When it comes to doing nursing math, which is essentially figuring out dosage amounts, the absolute best, easiest and most foolproof way to do it is by using dimensional analysis. You may remember it from your chemistry class and loved it even then ;-). In this post I will show you how to use dimensional analysis to solve any dosage calculation, even the tricky weight-based ones.

**Level 1 Dimensional Analysis: Piece of Cake**

We’ll start at level 1….super easy ones to give you a feel for the technique. Ready? Your order reads:

In dimensional analysis, you always start with what’s ordered. In this case it’s 650 mg of acetaminophen. You will write this as a fraction, with 650 on top and 1 on the bottom like so:

Next, you need to know what dosage amounts your medication comes in. This is known as the conversion factor. In the case of good ol’ Tylenol, we check our blister pack and see that it’s 325 mg per tablet. The next step is to add the conversion factor also as a fraction. Since mg is at the top of our first fraction, it will go on the bottom of our next one so that they cancel each other out. Like so:

As we move forward into more complicated calculations, it will become more evident that you know when to stop conversions when you are left only with the information needed to give the dose. In this case, we want to know how many tablets to give the patient. Everything except for tablets is crossed out, so we know we are ready to do some math.

1) Multiply across the top: 650 x 1

2) Then divide across the bottom: ÷ 325.

What answer did you get?

Let’s do one more easy one, then move on to something a little more difficult.

For this calculation, let’s assume midazolam comes in 5 mg tablets. I have no idea if it does, but we’re just practicing so it’s all good. Hopefully, you’ve set up your calculation like this:

Next, cross out the units that are the same…in this case it’s mg.

We are no weft with X # of tablets to give our 1 patient. Voila! Now, do your calculation!

OK…you have the hang of that now? That’s a simple calculation with one conversion. What if you have multiple conversions?

**Level 2 Dimensional Analysis: Multiple Conversions**

Rather than deal with actual drugs, I’m just going to make them up from here on out…easier and more fun that way! So, your order reads:

You go to the med room and see that Smartella comes in vials that have 4 mg of drug per ml. How much Smartella do you need to draw up? Start with the order:

Next, look at your vial…you know you need to ultimately get to mg, so let’s convert mcg to mg in our next step:

Notice how mcg are set up to cancel each other out…let’s continue as we add the amounts our medication comes in (4mg/ml)

Let’s get to canceling our units:

Now we do the math:

1) Multiply across the top: 2000 x 1 x 1

2) Divide across the bottom: ÷ 1000 ÷ 4

3) Press “=” on the calculator…what did you get?

Way to go! Let’s do one more…your order reads:

You check the vials of Awesome-Sauce and see they contain 1gm of drug in 10ml of solution. Because you yourself are taking Awesome-Sauce regularly, you know how to do the calculation:

Next you get to crossing things off that cancel each other out:

You’re just left with ml, which is exactly what you need to know. Now it’s time to do some math!

1) Multiply across the top: 60 x 1 x 10

2) Divide across the bottom: ÷ by 1000 ÷ by 1

OK! You’ve got this! Let’s move on to something a little more difficult.

**Level 3 Dimensional Analysis: Weight-based Calculations
**You will do a LOT of weight-based calculations in pediatrics and a fair amount in the adult population. For this first calculation, let’s assume your patient is 4 kg….awwwww a little baby!

You will start with the dose as always, which is:

Notice what we did differently here? We didn’t put the 1.4 mg over a 1…we put it over KILOGRAMS, because your dose is 1.4 mg per kilogram. By the way, Brainzy comes in a 10ml vial that contains 100mg of the good stuff, so you’ll write out your calculation like this:

As you write out your conversions, notice that we have 4kg over 1baby…that’s because your particular patient is a 4kg baby. Next we cancel things out:

We are left with X number of ml for 1 baby…which is exactly what you need to know. No Brainzy for you…you’re already super smart!

Sometimes your nursing professors will try to get all tricky on you and give you your patient’s weight in pounds, with the dosage in kilograms. No problem!

Your patient weighs 283 pounds and just had a BKA…you do NOT want him trying to get out of bed! You mosey on in to the med room and see that CalmDown (the greatest invention EVER) comes in a dosage of 1 mg per 10 ml bag. You diligently set up your equation, including all the conversion factors you need to end up with X number of ml per patient:

Notice how we start with the order (2 mcg/kg) then convert the kg to pounds, then input our patient’s weight in pounds, then convert the mcg to mg, then add in the number of mg in a 10ml bag. Easy!

