Dimensional Analysis: Study with Video Lessons, Practice Problems & Examples

Dimensional analysis is a systematic method for converting units, ensuring unwanted units cancel out. For example, converting 32 inches to centimeters involves using the conversion factor \(1 \text{ inch} = 2.54 \text{ cm}\), resulting in 81 cm after considering significant figures. In more complex conversions, like 115 minutes to years, multiple conversion factors (minutes to hours, hours to days, days to years) are used, leading to an answer of \(2.19 \times 10^{-4}\) years, also respecting significant figures. Mastery of conversion factors is essential for accurate unit conversions in scientific calculations.

Dimensional Analysis

So dimensional analysis is seen as a fail-proof process that allows you to convert from one unit to another. You design the problems to begin with your given amount and to finish with the end amount of your unknown. Now just follow the units to ensure the unwanted units are canceled out. In the same way that we did metric prefix conversions where we line things up on opposite levels to help them cancel out, we're going to take that and apply it here to the dimensional analysis. Now the strategy is, many conversion problems will utilize your given amount and conversion factors, and you're going to use those together to help isolate your end amount. So if we take a look here in this general format, let's say we're given 32 inches, so 32 inches is our given amount, and they're asking us to identify how many centimeters this equals. So centimeters is our end amount, what we're looking for. The conversion factor is a way of bridging these two ideas together. It's used to convert our given amount into our end amount. Now, to go from inches to centimeters means that we have to use a conversion factor that relates inches to centimeters. When we talked about conversion factors, we said that 1 inch is equal to 2.54 centimeters. I put inches here on the bottom so that they can cancel out with these inches up here on top. So you always want them on opposite levels. Doing that helps me isolate my end amount unit, which is centimeters. So then I would just multiply 32 times 2.54. Initially, I'll get 81.28 centimeters. But remember, when you're multiplying numbers together, your answer is the least number of significant figures (sig figs). 32 has 2 sig figs in it. Because, remember, when you don't have a decimal point, you move from right to left. You start counting once you get to your first non-zero number and count all the way through. So this has 2 sig figs. When you have a decimal point, you go the opposite way. Our first non-zero number is this 2. You start counting here, you go all the way through. So we have 3 sig figs. So you go with the least number of sig figs, which would be 2. So this would round to 81 centimeters as our end amount. But let's say we had to do a type of dimensional analysis with way more steps. What do we do then? Well, let's say here we are given 115 minutes. So that is our given amount, and we have to get to years. Years would be our end amount. To connect given to end, we have to utilize conversion factors. I need to cancel out minutes, so I'll put minutes here on the bottom. We know that minutes is connected to hours. We know that 1 hour has 60 minutes, and here, this will represent our first conversion factor, which I'll abbreviate that Minutes would cancel out. Now I have hours. We know that hours and days are connected. We know that one day has 24 hours. This would be my second conversion factor or CF 2. Hours cancel out. Now I have days. And then finally, we know that days to years can be connected as well. We put days on the bottom so we can cancel out with the days on top, and we know that 1 year has approximately 365 days. This is my conversion factor 3. So days cancel out, and now I'm left with years. So what we're going to have here is 115 on top multiplying with a bunch of ones, which doesn't change anything, but on the bottom, we have multiplying 60, 24, and 365. Now it's incredibly important you know how to plug this into your calculator. My suggestion when you have multiple things on the bottom multiplying is to just multiply them all together and get that sum total. When we multiply 60 times 24 times 365, their total is 525,600, and we still have the 115 on top. When we divide 115 by that total, we're going to get 2.19×10-4. We're going to say here that the number of sig figs within our final answer is based on our given information. Now our only given information was the 115. These other numbers were not given to us in any way. So we can't use them to determine the number of sig figs. Again, it's all based on the information that is presented. Look at those numbers presented to you and use those to determine the number of sig figs in your final answer. 115 has no decimal point, so we move from right to left. It has in it 3 sig figs. So our answer should have 3 sig figs, which it does. So 2.19×10-4 years would be our answer here. So everything we've learned up to this point, we're going to use in some way when it comes to dimensional analysis. And remember, our conversion factors are incredibly important because they're a way of connecting our given amount to our end amount. The whole point is to cancel out units and isolate the unit that you're looking for at the end. Now that we've done these example questions, let's move on in our discussion of dimensional analysis.

