Chapter 02: Measurement Systems and Unit Conversions
02
CHAPTER
Measurement Systems and Unit Conversions
This chapter will focus on 3 topics:
- Metric, Household, and Apothecary Conversions
- Weight, Volume, and Length Conversions
- Dimensional Analysis (unit factor method)
Metric, Household, and Apothecary Conversions
Metric conversions is the conversion between units within the metric system (i.e mL, L, dL). The metric system is a standardized decimal system that is used globally that is important in science and healthcare. There is 2 parts to a metric unit. the first part is the base unit such as meters – m, liters – L, and grams – g are common units healthcare. The next part is the prefix before the unit which directs the where decimal point. Here is a table of common prefixes and in the power of 10 form.
Metric System Prefixes
| Prefix | Symbol | Meaning | Factor (Multiply Base Unit by) |
|---|---|---|---|
| Kilo- | k | Thousand | 1,000 (103) |
| Hecto- | h | Hundred | 100 (102) |
| Deka- | da | Ten | 10 (101) |
| Base unit | – | Gram, Liter, Meter | 1 |
| Deci- | d | Tenth | 0.1 (10-1) |
| Centi- | c | Hundredth | 0.01 (10-2) |
| Milli- | m | Thousandth | 0.001 (10-3) |
| Micro- | µ | Millionth | 0.000001 (10-6) |
| Nano- | n | Billionth | 0.000000001 (10-9) |
Notice the bolded prefixes as there recognized as the most common prefixes a pharmacist will encounter. For instance, most liquid calculations like milliequivalents and other pharmacy doses with immunizations will have mL while most calculation that take in account of weight will use kg. while nano is not bolded, is still to have an idea of the scope when dealing with filtration of particles when studying pharmaceutics. The prefixes of hecto and deka are not as common in pharmaceutical calculations but are given for a better understanding of the metric system. Let’s do some conversion practice:
Example 1
A pharmacy receives a bulk supply of 2,500 mL of sterile water. What is 2,500 mL in liters?
We know 1 gram equals 1000 mg from the table above. We can set up a proportion to find the answer:
\(\frac{1\ L}{1000\ ml} = \frac{x\ L}{2500\ ml}\)
By setting up a proportion, we can get an answer of 2.5 L. You can also solve by know the factors of mL to the base unit L. we know mL is 10-3 of 1 L. you can then take 2500 mL and move the decimal place to the left because we are converting to a larger unit therefore the number will be smaller. And the 3 means the decimal place moves 3 times which gives us an answer of 2.5 L.
Example 2
You the pharmacist are treating a patient for hypothyroidism. The prescriber wants to put them on levothyroxine. You have calculated the dose of 0.075 mg starting out. Levothyroxine tablets come in micrograms, what the equivalent dose in micrograms (µg)?
We first look at what we have which is 0.075 mg. You can then set up a proportion going back to base unit grams which would move the decimal place 3 ties to the left making the unit now g and the digits equal to 0.000075 g. You do this because the exponent next to 10 which in this case is -3. The 3 indicates what the decimal place moves 3 times. Because the exponent is negative and we are going to a bigger unit, we know the number should get smaller which means the decimal place will move to the left. To convert it to micrograms, you know that micrograms is 10-6 from the base unit which means we will move the decimal place 6 times and because to a negative exponent, we will move the decimal place the right to get a conversion of 75 µg. Many students will do this in the beginning to get an understanding on how to convert. You can also convert by taking the difference of powers. In this case you take the mg factor of -3 and subtract it by -6 which is the gives a difference of 3. This means you can just take the original dose of 0.075 mg and move the decimal point to the right 3 times as the unit is getting smaller so the number should be larger. This will yield the same result of 75 µg. This way is much easier and cleaner for converting when the student understands the power of 10 factors relating to the base unit.
