Table 3-1. Common Unit Factors

Let's play a little game. The object of the game is to answer a question and you win the game by making the units on either side of an equation equal to one another. The first move of the game begins with the question, for example:

Q: How long does it take to earn enough money to pay off the shipbuilder when you buy a boat of 60 gur and rent it out?

The next step in the game is to translate this sentence into an equation. The verb in the sentence becomes the "equal" sign:

how long = money for a boat

We then assign units to the items in the equation:

days = 1 boat

Next we build a chain of unit factors connecting the unit of the answer to the unit of the question. A unit factor is simply unity, the number one, albeit written in a funny way. A boat of sixty gur, for example, costs 2 shekels according to the Code of Hammurabi. Therefore, 1 boat = 2 shekels. Therefore, (1 boat/2 shekels) = 1. Similarly, when I rent a boat the going rate is a shekel every six days. Therefore, 1 shekel = 6 days. Therefore, (1 shekel/6 days) = 1. The reciprocal of any unit factor is also a unit factor, so (2 shekels/1 boat) =1 and (6 days/1 shekel) = 1. Unit Factor Analysis works because we can always multiply by one without changing an equation; if 12=4x3, then 12=4x3x1 and 12=4x3x1x1. That's what makes "one" such an interesting number. So returning to the game, we build a chain of unit factors which cancel out the units we don't like and introduce units we do like. The game is over when the units are the same on either side of the equal sign. Here's the complete example:

Q: If a man buys a boat of 60 gur and rents it out, how long does it take to earn enough money to pay off the shipbuilder?

I took a couple of unit factors right out of the Code of Hammurabi. Can you find where they came from? Sometimes unit factors are given right in the problem. Sometimes, you have to look them up, but it helps if you have an arsenal of unit factors at your disposal. Table 3-1 gives you something to start with.

Notice that I actually wrote two equations, or mathematical sentences, one on top of the other. This is a way of keeping a long, complicated problem from getting out of hand. Here, "? day" equals (the first mess), and it also equals "12 days." Stacking the equal signs this way, I can show all the unit factors I used and then simplify the expression to get the answer. Let's try a more involved problem:

Q: How long does it take a car to go 100 yards when it is traveling 60 miles per hour?

To use UFA, you must identify the unit of the answer, and you may have some choice in this. For the current problem, what would be a reasonable unit? Obviously a unit of time: hours, minutes, seconds, any of these would do. Let's choose seconds. Translating the verb into an equal sign, our question could be rephrased:

Q: How many seconds equals 100 yards?


Now, a second is not the same as a yard, so we need a unit factor to get rid of this problematic unit on the right-hand side of the equation. There are many such unit factors, in fact, infinitely many of them. I won't agonize over the choice, I'll simply choose one and see if it gets me anywhere. Let's try (3 feet/1 yard).

Notice, I put the yard in the bottom of the unit factor to cancel the problematic yard in the top, the one I need to eliminate. This leaves feet, which are not the same as seconds, so I need another unit factor to get rid of feet. Let's try (1mile/5280 feet). This gets rid of feet but leaves miles, which are not the same as seconds. It seems that we are getting nowhere, but we are, in fact, on the verge of a breakthrough. Of all the unit factors we might write concerning miles, one stands out because it was given in the problem: (60 miles per hour) = (60 miles/hour). The word per is another way of saying "divided by", and in a word problem always indicates a unit factor. We need the miles in the bottom of the unit factor, so we just turn it upside down. Now we're getting somewhere! Hours remain on the right and this can be converted to seconds with two more unit factors. Unit factors have been used to connect the units of the question to the units of the answer.

All that remains is the arithmetic.

You may wonder how I knew to use (5280 feet/1 mile) rather than, for example, (1 foot/12 inches). The short answer is that I didn't know it for sure; I just tried it out to see whether it got me anywhere. Since unit factors are equal to one, and multiplying by one doesn't change a number, I can never go wrong with a unit factor. But some unit factors get me somewhere (toward the unit of the answer), while others don't. The slightly longer answer is that I knew from the statement of the problem, that "60 miles per hour" was a unit factor which contained both distance (the dimension of the problem) and time (the dimension of the answer) and so I knew that if I could get from yards to miles, I could get from miles to hours.

Try playing the game with these examples:

Q: A tap delivers 2 gallons per minute. How long does it take to fill a tub which measures 2 feet by 3 feet by 8 inches?
Q: "Investment" is a kind of plaster consisting of 1 part silica, 1 part plaster and 1 part water. How many grams of each are needed to fill a circular dish 15 cm in diameter to a depth of 5 mm? Investment has a density of 1.5 g/cm3.