Dimensional Analysis Example 1

Here are example states. A TA can grade 4 assignments per hour. If each assignment has 12 questions, how many questions can the TA grade in 130 minutes?

Alright. So if we're going to approach a question like this, let's look at our steps. If present, start with the given amount that is not a conversion factor. So remember, our given amount is when we have a single unit by itself that isn't tied to another. We're going to say within this question, our 130 minutes is our given amount.

Second, identify the end amount you want to isolate for your unknown. So here, this is our given amount, we have to figure out what our end amount will be. So our end amount, in this case, is questions, how many questions. Step two, write down all the conversion factors. So all our conversion factors, let's see. We have 4 assignments per hour. Remember, "per" is the word that connects different units together. So that's 4 assignments for every 1 hour. They also tell us each assignment has 12 questions. So that means one assignment is 12 questions.

The last part, find the connection between the given amount and the conversion factors in order to isolate the end amount. Let's look at the given amount. The given amount has minutes within it, but neither of the conversion factors has minutes involved. What we do have is hours. That tells me there's a conversion factor that's even before either one of these two. So conversion factor 1 actually involves us first changing minutes into hours. So we have minutes here on the bottom, hours here on top. 1 hour is 60 minutes. Minutes cancel out this way. And the reason we're doing that is because now that we have hours, we can connect it to the hours here within this conversion factor.

So that'll be my second conversion factor, bringing in the 1 hour for every 4 assignments. Now that we have hours lined up, they cancel out. Now we have assignments, and we need to get to questions. Here is our last conversion factor. Now it has assignments and questions within it, but we need assignments to cancel out. So assignments need to be here on the bottom. Right? And then we need questions, questions go here on top. And it is one assignment is 12 questions.

Remember, one of the first things we said is that conversion factors, we can flip them. They can be presented in two different ways. Here, we had to flip the initial conversion factor so that assignments can cancel one another out. They have to be on opposite levels to be able to do that. So in conversion factor 3, what I'll have left at the end is questions. So what we do now is we're going to multiply 130 times 4 times 12 divided by 60. When we do all of that, we're going to get here, as our final answer, 104 questions. So we have 104 questions as our end amount, but remember we need to worry about significant figures. 130 has 2 significant figures, 4 has 1 significant figure, 12 has 2 significant figures, 60 has 1 significant figure. So we need to go and have only 1 significant figure as our final answer. So we'd say roughly about 100 questions is what the TA could grade within 130 minutes, which is a lot of questions.

Alright. So now that we've done this example where we've set up the basic principles behind dimensional analysis, let's continue onward and do some practice questions.

If the distance between Washington, D.C. and New York City is 224.9 miles, determine the distance in centimeters.

• A.
• B.
• C.
• D.
• E.

The average human body is composed of approximately 160 fluid ounces of blood. How many quarts of blood does the average human body possess? (1 gallon = 4 quarts, 1 pint = 2 cups, 1 cup = 8 fluid ounces, 1 quart = 2 pints).

• A. 7.3 quarts
• B. 11.8 quarts
• C. 4.13 quarts
• D. 5.0 quarts
• E. 0.85 quarts

Lipitor, a pharmaceutical drug that has been shown to lower 'bad' cholesterol levels, while boosting 'good' cholesterol levels had over $12 billion in sales last year. Each pill weighs 2.5 g, which contains 4.0% of the active ingredient by mass. What mass in kg of the active ingredient is present in one bottle of 120 pills?

• A. 0.112 kg
• B. 0.020 kg
• C. 0.012 kg
• D. 0.024 kg
• E. 1.02 kg