How to cut pizza

The title of the post is pizza. But the real story is actually hamburger patties. Knowing how to cut a pizza, or rather knowing the relative size of a slice of pizza will make the story about beef patties interesting. The cutting of the pizza is shown in the following pictures.

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.

From the last picture, it is clear that one-third of a pizza is bigger than one-fourth or one-quarter of a pizza. The same relativity would apply for other things too. One third of a loaf of bread would be more than one-fourth of the same loaf. Sitting at the doctor’s office waiting for one-third of an hour would be a longer wait than sitting there for one-quarter of an hour. One-third of a gold bar would be more valuable than one-quarter of a gold bar.

At this point, I hope you agree that one-third of any thing is more than one-fourth of that same thing. In particular, one-third of a pound of ground beef would be more meat than one-quarter of a pound of beef. A hamburger patty weights one-third of a pound would contain more beef than a patty that weights one-quarter of a pound.

According to this article in New York Times, many people think the opposite, that one-third pound of beef is less meat than one-quarter pound of beef! The article is a long one about math educational reform efforts in the United States. The following are the two paragraphs relevant to our discussion.

    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.

The participants in the focus groups believed that 1/3 is a smaller number than 1/4 because the 3 in 1/3 is smaller than the 4 in 1/4. How about the public at large? The fact that the new one-third pounder of Whataburger (the parent company of Whataburger is A&W) was a commercial flop while the new burger was favored at taste tests makes it plausible that this was indeed arithmetic failings on the part of the American consumers.

There are two take-aways. One is arithmetic. The fraction 1/3 is bigger than the fraction 1/4. Or put it another way, the fraction 1/4 is smaller than 1/3. This is visually demonstrated in the series of pictures above. The fraction 1/n refers to the situation of dividing one unit of a thing into n equal pieces. More dividing means each fractional piece is smaller. You can view 1/n as the division of one thing among n people. The more people in the division, the smaller each share is for one person. Thus when the number n in the denominator 1/n gets larger, the smaller the share each person would get.

In the pizza example, the more people want to take a share of the same pizza, each person would get a smaller piece. With one million dollars shared by 2 people, each person would get half a million dollars. But with one million dollars shared by one million people, each person can only get one dollar! Again, when the denominator gets larger, the fraction become smaller.

Another way to know the relative size of a fraction is from using a calculator. One divided by 4 gives 0.25, while one divided by 3 gives 0.3333. Note that 0.25 is smaller than 0.3.

The other take-away is that this example is a vivid example of what author John Allen Paulos called innumeracy, which is the mathematical equivalence of not knowing how to read. In fact, he authored Innumeracy: Mathematical Illiteracy and its Consequences. This book is a good read for anyone who wants to improve his or her numeracy or for anyone who wants to understand the issue of innumeracy. It is not a cure for innumeracy, but is a good start.

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


Better math skills may help you keep your home

Math is in the news again. It’s not about the solution of a century old math problem. It is not about the math behind an earth shaking new technology. This time the focus is on the humble basic math skills such as how to calculate the price of a sofa in a clearance sale or how to calculate the earned interest in a savings account. The math news that is in the spotlight points to a tantalizing possibility that if enough people could practice this kind of simple basic math, the deep global recession that occurred a few years ago might have been averted, or its impact might have been greatly reduced.

According to a new study published in the National Academy of Sciences, there is a strong negative correlation between basic math and quantitative skills and the likelihood of defaulting on a subprime mortgage. In other words, borrowers with lower math skills are much more likely to default on their loans and borrowers with higher math skills are much less likely to default on their loans. It is well known that a massive increase in the volume of subprime mortgage defaults in years 2006 and 2007 helped push the economy into a deep global recession in 2008.


The Study

The study is based on data from subprime mortgages originated in 2006 and 2007 in greater New England area as well as data from a phone survey involving 339 holders of these subprime mortgages. The phone interviews assessed the numerical ability, financial literacy, verbal skills and cognitive ability of the 339 participants. Numerical ability in the study refers to the proficiency of a borrower for solving basic mathematical calculations.

One of the results of the study is that the numerical ability is a strong predictor of which borrowers will be more likely to be delinquent in loan payments or to be foreclosed upon (the lower the numerical ability, the higher the likelihood of default).

“Our analysis raises the possibility that limitations in numerical ability may have significantly contributed to the massive amount of defaults on subprime mortgages in the recent financial crisis,” concluded the researchers of the study.

Could it be that the loan delinquency of the participants of the phone survey was due to other factors such as income, education and credit scores? For example, could the results of the study be due to the low credit scores of the participants? If the participants of the study had defaulted prior to the study, they would be more likely to default again.

After the researchers of the study adjusted for these factors, the connection between poor math skills and higher default rates remain the same. As a result, the researchers concluded that it is likely the effect of the math skills that is driving the results and not the credit scores and other factors.

The researchers of the study also did not find any significant connection between the poor math skills and the types of mortgage loans among the participants of the study. The correlation between poor math skills and higher loan delinquency is present across all the loan types. So it is not that the borrowers in default happened to choose risky loan types such as adjustable mortgages or negative amortization mortgages. So the study suggests that restricting these risky types of mortgages may not prevent future mortgage default epidemics.


Some Comments

So all eyes now point to math and quantitative skills. What kind of numerical questions were asked in the phone survey? They are not calculus problems and they are certainly not advanced math in financial modeling. They are:

  1. In a sale, a shop is selling all items at half price. Before the sale, a sofa costs $300. How much will it cost in the sale?
  2. \text{ }

  3. If the chance of getting a disease is 10 per cent, how many people out of 1,000 would be expected to get the disease?
  4. \text{ }

  5. A second hand car dealer is selling a car for $6,000. This is two-thirds of what it cost new. How much did the car cost new?
  6. \text{ }

  7. If 5 people all have the winning numbers in the lottery and the prize is $2 million, how much will each of them get?
  8. \text{ }

  9. Let’s say you have $200 in a savings account. The account earns ten per cent interest per year. How much will you have in the account at the end of two years?

We know math is useful. In fact, this study provides empirical evidence that math pays. Having proficient level of math and quantitative skills can lead to favorable and optimal financial outcome in a person’s life. Having good number skills can help any person makes better financial decisions and better manage his or her financial affairs. At least, having good quantitative skills can help avoid costly financial dislocation that comes from defaulting on mortgages, as the study suggests.

The authors of the research study also indicated that their “results indicate possibly large benefits from increased financial education of homeowners.” Certainly, greater emphasis in financial education (and quantitative skills) in high school can have a positive effect on financial outcomes later in life.

Our view is that we do not need to wait for school reform to take place. We can make changes at the individual level. The above five questions are not math problems that can only be taught in school. These are problems that we encounter in our daily life. So in order to improve on quantitative skills, we do not necessarily need to go back to school to take a course in finance. We can improve on quantitative skills one purchase at a time and one bank transaction at a time. Come up with a right question. Then take out a calculator and a notepad and try to resolve the issue that is at hand.

Some students cannot wait for schooling to be over so that they don’t have to deal with math any more. This jaundiced view of math is unfortunate. There is school math and there is math in everyday life. This study published in the National Academy of Sciences shows that it pays to have skills in the latter.


Further Information

Here’s a few news outlets that reported on the same study.

This is the study published in the National Academy of Sciences.