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!