Now you cross things off…cancel out those matching units!

And you are left with X number of ml per angry patient…exactly what you need!

Now get to calculating!

1) Multiply across the top: 2 x 1 x 283 x 10

2) Divide across the bottom: ÷ 2.2 ÷ 1 ÷ 1000 ÷ 1

3) = ??

The next level of dimensional analysis is a bit more tricky…but you are amazing, so I’m not worried at all!

**Level 4 Dimensional Analysis: Weight Based Meds by Time**

If you precept or do clinicals in critical care, you will notice that meds are often dosed mcg/kg/min…woah! That’s a lot of conversions! But with dimensional analysis it’s a walk in the ol’ park. Your order is:

You start with the order…which is to start your med at 2mcg/kg/minute. So let’s start there!

Next, think about what you want to get out of this calculation. Your pumps run on mls/hr so that’s what we want! You check out the bag of Squeezalot and see that it delivers 250 mg of drug in 250 ml. Ready? Let’ do this.

You’ve started with your order, then added your patient’s weight, then converted mcg to mg (since that’s what unit measurement our med comes in). Next we converted minutes to hours (since our pumps are configured to run on mls per hour) and lastly we have 250 mg of Squeezalot in one 250 ml bag. If you’re sure you’ve got yourself covered, let’s cancel out!

As you can see, we are left with ml/hr/patient…which is what we want! Now let’s do the mathulations:

1) Multiply across the top: 2 x 45 x 60 x 250

2) Divide across the bottom: ÷ 1 ÷ 1000 ÷ 1 ÷ 250

3) = ??

One more tricky one and then you’ll be good to go! Your order is for a continuous infusion meant to keep intubated patients calm:You check the Copacetik IV bag and see that it provides 1000 mg in a 25o ml IV bag. Your patient weighs 180 kg and now you are ready to set up your equation. Hopefully it looks like this:

We’ve started with our dose of 3mcg/kg/min, then added in our patient’s weight, then converted mcg to mg, then converted minutes to hours, then added the information specific to Copacetik. What’s next? Cancel out those units!

Then it’s time to get down with yo bad self and math it out!

1) Multiply across the top: 3 x 180 x 1 x 60 x 250

2) Divide across the bottom: ÷ 1 ÷ 1000 ÷ 1 ÷ 1000

3) = ??

See how wonderful, versatile and ultimately easy this method is? It works for any kind of calculation, any time, anywhere, on any patient. In these examples, I had you multiply and divide all numbers, including the numeral 1. Obviously, you don’t have to do include the 1’s, but if it helps you to remember every step, then it doesn’t hurt to include them.

Coming soon…a dosage calc quiz to test your brainy-ness!

Be safe out there!

Dr. Rachel Silva DNPWhat an excellent resource you have created for nursing students! I certainly wish I could have had this to discover back when I was in nursing school (but we barely knew what computers were, haha). I appreciate the time and effort you devoted to explaining each of the different dimensional analyses necessary for medication calculations.

Happy Nurses Week and thank you for all you do to help the next generation of nurses!

Nurse MoPost authorThanks Dr. Rachel! Happy Nurses Week to you, too!

Joan @ TheNurseTeacher.comThis is AWESOME!! Love the clear language, easy to follow & TOTALLY practical! Will be bookmarking and sharing FOR SURE!! Thanks!!

Nurse MoPost authorThanks, Joan! Appreciate it!

Elizabeth ScalaHoly cow! Taking people through these calculations in real time like this is so helpful. I wish I found this resource when I was in nursing school. Great post, sharing with my tribe!!

Nurse MoPost authorGlad you like it, Elizabeth! I am debating making a video but hate how I look on camera! Plus, who has time to actually do their hair??? 😉

AllieI just want to say thank you this is the best resource for conversions I am starting pharmacology and I’m so scared but this has helped me out so much your steps are perfect and clear, excellent instructions! Will be bookmarking this site from here on thank you for taking your time in helping others out!!

Nurse MoPost authorYou’re so welcome! Good luck with school!

MarieLove it ,I’m just starting Pharm

Nurse MoPost authorGood luck with school, Marie!

Ms.AWThanks for explaining for dosage calculations formula you did wonderful .I am in dosage calculations class right now as we speak and I am completely new to this also it was seeming to be very hard to me at first.

Nurse MoPost authorThe more you do them, the easier it gets. I promise! Good luck!!

