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Jump to one of the following dimensional analysis parts:   Choose One Introduction to Dimensional Analysis Part 1- The Need for Units Part 2- Setting up Conversions Part 3- Basic Conversion Problems Part 4- Complex Conversion Problems Part 5- Problem Dissection I Part 6- Problem Dissection II Part 7- Density Problems Part 8- Quizzes

It's not enough to simply get an answer. You need to understand how you got the answer in order for it to be useful. Enough so you can apply the method to problems you haven't seen before and still think your way through the problem to the end.

If I were to ask you how many inches in 3.0 feet, you should be able to answer fairly quickly "36 inches". And you'd be right. However, let's break down the problem so you can have a method with which to work.

Step 1- what's being asked? I realize that this seems very obvious but that's only because of the simplicity of the question. "How many inches are there in 3.0 feet?"

Step 2- write down what's given. What numbers are you given to work with? In this problem, all that's been given is that you have 3.0 feet.

Step 3- write down any necessary formulas and conversions. There aren't any formulas to worry about (like density, the ideal gas formula, specific heat, etc.) but we do have a conversion to worry about-

Just a quick note that you now need to understand when a conversion is exact or not. Aside from the unit conversions' page that you can bring up, none of the conversions in this tutorial will explicitly state whether they are exact or not.

Step 4- set up the problem and do the calculation. The basic setup is the following-

Pretty easy, huh? Heh. Not always. The hardest part is deciding just how to put in the conversion. Remember in the previous section how there are always two choices when putting a conversion in a mathematically useful form? As in-

  or  

Which one? Simply put, which one allows us to have an answer with the proper units? This is where we use the idea of multiplying and dividing units. If you've taken any of my classes or if you happen to be one of friends, :), you know that I can be very anal about certain things. One of them is properly setting up a problem. One the one hand, I'm not suggesting that this is the only way to solve these problems. I always get asked the question, "do I have to do it in the same, exact way you do it?". And my answer is "nope". On the other hand, you need to think about why the question is asked. Sometimes it's because a student really does have a slightly different way to do the problem. Other times it's because someone's just too lazy to set it up with all the needed detail.

One thing I do is to keep close track of the numerator and denominator. So, I always write any values like 3.0 feet in fraction form with a numerator and a denominator. So, this is what we have at this point-

Looking at the needed conversion as a fraction and concentrating on the units, we need to be able to cancel the feet units and end up with units of inches. So, this is what we have-

Note that we have units of feet in the numerator and units of feet in the denominator so they cancel out. Just like 3/3 equals 1, feet/feet also equals 1. To reiterate- treat units as if they were numbers! We're left with units of inches in the numerator and no other units which is what we want.

One common method of solving problems is to ignore the units and slip them in at the end using the assumption that the problem was done correctly. Let's put the numbers in without regard to the units and then check the units to see what happens-

Step 5- look at your answer and make sure it answers the question. I hope you agree that the units make absolutely no sense in the way the problem was set up above. Checking your units is your first line of defense. In fact, I suggest you check your units before you whip out your calculator. Why go to the trouble of punching in the numbers if your setup is wrong? It's very useful to have a "common sense" check when you finish the problem but that's not always possible. If it isn't possible then you have to rely on the answer having proper units. This problem does have a common sense check, though. Since you are going from feet to inches and you know that there are 12 inches in one foot, you final answer, in inches, better be greater than the initial 3.0 we had for feet.

Let's make a list of rules.

Let's try one more problem. Let's take 2.56 quarts and find out how many liters it is.

Here's a basic list of unit conversions.

Step 1- what's being asked? We need to convert from quarts to liters.

Step 2- what's given? We have 2.56 quarts.

Step 3- what formulas and/or conversions do we need. Using the conversion list provided, find a conversion between liters and quarts.

Steps 4 and 5 (I'm doing them at the same time simply for display reasons)- set up the problem and work the calculation making sure your units work out correctly.

One more problem! I'm going to leave the steps up to you and simply give you a way to cycle through the problem setup, unit check, and final calculation.

Let's assume that gasoline costs $1.45 per gallon (ignoring for convenience the different grades available). Depending on when you read this, the price per gallon might be too high or extraordinarily low... You've got a ten dollar bill. How many gallons of gas can you get with that ten spot?

Press the "cycle forward" button to cycle the display from setup, to cancelling units, and to calculating the final answer. The "cycle back" button will move you back a step, if desired.

The next tutorial will introduce problem solving where there are multiple steps.