Welcome to dimensional analysis, aka the factor label method, which is a standard method for mathematically converting from one unit of measure to another unit of measure.
Dimensional analysis provides a straightforward algorithm for solving a wide variety of problems involving unit conversions.
In this video we will look at some examples of dimensional analysis, how to set up the conversion, and then some practice problems of increasing difficulty.
So a single measure can be expressed with different numbers depending on the unit being used.
A simple example would be measuring the length of a line. You could use inches, and this line would be 3.50 inches, or you could use centimeters, in which case the line is 8.89 cm.
The length does not change, but different units are used and so the numbers are different.
Let’s say you have 36 donuts. You could also express that as 3 dozen donuts. There is no change in the amount of donuts, just in the unit used in expressing the amount.
So we saw that 3.5 inches is the same as 8.89 cm.
Let’s take a look at how we can mathematically convert from one unit to another using what are called conversion factors and seeing how they are used in the specific mathematical structure called dimensional analysis.
Although most of the problems presented here are simple enough to do without the structure of dimensional analysis, we’ll use them to show how to construct the set-up of dimensional analysis, which will prove very useful down the road for more difficult problems involving multiple conversions.
The first problem gives us the mathematical basis for the inches to centimeter illustration shown earlier. The problem gives inches (circle), gives the numerical relationship between inches and cm (circle), and asks for centimeters. We will convert from inches to cm (arrow straight across from 3.5 in circle to cm circle). We’ll take a quick overview of what is involved, and then go back and review each step.
So what’s the mathematical set-up? Identify and write down the quantity given in the problem and multiply by a fraction in which the unit given is on bottom, unit wanted on top. Let’s put the first term into a fraction to make it clear what units will cancel. With inches both on top and bottom they will cancel, leaving the wanted unit which is cm.
The last step is putting in the numerical relationship between the two units, which is given in the problem— for every one inch there are 2.54 centimeters.
The fraction is called a conversion factor. The answer is 8.89 cm. Setting up the conversion factor correctly is the essential part of dimensional analysis which we will get to in a moment in more detail.
So this gives us the generic set-up for dimensional analysis: unit given times the conversion factor, unit wanted over unit given. Mathematically we multiply what’s on top and divide by what’s on bottom.
Let’s go back and review each step: Write the quantity given in the problem, and here we write it as a fraction to keep track of what’s on top and bottom.
Draw a line for the conversion factor, put on bottom the unit given, wanted unit on top, and then write in the numerical relationship between the two units.
After cancelling, you should be left with the unit wanted, and here we are left with cm, the unit wanted.
Plug into your calculator and we see the answer is the same as what our ruler tells us.
To illustrate another way of setting up a conversion factor, let’s convert centimeters to inches.
Here, we begin with 8.89 cm, multiply by a fraction with cm on bottom and inches on top.
Fill in the numerical relationship given, cm cancel leaving the desired inches, and the calculation gives the expected answer of 3.5 inches.
If we look at the two problems side by side we can see why dimensional analysis works: the conversion factors can be reciprocals—you simply choose the one that gets you to the unit desired. The conversion factor on left converts cm to inches, the conversion factor on the right converts inches to cm.
They are both the exact same relationship, just inverted for the purposes of the calculation, and conversion factors work like that because they are mathematical equalities.
The donuts provide another simple example, converting donuts to dozen.
Starting with 36, we convert with donuts on bottom and dozen on top, the relationship is 12 to 1, donuts cancel, we are left with 3 dozen.
Let’s apply dimensional analysis to something with several unit conversions in a single problem. It takes 14.5 seconds for you to run 100 meters.
How many meters will you run in 0.0520 hours? Let’s look at the conversions that will be needed. The problem already gives us the relationship of seconds to meters, but gives us a quantity of time in hours.
So we need to get from hours to seconds before finally using the relationship given to convert seconds to meters. We know in 1 hour there is 60 minutes, and we know there are 60 seconds in 1 minute.
Let’s see how we can use these relationships in calculating the answer.
We write down what’s given in the problem, which is the amount of hours, and so the first conversion has hours on bottom and minutes on top.
he answer is in minutes and will allow us to convert to seconds: the set-up cancels minutes, leaving seconds.
Finally we can use the relationship between seconds and meters given in the problem to convert seconds to meters.
Seconds on bottom meters on top, fill in the numerical relationship given in the problem, which then gives the answer in meters.
Note that with problems where multiple conversions are necessary, we can more efficiently write out the problem as a chain of calculations.
The first calculation converts hours to minutes, the second converts minutes to seconds, the third converts seconds to meters.
Let’s try one more problem. Pause the video and see if you can work it out. Note that one unit relationship, that of grams and cubic centimeters is given in the problem. You would also need to find the relationship of liters and cubic centimeters, which is 1 liter = 1000 cm3.
So let’s take a look. For the purposes of solving the problem, it might be easier to write out gold’s density as 19.3 g = 1 cubic cm. We start with the amount given in the problem, 250 g gold.
Note that 19.3 g/cm3 is NOT a specific amount, it is a unit relationship, which is used in conversion factors.
We want to get to volume, so the density gives us a way to do that, with grams, the unit given, on bottom, and cubic centimeters on top.
Grams cancel, we now have cubic centimeters, which we can convert to liters: cm3 goes on bottom so that it cancels, liters on top, fill in the numerical relationship: 1000 cm3 for every 1 liter.
The answer tells us that 0.013 liters of gold will have a mass of 250 g.
I hope you can see that dimensional analysis is a powerful method for solving unit conversions. Click here for applying dimensional analysis to metric prefix conversions and here for applying dimensional analysis to mole conversions. Seeya!