ChellyCan you help me solve this problem?

A patient is started on furosemide infusion. Lasix 2000mg in 200mL is prescribed at 1 mg per min. You should set the pump at ? mL per hour

Nurse MoPost authorOK..I’ll try to do this without pictures…maybe just follow along with these instructions!

1) What is the order? 1 mg per minute…so start there! Write 1mg/min as your starting number.

2) What is your conversation ratio? It’s the 2000mg in 200ml. So that’s next. Since mg was on top of your first number, you’ll put it on the bottom of this one…write 200ml/2000mg.

3) Next….what’s left? You still need to know how many ml/hr. Notice that you have “min” on the bottom of your first number…the next conversion you need is 60min per hour so write that next with the “min” on the top (since it was on the bottom in the first number). Write out 60min/1hr.

4) Now, cross off all your matching units. You’ll cross off all the mg and min…you’ll be left with ml and hr, which is what you want!

5) Now, do your math…multiply across the top to get 1 x 200 x 60 =12,000, then divide across the bottom (simply divide by 2000)….and you get 6ml/hr.

Hope that helps!

ShelThis is such a great resource!!!!! Thank you so much, I wish they would have simply taught dosage calculations using dimensional analysis the entire time I had been in nursing school. Theses steps are much easier to remember!!!!!

Nurse MoPost authorSo glad it’s helping you Shel! Did you take the dosage cal quiz? How’d you do?

K BerrI am a seasoned BSN, RN. I have made many specialty transitions throughout my career; seldom was dosage comp on my list of expertise. Recently, I was hired into a med-surg, hospital unit where dosage matters. I was very afraid, but I know to face my fears to achieve peace. I really want to thank you for showing me a way to tackle something so fearful accuratly. Easy, peasy and I GET IT.

Nurse MoPost authorThat is awesome, Kitty! Glad it helped 🙂

RaquelHey nurse mo,

I need help answering two questions. Please help.

Order: zofran 8mp p.o. T.i.d

Available: zofran in a 100mL bottle labeled 4 mg/tsp

How many mL will the nurse administer for each dose?

Order: Morphine gr 1/10

Available:Morphine 10 mg/mL

How many mL will the nurse give?

Nurse MoPost authorHi Raquel…sorry for the delay! I have been BURIED!!!

I realize this is probably too late for you, but maybe it will help you with your next dosage calc conundrum.

1) The trick with this one is to know how many ml per teaspoon (it is 5ml per teaspoon). This one is deceptively simple with the unnecessary conversion showing that 4mg/1tsp. But that’s your professors trying to trick you up! You’re not going to let them!!!

The simplest way to do it is to figure out how many mg your entire bottle of zofran contains. If 5 ml = 1 tsp, then if you have 100 ml you just divide by 5 and you have 20 tsp of liquid. Each tsp contains 4mg of zofran, so your entire bottle is 80mg/100ml. NOW you have your conversion factor, and it should be super simple to set up your dimensional analysis now…

8 mg/1 x 100ml/80mg = 10ml (which is two teaspoons…if each teaspoon is 4mg, then this answer makes perfect sense…you’d get 8mg total dose)

2) This is another bizarre calculation…who is making these up for you guys? By 1/10 do you mean one-tenth? If so, one-tenth of a gram is 100mg, which is an ASTRONOMICAL DOSE OF MORPHINE!!!! Regardless, set up your dimensional analysis like this:

0.1gram/1 x 1000mg/1gram x 1ml/10mg = 10ml morphine (which in this case, is 100 mg morphine and your patient would die).

Hope that helps!

CWgr 1/10

gr (grain) 1 = 60 mg

So, gr 1/10 = 6 mg

6 mg divided by 10 mg X 1 = 0.6 mL

Nurse MoPost authorHmmm…I saw this as “grams” but CW may be right…did the question writer mean for you to convert from “grains?” I’ve NEVER seen it done, but sometimes nursing school professors try to trick you! (did I just say that????) Thanks for this, CW! Good catch!

Kim FifisHello I have a case study.

Patient requires 4mg of Andensetron iv push over 2to 5 minites

how do I write the drug calculations to give in ml ?

I not sure if my last comment sent sorry if you get this twice

Regards

Kim

Nurse MoPost authorHi Kim…in order to answer this question, you need more information. How many ml fluid is your 4mg of Ondansetron in? It should say somewhere in the question something about the vial it comes in, or how to reconstitute it. Check and let me know!

