Dimensional analysis, also called the factor/label method, is a method for converting between units using ratios between different unit systems.
Alright, when you are in Chemistry class you're going to be doing a lot of converting. A lot of converting from moles to grams, grams to moles, ions, molecules a lot. So we want to make sure we can do that in a way that makes sense.
So we're going to talk about dimensional analysis, it's just a method of converting from one unit to another. You might actually call it Factor Label Method depends on what your teacher likes to use. Alright, so we're going to actually do an example rather than explain what it is cause it's actually easier to explain via example rather than just a bold explanation.
Alright, so the first thing you're going to do when you're converting from one unit to another, in this case 7 days to seconds, we are going to start off with our given and our given in this case we're given 7 days, okay. So we're going to write that down 7days. Alright then we're going to write what we call some teachers call the picket fence and this is actually used and it's quite easy to when you're converting from one thing to another. This method is just the easiest way to organise your data. Okay. So we're going through 7 days and we want to end up in the unit of seconds. So the things I know about days, in one day we have 24 hours. So one day, and we put down here and 24 hours up here. Now the reason to put day in the bottom of this ratio, these are equal to each other, they're equal units one day is equal to 24 hours. The reason to put a day on the bottom is because I can just like I can cross out when I multiply one half and two fifth, I can cross out the numerator and denominator. I can do that with the units as well, so I can cross out day and so now, if I multiply these guys together I'm actually now on the unit of hours. That's what makes this method so convenient.
Okay, but I don't want hours, I want seconds. So, I'm going to keep going. In one hour, I, also know that there are 60 minutes. Okay? So now I have hours in the numerator and hours denominator so I can cross those guys out as well. But again, I don't want minutes, I want seconds so I got to keep going. So I'm going to say, okay. In one minute, I know that there are 60 seconds. Okay. These minutes cross out and I'm left with seconds which is exactly what I wanted so now I can stop. Awesome. So now I can actually multiply all the numerators together and divide by all the denominators and my answer should be in seconds. That should be the correct answer. So if I multiply 7 times 24 times 60 times 60 divided by 1 divided by 1 divided by 1, I get 604,800 seconds. And this makes sense because in 7 days there are a lot of seconds so you can imagine. And 604,000 is a lot of seconds. So this actually would make a lot of sense. Okay. This is a method, this is dimensional analysis in a nutshell.
Let's do something a little bit harder, let's just say we're changing two units, okay? So if you're going 50 miles an hour, how many feet, feet per second are you going? Okay. So we know we're going from miles to feet and from hour to seconds. Okay, so first thing we're going to do is to write down our given. And we're going to write down 50 miles per one hour. Okay. So here's a conversion that might be helpful for us to continue this problem. Alright, so, we want to, first thing I'm going to do is to change these miles. I want to change the miles to feet. So I know that 5,280 feet equal one mile. So I want to put the mile in the bottom because I want to cross these guys out. So I have one mile equals 5,280 feet. Cross the miles out, and right now if I'd stopped if I stopp this problem, I would be in feet per hour. But I don't want to be in feet per hour I want to be in feet per second so I got to figure out how to change these hours. So I'm going to keep going. I don't want to mess up the speed so I'm just going to deal with the hour. So I know in one hour, I have there are 60 minutes. So I can cross off that hour so if I were to stop now I'd be in feet per minute, but I don't want it feet per minute, I want it feet per second. So I'm going to keep going.
I know in one minute, I have 60 seconds. Okay. These minutes then crossed out and I'm with seconds. So I'm now left with feet at top which is what I want and seconds on the bottom to what I want. So now I'm in the units that I want, fantastic. I multiply all the numerators and divide by the denominators. 50 by times 5,280 times one times one divided by one divided by one divided by 60 divided by 60 is going to give me 73.3 feet per second. Okay. Now notice the whole point of this is actually just to make sure units all cross out and make sure that we're left with the units that we want.
Alright, let's do something that deals with Chemistry. Okay. One question is how much weight would you need to make a quart of 5% bleach solution? Okay. So we're given is we want a quart of this and we want a quart of this. So I'm going to say our given is one quart and always start with your given. Okay. So I know this unit. If one quart is 32 ounces. So I'm going to say that one quart is 32 ounces. Okay. So quart crosses out and if I were to convert this that would say how many ounces are in the quart. Okay. Then I want 5% bleach solution. So now the difference is, I'm going to say if I have one quart of bleach solution, this is where your units actually could become very important. You're going to get 32 ounces of bleach solution for every one quart of bleach solution.
Now, I'm going to want 5 ounces of just straight bleach for every a hundred ounces of bleach solution, okay? So that's where this crosses out. So, because we know percent means out of a hundred. So I can actually just do this math and say one times 32 times 5 divided by one divided by a hundred is going to give me 1.6 units ounces of bleach to put in a quart of water to make it 5% bleach solution.
This type of question you are going to be asked when dealing with Chemistry. The dimensional analysis is extremely useful when dealing with your changing of units and going from one unit to another and trying to figure out how to get from one place to another.