Multiple units word problem: drug dosage (advanced)

I just received this drug calculation problem from a nursing student, and I think it's essential that the nursing students out there are able to do this, just in case I'm the patient receiving the drug. So let's do it. And I think it's an interesting unit conversion problem for pretty much anyone who wants practice with unit conversion. So the question is that we have a doctor. The doctor orders drug x. And this is the dosage that the doctor's requesting. They're saying 5 milligrams per pound of patient weight-- I'll just write per pound of patient weight-- every 12 hours. This is what we're supposed to do. But our supply of the drug-- it isn't just, you know, not just nuggets and milligrams. It's a solution. There's a certain amount of grams for every milliliter that we have of the solution. It's dissolved in some water. So this is our supply of drug x. We have 0.9-- I'll write a 0 in front. My wife, who is a doctor, says it's essential to write the 0 in front of the decimal. We have 0.9 grams per milliliter of solution. So if I were to take 1 milliliter out of my solution and give it to someone, I'm essentially giving them 0.9 grams of this drug. And the final piece of information we're given is that the patient-- they weigh-- and maybe we should say they mass, because kilograms is mass, but we get the idea. The patient is 72.7 kilograms. So there's a couple interesting things here. We have to figure out the dosage in terms of milliliters. We have to-- oh, actually, I didn't even tell you the question. The question is, how many milliliters of solution do we have to give to the patient per dose? So milliliters of solution per dose. That's our question. And there's a couple of things. We have to go from milligrams to grams. And then convert that to milliliters. And then they tell us per pound, but then they gave us the patient's weight, or their mass, in kilograms. So we have to do some conversion there. So I definitely can appreciate how this can be a little daunting and maybe confusing at times. So let's just do it step by step. So the first interesting thing-- and this is just something that you might need to know, or you might have written down on paper, or you might have a calculator that does this-- is just how to convert kilograms to pounds. And it's good to know in general, if you're converting between the metric and the English systems. So 1 kilogram is approximately equal to 2.2 pounds. Not exactly, but that's a pretty good approximation. And 1 pound-- if you just take 1 over that-- 1 pound is approximately equal to 0.45 kilograms. So we'll just put this in a box. This is the only kind of outside conversion information we'll need to do this problem. Everything else, we'll just need a calculator, unless we just want to spend a lot of time doing some arithmetic. So the first thing. Let's figure out our dosage in terms of per kilogram. This is per pound, and we really don't need to know every 12 hours. Because they're saying, how many milliliters of solution do we do per dose? A dose is every 12 hours. So we just really, you know-- the every 12 hours is kind of extra information. So we want to figure out this 5 milligrams per pound. How do we convert that to how many milligrams per kilogram? So let's do 5-- I'll write it down here in magenta-- 5 milligrams per pound. And then we want to convert this to per kilogram. So we can multiply this times the number of pounds per kilogram-- I'll do it in yellow-- times this information up here. Times 2.2 pounds per kilogram. And if you ever get confused-- you know, gee, how did Sal know to multiply by 2.2 instead of dividing by 2.2? Which is the same thing as multiplying by 0.5. You can pay very close attention to the units. Notice, I wrote 2.2 pounds per kilogram. 2.2 pounds per 1 kilogram. And you know this'll work out, because you have a pound in the numerator and you have a pound in the denominator. It's called dimensional analysis. If you ever get confused with these things-- and I think, once you do enough practice, you'll find that you won't have to pay too much attention to this. But at first, when you're getting started, just to make sure you're not multiplying or dividing by the wrong thing, just make sure the dimensions cancel out. Pounds in the numerator, pounds in the denominator. So let's do that. Pounds in the numerator, pounds in the denominator cancel out. And you multiply 5 times 2.2. This is equal to-- let's see. 5 times 2 is 10. 5 times 0.2 is 1. So this is equal to 11. And then in our numerator, we have milligrams. 11 milligrams per kilogram. So we just converted our dosage information to a pure metric system. It was actually a mix between the metric and the English system before. Now let's see what we can do. Well, let's see if we can get it in terms of how many milliliters we have to deliver per pound. So once again, we want this-- well, actually, let's go to grams first. Because we have milligrams here. We have grams up here. So let's see if we can convert this thing to grams. So just like we did before, we want a milligrams in the denominator. I'll do it in orange. We want a milligrams in the denominator and we want a gram in the numerator. Why did I say that? Because I want this and this to cancel. And I want a grams in the numerator. So how many grams are there per milligram? You can just think it through. There's 1 gram per 1,000 milligrams. Or 1,000 milligrams per gram. And you just multiply it out. So the milligrams cancels with the milligrams, and then we get-- this is equal to 11/1,000 grams per kilogram. So now we have everything in terms of grams, but we want it in terms of milliliters. The question is, how many milliliters of solution per dose? So let me go down here on this line right here. So we had this result. We have 11/1,000-- I won't do the division just yet-- grams of drug x per kilogram. This is really just a re-- we've just rewritten this dosage information in different units. And let's see how much solution we need per kilogram. So I want to cancel out the grams here and have a milliliters there. So to cancel out that grams, I'm going to have to have a gram in the denominator and a milliliter in the numerator. So in our solution, how many grams are there per milliliter? Well, they told us. There are 0.9 grams per milliliter. Or for every 1 milliliter, there are 0.9 grams. Notice, I just took the inverse of that. Because we want a milliliter in the numerator, grams in the denominator, so that these two cancel out. And let's do this multiplication now. So our grams cancel out. We have milliliters per kilogram. And then we multiply it out. 11/1,000 times 1 over 0.9. So I'll just keep-- let me just write it like this. So there's going to be 11/1,000 times 0.9 milliliters of our solution per kilogram. So we've gotten this far. So this is per kilogram of patient body weight. And then finally, they tell us how many kilograms our patient weighs. So let's do that last multiplication, and then we can actually get our calculator out and do it all at once. So let's multiply this times-- we want to know how many milliliters per patient. We want the kilograms to cancel out. So we want kilograms per patient. Now we're talking about this particular patient. Not every patient is going to be the same number of kilograms. But if we do this, kilograms will cancel out. We'll have milliliters per patient-- milliliters of solution per patient-- which is exactly what we want. We want milliliters of solution per dose per patient. But everything we've assumed so far has been per dose. So how many kilograms does the patient weigh? Well, there's 72.7 kilograms per patient. That's how much the patient weighs. So we just do this final multiplication and we'll be done. So our answer-- and as these two things are going to cancel out-- so our final answer is going to be 11 times 72.7 divided by 100 times-- actually, 100 times 0.9 is pretty easy to figure out. That's 900. Divided by 900 milliliters per patient. Or you can just say milliliters per dose. However you want to say it. Per dose per patient. Let's get our calculator out and do this. So we have 11 times 72.7 is equal to 799 divided by 900. Is equal to 0.88-- well, we could round up. 0.889. Hopefully the doctor won't mind. So that is equal to-- I'll write it in a nice, vibrant color-- 0.889 milliliters of solution per dose. So this is what we're going to give every 12 hours. If they ask, how many total milliliters over the course of 2 days? We would have to say, oh, there's 48 hours. We'd multiply it by 4. But that 12 hours was extra information in this problem. But anyway, hopefully this is useful, and it'll ensure that any nurses serving me in the future are giving me my proper dosage. And hopefully, the doctor even got the right dosage to begin with, because otherwise it's all for naught. Anyway.