LLOndansetron usually comes 4mg / 1 cc or ml.

Note, if you are giving this for the first time, that is this is the first dose for the patient, be sure to give it over 5 minutes, and never give more than 4 mg at a time! The side effects can be severe such as LOC, tongue swelling, TIA. I saw this happen once when the doctor ordered 12mg IVP. Nurse checked with pharmacy and they approved it! She gave it. NEVER DO IT! No matter what the order is, only give 4 mg IVP at a time. Some IV meds only last 1/2 hour, perhaps after an hour, if patient is ok, but still symptomatic, one could repeat the dose, but NEVER EVER give 12 mg at the same time as a first dose. One must recognize that although this doctor may have been used to prescribing this dose in a certain population… in other populations people are very sensitive to medications! Several days later, after patient was admitted to ICU but improved and was discharged, the pharmacy admitted it was an OD. Go with your gut! This nurse said she had never seen or given a dose this high before, and had worked oncology. Play safe always, practice safely always.

Nurse MoPost authorOMG! I can’t imagine!!! I’ve never seen anything but 4mg ordered…yikes! And yes…ALWAYS go with your gut. If something seems “off” then investigate! Great advice 🙂

Katie FlorczykCan you show me how to set this one up?

pt has infusion of 1000ml D5W infusing at 100ml/hr. At 0700 there is 500ml of infusion remaining. The completion time for this infusion is?

Nurse MoPost authorIs the question asking you how many hours total for the infusion to run, or what time will it complete?

If you want to figure out how many hours total, you just need to divide 1000 by 100 and you’ll get 10 hours. If you want to know at what time the infusion will be complete, then you just look at what time it is now (0700) and how much fluid is gone (half of it). If it takes 10 hours to run the whole bag, then you have 5 hours left on the bag and your infusion will be complete at 1200. This isn’t really a dimensional analysis question…just a time question!

Hope that helps!

Prof TWhat a great site! I’m glad I’m here too! I am a Professor teaching basic chemistry to incoming freshman nursing students. I am not a nurse, I have been a research scientist and teacher for a long time. I have the science part down, but I need insight into the daily lives of nursing students and nurses in general. I want to emphasize good habits, etc for them in addition to the science.

Perfect example: I find that many of these students are math/chem- phobic. I will do my best to explain the science, but how can I convey how important things like conversion factors and calculations are for dosages, etc? Any insight will really be helpful.

Thanks!

Nurse MoPost authorI think it’s important for new students who are afraid of math and chem to understand that these are principles they will use daily as an RN…so basically, they need to accept it and move forward! The best way to teach this, in my opinion, is to utilize examples that are related to the field of nursing. So, when you are teaching dimensional analysis maybe throw in a few samples that are related to med dosing…when discussing chemistry, try to relate it to the field of nursing whenever you can. That way, they see that it’s applicable and might even find it interesting. I loved chemistry…it’s so fascinating!

dairian bolleyin the last problem what happened with the unit PT. It doesnt cross cancel with any and it not labled at the end. where does it go? thank you this makes it all five times easier!! I appreciate you taking the time do this! that one just through me off

dairian bolleyOh my goodness nevermind.. it stands for patient lol Im sorry this is my second glance into dosage calc and I am TOTALLY overthinking it!!

Nurse MoPost authorThat’s quite alright! Nursing school has a tendency to do that to your brain 🙂

Maddrey CooverSo, total thanks for this tutorial and your site!

In your calculations as listed, does one not multiply all of the numerators, then multiply all of the denomenators together then divide the numerator by the resultant denominator? I couldn’t get your numbers to work with sequential dividing of the denominators, which is as written. Implying we were dividing all of those numbers by each other straight across, but they work when multiplied. Math is my weak spot, so feel free to school me. And I absolutely love the 1/ angry pt. Perfect.

Thanks so much!

Nurse MoPost authorMaddrey…I always multiply across the top, hit the equal sign, then divide all across the bottom and hit the equal sign one last time. Not sure why it’s not working for you…but I tried it your way and that works as well!

Daniel P.Hi Nurse Mo! I’m loving your site! I have a dosage question that is throwing me through the loop. I can’t seem to figure out how to set it up, or solve it for that matter. The question reads…

A 30 mL bag of solution contains 3.5 grams of ampicillin and is to be infused over 25 minutes.

What should the rate of flow per minute be if each milliliter contains 60 drops of solution?

When the intravenous solution has infused at the correct rate for 15 minutes, how much ampicillin will the patient have received (rounded to the nearest tenth of a gram)?

