Business Math

by breeve 4. April 2010 10:22

In my family—in laws included—I am the engineer weirdo performing voodoo magic to make computers do things. The majority work in business related fields and tout MBA degrees with years of experience to back them up but somehow still think that no matter where I work the company must be creating some kind of operating system. In this type of family environment discussions consist not only of how are the kids or pass the mashed potatoes but things like calculating percents—like say 35% of 60—or calculating how much money will appreciate at 8% over 30 years. If there is one thing I have learned it’s that nothing will bring quicker looks of disdain than to trip up calculating percents to the exact number. Close enough doesn’t cut it.

At first, I didn’t know whether to feel disturbed or envious at the speed at which these calculations came spewing out of their mouths like hungry bats emerging for a night feed. The disturbed part of me felt they had spent way too much time practicing the skill; after all, anyone could grab a calculator to do the same thing. The envious part of me couldn’t help but admire it for a computer program would be hard pressed to beat them. Still, I couldn’t pinpoint why this was so important.

Years later, while reading business books, it dawned on me that in the world of understanding how businesses work calculating percentages is essential. Not knowing them would be like coming across a program without for loops causing the intelligence of the author to be questioned.

And like for loops in programs, percents are everywhere in business. What percent of revenue is spent on R&D vs. Sales vs. Marketing? How do those percents compare to our competitors? What percent is our revenue going to increase? Every thought and comparison is expressed as a percent. Spend any time studying financial statements and there is no way to escape them.

Despite this, it took me—who took many courses in calculus and physics—years before I understood how to calculate a 15% tip in my head. Shamefully, I have found that many engineers also struggle with this skill. It is time we stop being shown up by our business associates and learn how to calculate percents and value appreciation in our heads without the help of our favorite programming language.

Percent calculation

Remember Chunk from the 80’s movie "The Goonies". His name not his character—the paranoid husky kid who would rather be eating rocky road ice cream than contributing to the goal of finding the treasure of one-eyed Willie—holds the secret to calculating percents in your head. The technique that resembles his name, called chunking, is commonly applied to memorizing numbers by breaking the problem into groups. Like numbers, percent calculation can be broken down into groups to simplify things.

First rule of percent chunking is 10%. 10% is easy to calculate because you move the decimal one place to the left. To reach 1% you simple do the 10% calculation twice in your head resulting in the decimal place moving two times to the left.  Calculating 5% is easy once you have the 10% number because you just divide it in half.

In our example above of taking 35% of 60 we first take 10% which is 6—simply move the decimal place of 60.0 one to the left. Now, we multiply that number by 3 to get to the 30% number of 18. We know that 5% is just half of 10% which is 3 in our case. Add that to 18 and we get the final answer of 21. So mathematically this can be chunked like: .35x = .30x + .05x = (.10*3*x) + (.10*(½)*x) = [(.10*x)*3] + [(.10*x)*(½)]. Substituting 60 in for x using the last equation gives us the same thought process previously worked through. The key is to master this chunking technique in your mind.

You can use subtraction as well. Say we want to calculate 18% of 95—which I just made up and figured to be 17.10. Here is the chunking equation: .18x = .20x - .02x = [[(.10*2)*x] - [(.10*.10)*2*x] = [(.10*x)*2] - [(.10*x)*.10*2] = (9.5*2) - [(9.5)*.10*2] = 19 – (.95*2) = 19 – 1.9 = 17.10.

Practice doing random problems in the shower every morning and at the very least you will amaze the helpless Gap employee who can’t seem to calculate 20% of anything without the register.

Value Appreciation

Calculating how your principle will grow after x years at y percent is easy if you know the secret Rule of 72. After discovering it on my own I would be lying if I didn’t admit I was angry at my family for not informing me of it.

The rule, which is surprisingly accurate and powerful, is simply: take 72 and divide it by the interested rate to get the number of years it will take to double. So back to our example of 30 years at 8% which is a common retirement account calculation. The rule of 72 says that at 8% your retirement money will double every 9 years—72 / 8 = 9. So if you have 20k in your account, in 27 years you should have 160k—3 doubles: 40k, 80k, and finally 160k. If the calculation is done manually—20k*1.08^27—it gives 159,761.

Thinking more about the rule will reveal some interesting facts beyond retirement appreciation. For example if inflation is 4% a year, how long will it take before prices double? The rule of 72 says 18 years; so in 18 years if your money just sits in an account somewhere earning meagerly interest its purchasing power will be cut in half. Even more depressing, that sweet little baby you hold in your arms now will grow up and when they are 18 years old and ready to go to college everything will be twice as expensive.

Bonus - Multiplication

Cost per month is a common price scheme; like a car payment of $250 per month. To calculate the price per year effectively in your head, it is important to move in chunks of 10. To do this, factor 250 into tens like 100*2.5 and then multiply by 12 to get final chunks of 12*100*2.5. Taking 12*100 gives 1200. We need 2 and a half of those. A half is 600 and 2x is 2400. Add them together to get 3000.

A better approach to the same problem is to rearrange the numbers like 100*(12*2.5). This allows the more complicated math to be done on small numbers which becomes more important as the numbers get larger. Thinking 2.5 times 12 is 30—[2*12]+[(½)*12]—then multiplying by 100 (or equivalently 10 two times) to get 3000 is easier than multiplying 12 by 100 and taking 2.5 of that.

A third option is to use elementary algebra and the distributive law like 12*250 = (10 + 2)*250 = (10*250) + (2*250). This gives us the nice number 10 in one factor and the low number of 2 in the other. The math from here is easy to do in your head. 2500+500 = 3000.

The three methods above were given not to show which one is better but to give three different methods for attacking multiplication. For certain numbers, a particular method may work better which makes knowing as many methods as possible essential for doing fast multiplication in your head.

One more example. Employees get paid $1500 a month and there are 550 of them. What is the expense per month? 100*15*100*5.5 = 100*10*10(15*5.5) = 1,000*10[(15*5) + (15*(½)] = 1000*10*82.5 = 10*82,500 = 825,000.



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About Me

I am a Principal Engineer with 16 years experience developing and releasing software products. I started developing in C/C++ then moved into .NET and C#. Currently working in Python/Flask and Docker. Have tech lead multiple projects. I have developed products in Windows Forms, ASP.NET/MVC, Silverlight, WPF, and Python. I currently reside in Austin, Texas.

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