Fourth Grade Math or Advanced Engineering Math… “It’s all in the application"

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Tony Catalfomo, M.S.E.E., P.E.

Fourth Grade Math or Advanced Engineering Math… “It’s all in the application"

This topic is near and dear to my heart. You see, my wife is a Math Professor at a local technical college. She is the best kind of math professor because she has an engineering degree, an advanced math degree, and understands the proper application of simple and complex functions in real-world applications. As grounded a math teacher as there ever was, she will tell you, the difference between simple and complex math functions is not always obvious. Take, for instance, “the case of the missing dollar.” I’m sure some of you are familiar with the story, but for those of you who aren’t, here it is:

“The Case of the Missing Dollar”

Three tool salesmen were traveling in a remote part of the country when their car broke down. Fortunately, the car broke down near a small hotel. It was late, so the salesmen decided to spend the night at the hotel before trying to repair the car. The hotel only accepted cash payments. Each salesman was carrying only ten dollars. The innkeeper told them they could get a room for thirty dollars, which worked perfectly for the salesmen. They booked one room for the night.

Being tool salesmen, they were each carrying heavy luggage containing samples of the tools they sold. The bellhop carried their heavy bags to the room. The salesmen saw he was exhausted from carrying the luggage but had no tip to give. They apologized saying they gave all their money to the innkeeper.  

The bellhop returned to the main desk, exhausted and covered in sweat. The innkeeper realized he had made a mistake and overcharged the salesmen. The room was only $25 per night. He gave the bellhop 5 dollars in singles to return to the salesmen. The bellhop was not impressed by the salesmen and felt they would take the money and not tip him for his services. He also figured they wouldn’t be able to split the 5 dollars evenly, so he pocketed 2 dollars and would return 3 dollars to the salesmen. He would get his tip and they would get money back from being overcharged. His assumption was correct as the salesmen took the change but gave some lame excuse for not wanting to tip him.

Everybody was happy. The innkeeper had been paid for the room. The salesmen had a room for the night for less than expected. The bellhop got his well-deserved tip. So, how much money did each salesman contribute for the room?  I’m thinking nine dollars...

{$10 (original) -$1 (change) = $9}, therefore, the room cost the salesmen $27 dollars 
($9 x 3 salesmen = $27).  The bellhop kept $2 dollars. If we total the two amounts; $27 (for the room) + $2 (retained for the tip) = $29.  But wait, they gave the innkeeper $30.00.  $30 – $29 = $1, where’s the missing dollar? 

Let’s review the math:
10 – 1 = 9
9 x 3 = 27
27 + 2 = 29
30 – 29 = 1

Simple fourth-grade math with addition, subtraction, and multiplication, but for some reason, the math doesn’t seem to add up. There is a very “simple” solution to this “complex” problem.  It’s all in the application of the math principles. There is nothing wrong with the math equations, it’s all in how the math is being applied. So again, is the math correct? Where is the missing dollar?  

My father-in-law, also a pretty sharp math guy, had a saying for this conundrum, “figures don’t lie, but liars figure.” Now that may be a bit cynical, but he’s not completely wrong. His premise was someone can use simple math, or principles, to guide others, who are not as well versed in a topic, to get a desired conclusion. Sometimes the desired result may be nefarious. This is the art of the “flim-flam.”

A Brief Case Study In Code Application

To be fair, sometimes the improper application of math concepts, or principles, may not be nefarious. It may simply be subconsciously guided by our desire to obtain a desired result. We often see this when applying codes. We can find codes where the words say exactly what we want, or need, but unfortunately, the code may not be applicable to our specific situation.

Take for instance the time I was notified of an issue regarding conductors overheating in conduits contained in an underground duct bank at a large commercial building. It didn’t take long to notice the cables in the duct bank were undersized and the conduits were overfilled, meaning they had too many cables. The cables were overheating causing degradation and breakdown of the insulating material. The result was a catastrophic failure of the conductors. The cost associated with the failure resulted in several millions of dollars of property damages and lost business.

The issue was discussed with the designer who emphatically insisted the cable ampacity ratings and conduit fill was per the National Electrical Code (NEC). The designer proceeded to support his claim by providing tables showing cable ampacities and the number of cables in the conduit was per the NEC. He was right, the cable ampacities and conduit fill were per the tables he referenced in the NEC. The only problem was he referenced tables that considered “cables in free air.” These cables were underground in a duct bank and required derating. A completely different set of tables and calculations were required for this application. He had misinterpreted the tables and misapplied the code.

Understanding the proper application of regulatory compliance issues and codes is important. We have seen on numerous occasions where code issues are taken out of context or improperly applied, resulting in physical damage or incurring unnecessary costs. There are numerous examples that can be discussed. At ED&T we pride ourselves on the ability to assist our clients in the interpretation and application of simple and complex issues. Our vast experience and knowledge in varied engineering disciplines has been and will continue to be, a valuable asset to our clients.

So back to the case of the missing dollar.  If this little riddle still perplexes you, please call me and we can discuss the “solution.” Don’t be embarrassed to call.  Trust me, I have presented this riddle to all kinds of people over the last 30 years, you are in good company. Until next time, try not to break down near a cheap hotel with a cynical bellhop, or at least remember to tip him.

About the Author

Tony Catalfomo, M.S.E.E., P.E. is a Consulting Engineer in our Charlotte, NC Office. Mr. Catalfomo provides consultation in the areas of electrical engineering, with an emphasis in industrial power distribution and controls including electrical failure analysis, damage assessment, and repair/replacement analysis. You may contact him for your forensic engineering needs at or (704) 523-2520.

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