn't ever become an American citizen. His charismatic confidence had faded, and when he left the prison gates, he was met by an angry crowd. He told reporters before he left, "I went looking for trouble, and I found it."

Video Lesson on Percent Change

What this quick video lesson about Percent Change


(1) Check yourself!

Who created the Dow Jones Industrial Average?

  • Charles Dow
  • Thomas Jones
  • Dowen Jones
  • Chester Dowen

(2) Check Yourself!

A ___________ is a fraudulent investment operation that pays returns to investors from their own money or money paid by subsequent investors, rather than from profit? 

(3) Check Yourself!

  • Benjamin Franklin has adorned all the variations of American $100 bills since 1928

Charles Ponzi was responsible for ____________________________

  • Put your answer option here
  • Put your answer option here