Conversion between metric units

We're asked how many centiliters are in one dekaliter? So the first thing we want to do is just think about how much is a centiliter relative to a liter, and how much is a dekaliter relative to a liter? And I'll write the prefixes down. And really, you should have these memorized because you're going to see these prefixes over and over again for different types of units. So the prefix, kilo, sometimes [? ki-lo, ?] this means 1,000. If you see hecto, hecto means 100. Deka means 10. If you have nothing, then that just means 1. Let's put that there. Then if you have deci, this means 1/10. If you have centi, this means 1/100. If you have milli, this means 1/1,000. So let's go back to what we have. We have centiliters. Let me write this in a different color. If you have a centiliter, this is equal to 1/100 of a liter, Or you could say 1 liter for every 100 centiliters, so you could also write it like this: 1 liter for every 100, or per every 100, centiliters. So we got the centi, now let's think about the dekaliter. So the deka is right over here. So a dekaliter means 10 liters. Or another way to say it is for every 10 liters, you will have 1 dekaliter. Now, before I actually work out the problem, what's going on here? We're going from a smaller unit to a larger unit, so there are going to be many of the smaller units in one of the larger ones. And we can do it multiple ways. So we want to essentially convert 1 dekaliter into centiliters. Now, we could just do it by looking at this chart, or we could do it with the dimensional analysis, making sure the dimensions work out. Let's do it the first way. So if you have one dekaliter, how many liters is that? 1 dekaliter over here would be the same thing as 10 liters. That's liters. We're assuming that our unit is liters here. And then 10 liters is going to be how many deciliters? It's going to be 100 deciliters, right? Because each of these is 10 deciliters, and you have 10 of them. So every time you go down, you're going to be multiplying by a factor of 10. 100 deciliters is how many centiliters? Well, 100 deciliters, each of them is going to be worth 10 centiliters, so that's going to be 1,000 centiliters. So we have our answer: 1 dekaliter is 1,000 centiliters. Now, the other way to do it is you could convert a dekaliter to liters, and then convert a liter to centiliters. So if we have one dekaliter-- and whenever you do unit things, just make sure that it makes sense. Sometimes, people, instead of multiplying, they would divide, and then they'd get, oh, 1 dekaliter is equal to 1/1,000 of a centiliter. And they say, no, no, no, no. A dekaliter is a much larger unit that a centiliter. So 1 dekaliter has to be a bunch of centiliters. This should be a large number, so you should always do that reality check whenever you're dealing with units. Now, let's do it the dimensional analysis way. We're starting with one dekaliter. We want to convert it to liters. So if you're converting it to liters, you want the dekaliter in the denominator and you want liters in the numerator. Now, how many liters are 1 dekaliter? Well, you could say 10 liters is equal to 1 dekaliter. So 1 dekaliter is equal to-- these cancel out. 1 times 10 is 10 liters. Now, if we wanted to convert this to centiliters, we're going to want the liters in the denominator, and you want the centiliters in the numerator. Now, how many centiliters are there per liter? How many centiliters? Well, 1 liter is 100 centiliters. Centiliter is 1/100 of a liter. Notice, this and this are the inverse statements. They're saying same the exact same thing. 1 liter per 100 centiliters. Here, writing 1 liter per 100 centiliters. We've just flipped it, but they're giving the same information. And the reason why we flipped it is so that the liters cancel out, and then we're just left with 10 times 100 is 1,000 centiliters. And we are done!