What do you make of the following restaurant receipt? The diner left $\pi$ as tips. A few things conspire to make this a “cute” receipt. The amount of the dinner ($26.86) is such that when it is rounded up to the nearest ten, the difference is the common approximation of the number Pi (3.14). Maybe that dining experience happened on the 14th day of March or maybe not. However, the number Pi is in the news today (with today being March 14). This is a piece from CNN.com. This is a piece from Time.com. This is a mention of Pi from npr.org last year. The number Pi is purpose driven math, from mundane tasks such as building a dining room table to advanced calculation of orbits for space travel. In general, pi is used to study everything that cycles, e.g. movements that repeat at fixed intervals (think heart pulses and many periodic motions in the natural world). There are other uses of Pi that are less obvious. One is that Pi is used is used in testing computer precision. The other is that the digits of Pi are used as random numbers since they behave like random numbers. Here’s is a discussion of the digits of Pi as random numbers in a companion blog. __________________________________________________________________________ $\copyright$ 2017 – Dan Ma Advertisements # Knowing fractions helps find better deals Fractions may get a bad wrap. It is undeniable that one must have a good understanding of fractions in order to be good with with numbers. Here’s an example where knowing fractions will help a consumer find a better deal. According to this article in New York Times, many consumers in the United States funked an “arithmetic test.” The article is a long one on math educational reform efforts in the United States. The following two paragraphs are crucial for the story here. One of the most vivid arithmetic failings displayed by Americans occurred in the early 1980s, when the A&W restaurant chain released a new hamburger to rival the McDonald’s Quarter Pounder. With a third-pound of beef, the A&W burger had more meat than the Quarter Pounder; in taste tests, customers preferred A&W’s burger. And it was less expensive. A lavish A&W television and radio marketing campaign cited these benefits. Yet instead of leaping at the great value, customers snubbed it. Only when the company held customer focus groups did it become clear why. The Third Pounder presented the American public with a test in fractions. And we failed. Misunderstanding the value of one-third, customers believed they were being overcharged. Why, they asked the researchers, should they pay the same amount for a third of a pound of meat as they did for a quarter-pound of meat at McDonald’s. The “4” in “¼,” larger than the “3” in “⅓,” led them astray. It is undeniable that many of the consumers who participated in the focus groups conducted by A&W flunked the arithmetic test in question. How about the public at large? The failure of the new burger of Whataburger in the market place suggests that a large swath of the consuming public probably flunked the test too. The take-away for any company that conducts a marketing campaign: don’t give the consumers a math test. The take-away for the consumers who are confused with fractions: 1/3 is indeed bigger than 1/4. The following pictures show why. $\text{ }$ $\text{ }$ Figure 1 This is a whole pizza. Only one slice. The relative size of a slice is 1. Figure 2 Cut the pizza in two equal slices. The relative size of a slice is one-half or 1/2. Figure 3 Cut the pizza in three equal slices. The relative size of a slice is one-third or 1/3. Figure 4 Cut the pizza in four equal slices. The relative size of a slice is one-fourth or one-quarter or 1/4. Figure 5 Put the pizza slices together from the smallest to the largest. In the last picture, it is clear that one-third of a pizza is indeed larger than one-quarter of a pizza. Cutting a pizza, real pizza or pictures of pizza as we are doing here, makes this point clear. This relativity holds true in other settings too. One-third of a loaf of bread is bigger than one-fourth of the same loaf. Waiting at the doctor’s office for one-third hour is a longer wait than waiting for one-quarter of an hour. One-third of a gold bar is more valuable than one-fourth of a gold bar. One third of a million dollars is more money than one-quarter of a million dollars. Of course, one-third pound of beef has more meat than one-quarter pound of beef. The list can go on and on. The insight about fractions can follow from a general reasoning too. The fraction 1/n refers to a division of a thing into n equal pieces. The more units the division produce, the smaller each piece will be (not bigger). Take pizza for example. If more people want to share a pizza equally, the smaller the slice will be for each person (not bigger). In dividing a million dollars ($1,000,000) equally between two people, each person gets half a million dollars (\$500,000), a very large sum of money. If the same million dollars are divided equally among one million people, each share will be just one dollar. The more people sharing a quantity will lead to a smaller share for each person.

So the larger the denominator in the fraction 1/n, the smaller the number 1/n.

Knowing how to work with numbers, fractions in particular, is purpose driven math. Such skills are indispensable to academic success and career success. The example with Whataburger shows that understanding fractions can help a consumer spot a good deal too.

This blog post is adapted from a post in a companion blog.

Elizabeth Green is the author of the New York Times article mentioned above. Here’s another NY Times article that discusses the article by Elizabeth Green.

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$\copyright$ 2017 – Dan Ma