Michael B. answered • 11/13/12

I can provide your 'A-HA' moment

Units (also called "dimensions") are funny things, aren't they?? They actually change the MEANING of the numbers that they are attached to... So "3 cups" is obviously not the same as "3 gallons". It's something most people take for granted.

But when attempting to convert units, there is ONE overriding principle that you should remember - you are just trying to change what they are called, and at the same time, NOT change how much stuff you have. With this as your perspective, it's easier to see how to convert units using a method called "dimensional analysis".

The basic idea of dimensional analysis is that you are converting the units, one at a time, in a methodical manner, in a way that "preserves" the original value if the units are accounted for. As you proceed, the units that you don't want are replaced by the units that you do want.

Let's start with a more complicated example than what you are asking, then we'll get to your answer... For example, how fast is 55 miles per hour in inches per second? We know that 1 mile has 5280 feet, 1 foot has 12 inches, 1 hour has 60 minutes, and 1 minute has 60 seconds. If you wanted to know how many inches are in a mile, you would do the following:

`55 miles | 5280 feet | 12 inches | 1 hour | 1 min`

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1 hour | 1 mile | 1 foot | 60 min | 60 sec

Each number is just a standard "multiply" or "divide" depending on whether it's in the top row or the bottom row... Thus, ignoring all of the 1's, you would start with 55, multiply by 5280, multiply by 12, divide by 60, then divide by 60 again. Note also that each unit "cancels" until we end up with what we want - miles in the top cancels miles in the bottom, feet in the top cancels feet in the bottom, etc. What is left over (i.e. what doesn't cancel) is "inches" in the top and "seconds" in the bottom, or inches/sec, which is what we are looking for. The numerical answer in this case is 968 inches / sec.

Furthermore, * and of crucial importance to understand*, each column (eg 5280 feet / 1 mile) is a ratio that is "equivalent to 1" if you account for the units. In other words, "5280 feet is equivalent to 1 mile" and so this number is "equivalent to 1" when the units are accounted for. Because these ratios are all equal to 1, we are multiplying the original value by a bunch of 1's, we aren't changing the PHYSICAL value of the answer. We are changing the numerical value, but it doesn't matter because we are also changing the units in just the right way so that the two "cancel out". The only time you don't use a factor that is equivalent to 1 is if you ACTUALLY want to change the amount. For example, in your problem, the question says to "triple" the amount of flour.

The hard part (if you can even call it that) of dimensional analysis is simply knowing the right conversion factors, but these are often easily found in a reference book or by looking them up online.

Thus, to answer your question using "dimensional analysis", we would need to know the conversion factor from "cups" to "tablespoons", and the answer is that 1 cup is equivalent to 16 tablespoons. So we use that to solve:

`3 cups | 3 | 16 Tbsp`

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1 | 1 | 1 cup

And we are done... We start with 3 cups, then multiply by 3 to triple the recipe, then convert with the "equivalent to 1" factor 16 Tbsp = 1 cup by multiplying by 16. Cups cancels with cups and we are left with tablespoons. When you multiply the numbers together, you get **3 * 3 * 16 = 144 Tablespoons**.

The nice thing about using this method is that it helps to convince you that you are multiplying and dividing correctly. That is, sometimes people get confused about whether to multiply by a conversion factor or divide by it. Well, if you write it as an "equivalent to 1" factor with the units, then it is much more clear whether you multiply or divide.

Michael B.

In addition to this response, I've created a blog entry for it:

http://orlandotutor.wordpress.com/2012/11/13/dimensional-analysis/

11/13/12