Hi, this is Cathy with Level Up RN. In this video, I will be going over several dosage calculation problems that involve dilution. I'll be solving these problems on my whiteboard using three different methods, including dimensional analysis, ratio and proportion, and the formula method. You can find all of the information that I'll be covering in this video in our Level Up RN dosage calculation workbook. If you are in nursing school, then you know how important it is to master dosage calculations. And our workbook will help you do just that. For more information about our workbook, you can click here. But in a nutshell, our workbook contains all the different types of dosage calculation problems that you are likely to encounter in nursing school. And we demonstrate how to solve each problem using multiple methods so you can pick the way that makes the most sense to you.
So with this first problem, we have an order for gentamicin 200 mg IV q8h using a vial with a concentration of 800 milligrams per 20 mL. Then we need to dilute the ordered dose with 50 mL of normal saline and administer it over two hours. We need to make sure we round all our answers to the nearest tenth. And part A of this problem says, how many milliliters will you remove from the vial to dilute with 50 mL of normal saline? So this part is just asking, how much do we take out of the vial before dilution? So we can solve this with dimensional analysis, ratio and proportion, or the formula method.
So what is ordered is 200 milligrams. So I'm going to solve it with dimensional analysis first. So we're going to write down what is ordered. And then I'm going to check my units of measure against the available concentration. And they're both in milligrams, so we're good to go there. So I just need to take this ordered dose and multiply it times the available concentration, making sure I put the milliliters on top and the milligrams on bottom such that these cross out. And then we are left with milliliters. So 5 milliliters is what we're going to need to take out of that vial, and then we'll worry about dilution next. I could solve this same problem using ratio and proportion. So I would just write down my available concentration on one side of the equation, what the unknown value is here for X, and the ordered dose. So we're going to cross multiply here. So 800X equals 20 times 200. 800X equals 4,000. And then X equals 4,000 divided by 800, which equals 5 mL. Okay. That's how we would solve it with ratio and proportion.
And then with formula, we can do that too. Formula is desired over have times the vehicle. So what is desired is 200 milligrams. That's what's ordered. What we have is 800 milligrams in 20 mL of solution. That's our vehicle for getting those 800 milligrams. So we calculate this out. Again, we end up with 5 mL. 5 mL is what we need to remove from the vial. But now we need to dilute that 5 mL with 50 mL of normal saline.
Part B asks how many total milliliters will be administered over two hours. So I'm going to take my 5 mL. I'm going to add in that 50 mL of normal saline, and that's going to give me 55 mL. That is how much mLs I will administer over two hours after dilution. And then the last part of this problem asks, what is the concentration of the final solution? Okay. So the concentration, again, is milligrams in milliliters. So we have 200 milligrams, right? Because that's how much we took out of that vial. And we have 200 milligrams in 55 mL of solution now that we've diluted it. And if we do this math, we end up with 3.6 milligrams per mL. And this is rounded to the nearest tenth, which is what we are instructed to do. So just double-checking all our rounding, we ended up with 5 mL for part A, which is rounded already. And then we ended up with 55 milliliters for part B. That's rounded. And for part C, we have it rounded to the nearest tenth. So we're good to go there. So this is the final concentration of the solution that we're going to give the patient after dilution.
All right. I'm now going to work through a second dilution problem. This problem is a little trickier. Your school or program may not require you to do dilution problems at all. And if that's the case, then lucky you. But if they do, then I'm here to help. So with this second example problem, we have an order for ranitidine 50 milligrams IV every six hours using a vial with a concentration of 1,000 milligrams per 40 mL. And then we need to dilute the dose in 0.9% NaCl to achieve a concentration of 2.5 milligrams per milliliter and inject it at 4 milliliters per minute. And we need to round all our answers to the nearest tenth. So there's a lot of numbers going on here. And spoiler alert, we don't care about some of them. So for example, when we go through these questions and read these questions, we don't care about the rate at which we're going to inject it at. So we can kind of ignore that last sentence. And I know all these numbers can be very overwhelming, and you're trying to figure out how you factor those in. You really just want to concentrate on the question being asked and only use the information you need to answer that question.
All right. So part A, how many milliliters of ranitidine will be diluted with 0.9% NaCl? So this is before dilution. How many milliliters do we need to get out of this vial? So we can solve this using dimensional analysis, ratio and proportion, or the formula method. So using dimensional analysis, we want to first write down what's ordered, which is 50 milligrams. And we want to make sure that we have the same units of measure from what's ordered and our available concentration. And we do. They're both in milligrams. So we're good to go there. We don't have to multiply times any conversion factor. Now, we're going to take what's ordered, 50 milligrams, multiply that times the available concentration. I'm going to make sure I put 40 mL on top and 1,000 mg on the bottom, such that my milligrams cross off, and I'm left with milliliters, which is what we are looking for. So when you calculate this out, we end up with 2 milliliters. So that's how much we need to take out of that vial to then dilute it.
All right. And we can do the same thing with ratio and proportion. We would write down our concentration on one side of the equation. And then we would write down the unknown value and what is ordered on the other side of the equation. And then we would do our cross multiplication. So we have 1,000X equals 40 times 50. And then we're going to solve for X. So 1,000X equals 2,000, and then X equals 2,000 divided by 1,000. So X will equal 2. And of course, we can do the same thing with the formula method. So the formula method is desired over have times the vehicle. So we are desiring 50 milligrams. And what we have is 1,000 milligrams in 40 mL of solution. That's our vehicle. That's how we get the 1,000 milligrams. So we do this math. Again, we end up with 2 mL. So we're going to pull 2 mL out of the vial and then dilute it in normal saline.
So part B of this problem asks us, what is the total volume that is needed to achieve a 2.5-milligram per milliliter concentration? Because that's the concentration we're looking for. And right now, we have our 50 milligrams in 2 mL of solution. But we want a different concentration. So we can solve part B of this problem using dimensional analysis and ratio and proportion. Formula method really doesn't apply for this problem. So this was part A over here. Now we're going to do part B. So for part B, our dose is 50 milligrams, and we want to achieve a 2.5-milligram per milliliter concentration. So we can figure out how many milliliters that's going to take to achieve that concentration by multiplying the dose times that desired concentration. So I'm going to put my milliliters on top, and I'm going to put my milligrams on the bottom so that my milligrams cross off. And if we multiply this out, we end up with 20 mL. So in order to achieve this concentration, I need to make sure my 50 milligrams is in 20 mL of solution. And I just solved this using dimensional analysis.
I can do this with ratio and proportion as well. So this is our desired concentration. And what I have is 50 milligrams, and I'm looking for how many milliliters I need to have that dose in to have that desired concentration. Cross multiply, 2.5X equals 50. X equals 50 divided by 2.5. X will equal 20 mL. So we need to make sure our 50 milligrams is in 20 mL of solution so that we have the right concentration.
So then the last question here is, how much diluent should be added to achieve that concentration? So I need a total volume of 20 mL. But when I pulled my dose out of the vial, it came out with 2 mL already. So my dose is in 2 mL. I need a final volume of 20 mL, so I only need to add 18 mL to that 2 mL to get a total volume of 20 mL. Again, just to kind of repeat that, when I pulled my dose out of the vial, I got 2 milliliters. Our desired concentration is here, and we calculated that in order to achieve that concentration, we need 20 mL of volume. So in order to achieve that, I need to add 18 to that 2 mL to get the 20 mL. So the answer to Part C is 18 mL. And I'll have to say that this type of problem is probably one of the more harder type of problems that you'll see when it comes to dose calculations. And I hope my explanation was helpful.