Notice that I stacked three equations here, each one equal to "? min." But I ran out of room on the first line, so I continued it on the second, just as I would continue a sentence that ran longer than one line. The first three lines are all part of one long equation. The fourth and fifth lines are a second equation, equal to the first, and the sixth line is a third equation, equal to the other two.

Why did I go to the trouble of writing the second equation at all? Because I wanted to show how you can combine two or more unit factors that are identical. For example, (12 in/1 ft) appears twice. I can simplify my equation by writing (12 in/1 ft)2. I just have to remember that everything inside the parentheses is squared, both the numbers and the units. By contrast, in (1 mL/1 cm3), only the cm is cubed, since the "3" is inside the parentheses. It's really not so hard, but you have to pay attention. Get out your calculator and make sure you can do the arithmetic both ways.

Either you're bored at this point because you've seen this before, or you're bewildered because anything mathematical scares you. If you're bored, you've probably skipped to the next chapter by now anyway, so I'll just add a little more for the bewildered guys. How did I know to start with "2 ft" instead of "2 gallons per minute?" The short answer is that I could just as well have started with (1 min/2 gal), turning this unit factor upside down so that "minutes," the unit of the answer, is in the top. The slightly longer answer is that I started with "2 ft" because I knew, intuitively, that if I doubled the length of the tub, it would take twice as long to fill. When doubling something in the problem would double the answer, we know it goes on top. The same thing was true for the width and the depth, so they also went on top. Conversely, if I were to double the filling rate from 2 gal/min to 4 gal/min, I know, intuitively, that it would take half as long to fill the tub. When doubling something in the problem would halve the answer, I know that it goes in the bottom, i.e. for this unit factor, upside down. One more clue is the word "by" in the problem. Just as the verb of a sentence translates into an "=" sign, by translates into multiplication.

One of the little tricks memes use to ensure their own survival is the mnemonic, a little phrase or jingle intended to jog the memory. So if you forget how Unit Factor Analysis works, here's a little mnemonic to help you remember:


That which is above corresponds to that which is below and that which is below corresponds to that which is above in the accomplishment of the miracle of One Thing. And just as all things come from One, so through the mediation of One, all things follow from this One Thing in the same way.

 The Emerald Tablet of Hermes Trismegistos

That is, the numerator of a unit factor is equal to the denominator and vice versa. Everything in the answer follows from a string of "one things," or unit factors. If you're still bewildered, don't give up. I promise you that I will teach you as little as possible. Instead of giving you different methods for working all the complicated problems you'll run into in this book, I'm going to teach you one method, and we'll use it over and over. Think of it as the Swiss Army Knife for numerical problem solving.

WarningMaterial Safety

So far, this book has talked about sticks, stones, and other materials with which you have some familiarity. The only hazard in completing this project is the risk of banging your head against a wall as you struggle to learn the intricacies of UFA. But as long as I have your attention for a moment, let me introduce you to Material Safety Data Sheets. MSDS's are required by the United States Occupational Health and Safety Administration (OSHA):


Chemical manufacturers and importers shall obtain or develop a material safety data sheet for each hazardous chemical they produce or import. Employers shall have a material safety data sheet in the workplace for each hazardous chemical which they use.[1]

While required only for hazardous chemicals used in the workplace, many retailers provide MSDS's to consumers upon request. MSDS's are also available online for hazardous and non-hazardous chemicals alike. Since many chemicals have more than one name, the reliability of online searches may be improved by using the Chemical Abstracts (CAS) number instead of, or in addition to the name. CAS numbers for the chemicals discussed in this book are given in each Material Safety section. Though MSDS's include more technical information than most consumers require, they're one of the most readily-available sources of information on hazardous materials and so you should familiarize yourself with them.

Since this project uses no materials at all, you'll introduce yourself to MSDS's by finding them for some common items. Look up charcoal (CAS 7440-44-0), silica (CAS 14808-60-7), and sodium chloride (CAS 7647-14-5). By familiarizing yourself with the hazardous properties of relatively safe materials, MSDS's for materials we will meet later won't seem so intimidating. You may request your sheets from a retailer or you may search for them online using the keyword "MSDS" and the CAS number for the chemical in which you are interested. Particularly on the Internet, there is nothing to prevent the posting of bogus information, so you should always consider the source of your information; a genuine MSDS must include a way to contact the manufacturer.

There is a lot of technical information given on a typical MSDS and we'll look at several of these as the book continues. But every MSDS includes a section—usually the third section—which summarizes the hazards of the material. For this project, summarize the hazardous properties of charcoal, silica, and sodium chloride in your notebook. Include the identity of the company which produced the MSDS and the potential health effects for eye contact, skin contact, inhalation, and ingestion.

NoteResearch and Development

So there you are, studying for a test, and you wonder what will be on it.



Reference [38], Regulation 1910.1200(g).