Can you help me set this type of problem up using dimensional analysis?

Nurse MoPost authorThis is a great example of a dosage calc question designed to trick you! The key is knowing which information you need for each calculation…once you figure that out, the calculations are pretty simple!

For the first question, set your equation up like so:

30ml/25min x 60 gtt/ml…multiply across the top (30 x 60), then divide across the bottom (divide by 25) and you get 72gtt/min.

For the second part, the question is really asking…how much ampicillin will the patient receive each minute? So, set it up like so:

3.5g/25 min…do the division (3.5 divided by 25) to get 0.14 g per minute. Now just multiply by 15 to get a total of 2.1 grams in 15 minutes.

Hope that helps!

Daniel PThank you so much Nurse Mo! I had a sneaky suspicion I was making it much more difficult than it needed to be. ?

Nurse MoPost authorThat’s what they’re trying to do! We’re on to their sneaky ways!

Tayla RHi Nurse Mo!

Thank you for this post – I had watched about 40 different Youtube videos covering nursing numeracy but nothing seemed to click aha! in my brain until I saw this post. Its safe to say I have some basic dosages down.

However, questions with g, mg, mcg make my brain hurt. I may be overthinking it but I feel stumped as how to work it out!!

A client is ordered 62.5 micrograms of Benztropine Mesylate. 0.25 mg tablets are available. How many tablets will you give?

I set out my problems as shown above, (is this wrong? Perhaps they are both divided?) but how would one work this out? I know mcg is smaller than mg…but…HELP?! Please, Id love to hear how you might work this out.

Thanks in advance, look forward to hearing back. 🙂

Nurse MoPost authorHi Tayla…this is just a simple conversion from mcg to mg. There are 1000 mcg in each milligram, so you’ll work that into your dimensional analysis. Write it out like this:

62.5mcg/1 X 1mg/1000mcg X 1 tablet/.25mg

Then cross out your units…mcg and mg (this leaves you with tablets, which is what you want).

Then multiply across the top and divide across the bottom to get 0.25 tablets.

Hope that helps!

Tayla RHi Nurse Mo,

Thank you – I actually worked it out pretty soon after posting this and felt like such a silly billy!

Thank you anyway, hopefully my brain remembers now!

Daniel PHey Nurse Mo! I’ve found another problem that is giving me some issues. This comes from the pediatric section.

Your patient weighs 18.62kg. The doctor has ordered the patient to receive 14mg/hr of medication which comes in a concentration of 2.8 mg/ml.

How many mcg/min/kg will the patient receive?

I am at a loss trying to figure out how to set up this problem. I know there is going to be a lot of converting, I’m just unsure on where to begin.

Simone C.This IS AWESOME ? THANKS

Nurse MoPost authorFantastic! Glad you like!!! Hope it helps and that you rock nursing school 🙂

SandraHi there Nurse Mo

Firstly thanks for your fab site – as a fifty something mature aged student l am struggling with the drug calculations side of it – my problem is l am overthinking everything and l am surrounded by young bright things who appear to ‘get it’ no problems.

This has exacerbated my maths anxiety – there seem to be numerous different formulae for intravenous delivery – and l have become totally confused – l might be right off track here Nurse Mo but does your method work for All the infusion calculations? Any advice you have in approaching intravenous drug calculations for this mature aged nursing student from Australia who’s maths abilities were never brilliant to start with would be so very very welcomewelcome

Nurse MoPost authorHi Sandra….YES, this method works for ALL calculations ALL THE TIME! My advice is to work through the tutorial step by step, then check out the other posts on dosage calculations and the sample quiz. If you are still having trouble with a SPECIFIC type of question after all that, let me know. YOU GOT THIS!

MichaelThanks Nurse Mo,

This is a great resource for this 50 something on his 3rd career. I have the “desire” over “have” times “volume” (D/H X Vol) down with no problem. Dimensional analysis just had me confused. my instructor showed us but went thru it very quickly and D/H x V worked for everything, but now on to IV Calculations, I was told that this would be easier. Now I think I have this.

Thanks A lot for your expertise,

Michael

Nurse MoPost authorThe nice thing about dimensional analysis is that it works for EVERY type of med math problem EVERY time! I don’t even know what D/H x V is…looks confusing 🙂

Glad it helped! Keep on trucking, Michael…50 is the new 30 and having a third career just makes you experienced and curious about life. All good things!