How is solving a chemistry problem like playing dominoes? Watch this lesson to find out how you can use your domino skills to solve almost any chemistry problem.
How many minutes are in 2.45 hours? I know this doesn't seem like a chemistry question, but in this lesson, we're going to learn a technique that is used to solve nearly every type of chemistry problem. This technique has several different names, but the most common are dimensional analysis and the factor-label method. Dimensional analysis is a simple tool for solving problems not just in chemistry, but in everyday life. It allows us to convert a number from one unit to another unit. If you ever need to figure out how much carpet to buy for your living room, how much it will cost, or how long it will take to travel somewhere, you can use dimensional analysis. Not only will using it help you stay organized as you solve problems, but it will likely reduce the risk of errors by providing a way to double-check your work.
So let's go back to our initial question: how many minutes are in 2.45 hours? My number-one piece of advice for answering a question like this is to try not to do it in your head or take any shortcuts. It may seem like an easy question to answer by punching numbers into a calculator, but when the problems start getting more complicated and more abstract, you will be more likely to make mistakes if you don't use this method to solve problems.
Examples of some conversion factors
Let's start the process of answering this question by first reviewing three basic math rules:
1. First of all, what do you get when you multiply a number, let's just say x, by 1? Hopefully, you said 'the original number' or x. Multiplying by 1 does not change the value of the original number. You can always multiply by 1 without changing a number's value.
2. Next, let's take a look at some of the properties of fractions. You would solve 2/4 * 5/2 by multiplying across the top (to get 10) and across the bottom (to get 8), but because you have the same value (2) in both the numerator and the denominator, these numbers can cancel out, reducing the amount of work you would need to do in the end by simplifying the 10/8 fraction to 5/4. But what if you have something like 5 cm * 1 m/100 cm? Can you cancel out the units as well? Yes! Identical numbers and units on the top and bottom of multiplied fractions will cancel each other out.
3. Finally, we need to examine what a measurement really is. If a reaction takes 1 hour, there are several different ways we can record this: 1 hour, 60 minutes, 3,600 seconds, and so on. There are multiple ways to write a measurement without changing its value. In dimensional analysis, we will use conversion factors to express these equalities. A conversion factor is a relationship in the form of an equality. For example, 7 days/1 week, 60 seconds/1 minute, or 12 inches/1 foot are all examples of conversion factors. Some conversion factors may be difficult to identify. For example, the density of aluminum is 2.7 g/cm3.
Solving a One-Step Problem
How do all these rules help us solve our problem converting 2.45 hours into minutes? We will first want to start out with the quantity we are given. Then, we will need to set up a conversion factor (or a group of conversion factors) that will allow us to solve the problem. When setting up a conversion factor for this problem, we need to identify our known equalities. In this situation, we know that 1 hour is equivalent to 60 minutes. We can either write our conversion factor as 60 minutes/1 hour or 1 hour/60 minutes. We are going to need to choose one of these conversion factors to help answer our question, so which one do we use?
Let's see what happens with each one. If we take 2.45 hours and multiply by 1 hour/60 minutes (which is equal to 1), we get a value of 0.041 hours2 /minutes. If we take 2.45 hours and multiply by 60 minutes/1 hour (also equal to 1), we get a value of 147 minutes. If we compare these two answers, we need to understand that both of them are equivalent values, but one gives you the units you want, and one does not.
Even though these answers are equal, use the one with the right unit of measurement.
Solving a Multi-Step Problem
Let's take a slightly more challenging example: convert 2.3 miles into centimeters. First, let's list our conversion factors (the information we will most likely be using to solve this problem). We have 1 mile = 5,280 feet, 1 foot = 12 inches, and 1 inch = 2.54 centimeters. There may be some conversion factors you will need to memorize, but we will discuss those as needed throughout the course. For now, just focus on setting up the problem using the conversion factors. This is where it gets fun.
We can solve this almost like we would play a game of dominoes. If you are given the tile shown (3/6) and you have tiles a (0/5), b (6/0), c (2/5), and d (4/5) in your possession, you will need to place them in an order that allows you to connect to the 2/3 on the right side of the sequence. This may take a little trial-and-error and a little problem-solving, but you should come up with this sequence: tile b, tile a, tile c. You may think you could use either tile c or tile d for the last-placed tile, but tile d wouldn't work because it doesn't connect up with the 2 on the last tile.
We can use the same exact process to convert 2.3 miles into centimeters. We start out with 2.3 miles, but because we don't have a direct connection (in the form of a conversion factor) to centimeters, we need to plan out how we are going to set up our fractions so that the units we want canceled out will end up being on opposite sides of each other. So we will start out with 2.3 miles. The conversion factor that includes miles is 1 mile = 5,280 feet. Are we going to write the fraction so the miles are on the top or the bottom? Miles should be on the bottom in this situation because we are trying to cancel out miles. Placing it this way allows us to cancel out miles, but the remaining unit is feet (which we don't want). We will have to use another conversion factor to convert from feet. If we use the conversion factor 1 foot = 12 inches, would we want to place feet on the top or bottom of the fraction? In this situation we would place it on the bottom, so we would multiply by 12 inches/1 foot. This cancels out feet, but we are still left with inches. We need to convert to centimeters, so we place our final conversion factor in the line in a way that allows us to cancel out inches and convert to centimeters, so inches would need to be on the bottom. If we multiply everything out, we get 370,000 centimeters (using the proper number of significant figures).
The answer to the sample problem is reached by canceling out like terms.
The biggest issue people have when converting units is that they're not sure whether to multiply or divide. If you set up every problem just like this, it should become clear whether you will need to multiply or divide, and you will reduce your risk of error.
Canceling Numerator and Denominator Units
Let's finish with a slightly more challenging problem. For this, let's use the density of ethanol, 0.8 g/mL, and convert it to kg/L. First we will list our conversion factors; we know that 1,000 mL = 1 L and 1,000 g = 1 kg. So how will we set this up? Always start with the value that the question gives you. In this case, it would be 0.8 g/mL. We may see right away that grams is in the numerator, and we would like it to be kilograms. So we're first going to multiply by 1 kg/1,000 g, which allows us to cancel out grams. Next, we will convert milliliters on the bottom to liters on the bottom. To do this we will need to manipulate our conversion factor so that milliliters is on the top. We will multiply by 1,000 mL/1 L. This allows us to cancel out milliliters, leaving liters in the denominator. You may also see that the 1,000s will cancel out, leaving us with 0.8 kg/L as a final answer. Keep in mind that when you multiply numbers, the order does not matter, so if you wanted to cancel out milliliters first and grams second, that would work just as well.
When you're solving these types of problems, just remember it takes a little planning and a lot of practice. We will be solving problems using this method for the rest of the course. Even if you think you can solve these types of problems quickly and in your head, you should still learn this method, because it will greatly reduce the number of errors you make in calculations. Also remember that even though it doesn't look like it, all you're doing is multiplying by 1, which is okay, because it doesn't change the original value of the number.
After watching this lesson, you should be able to:
- Solve one-step and multi-step problems using dimensional analysis and conversion factors
- Explain how to cancel out numerator and denominator units