Household units are everyday measurement units commonly used to measure liquid or solid matter. Household conversions are non-standardized unlike metric conversions. Some common household units are teaspoon, tablespoon, cup, and an ounce. Here is a table of some household units you’ll be expected to know for the NAPLEX and in practice.
| Household Unit | Abbreviation | Metric Equivalent | Household Equivalent |
|---|---|---|---|
| Teaspoon | tsp | 5 mL | 1/3 tbsp |
| Tablespoon | tbsp | 15 mL | 3 tsp |
| Fluid ounce | fl oz | 30 mL | 2 tbsp |
| Cup | – | 240 mL | 8 fl oz |
| Pint | Pt | 473 mL | – |
| Quart | qt | 946 mL | 2 pt |
| Gallon | gal | 3785 mL | 4 qt |
| Ounce | oz | 28.4 g | – |
| Pound | lbs | 1 kg | – |
Notice the difference in oz as fl oz relates to liquid while oz relates to solids. fl oz is not exactly 30 mL, but we round to 30 mL because household units are for conveniency. An example of this is when working at a pharmacy, a prescriber might order cough syrup for ill patients that say 2 fl oz. A pharmacy technician will measure 60 mL instead of 59.2 mL because it is more practical. A Pharmacist would use precise measurement when compounding as more precise and accurate equipment is available.
Lets do some examples problems
Example 1
A patient is instructed to take 20 mL of cough syrup every 6 hours. What is 20 mL equivalency in teaspoons and tablespoons?
For conversions like these you can set up a proportion. Let’s start with finding the equivalency in teaspoons.
\(\frac{1\ tsp}{5\ mL} = \frac{x\ tsp}{20\ mL}\)
With a simple proportion, we get an answer of 4 tsp. Now we do the exact same thing for tablespoons and get an answer of 1.33 tbsp. This example is very practical as some patients don’t have syringes around to measure mL almost every patient has a measuring cup to measuring spoons.
Example 2
A pharmacist a mouth wash rinse and pours 1¾ cups into a bottle. When dispensing, the patient asks how many fluid ounces this is because their measuring cup is marked in ounces.
To solve this problem, it is more straightforward to first convert cups to milliliters then to ounces. This can be done by setting up 2 proportions: one converting cups to milliliters, and another for converting milliliters into ounces.
\(\frac{1 \ cup}{240 \ mL} = \frac{1.75 \ cups}{x \ mL}\)
\(\frac{1 \ fl \ oz}{30 \ mL} = \frac{x \ fl \ oz}{420 \ mL}\)
Using proportions, we should get an answer of 14. Now another way of solving the problem is knowing how many ounces is in a a cup which is 8. Using this knowledge we can set a proportion converting cups to ounces.
\(\frac{1 \ cup}{8 \ fl \ oz} = \frac{1.75 \ cups}{x \ fl \ oz}\)
This will yield the same answer while avoiding an unnecessary conversion of cups to milliliters. Just like the metric system, the more you practice and memorize common conversions, the easier to recognize patterns within different household units such as cups and ounces.
Apothecary units are historical system used in pharmaceutical calculations that is traditional units for weights, and measures. Apothecary units are primarily used for pharmaceutical context. Some units like pints and quarts are categorized also as apothecary units, but are more typically associated with household measurements. Here a list of apothecary units
| Category | Unit | Abbreviation | Equivalent in Apothecary System | Equivalent in Metric System |
|---|---|---|---|---|
| Weight | Grain | gr | – | 1 gr ~ 64.8 mg (rounded to 65 mg) |
| Scruple | ℈ | 1 ℈ = 20 gr | 1 scruple ~ 1.3 g | |
| Dram | 3 | 1 ʒ = 3 scruples or 60 gr | 1 dram ~ 3.9 g | |
| Volume | Minim | m | – | 1 minim ~ 0.0616 mL |
| drop | gtts | – | 1 gtts ~ 0.5 mL | |
| Fluid dram | fl3 | 1 flʒ = 60minims | 1 flʒ = 3.7mL |
The only units to know for practice and for the NAPLEX is grains and drops as they are widely used in pharmaceutics such as compounding. You will probably see some apothecary units in historical pharmacy orders primarily before the 20th century. Let’s do some practice with apothecary units
Example 1
A prescription calls for aspirin gr V to be taken daily. What is the dose in milligrams (mg)?
The V is a roman numeral that is used to describe the number of grains. V is the roman numeral for 5 which means the prescription calls for 5 grains. To solve this problem, we would set a proportion of grains to milligrams and for you can use the rounded number for grains as it is commonly accepted.
\(\frac{1\ gr}{65\ mg} = \frac{5\ gr}{x\ mg}\)
By setting up a proportion, we can determine that the equivalent dose is 325 mg.
Example 2
A prescription instructs the patient to take 60 drops (gtt) of a liquid supplement per dose. The patient wants to know many teaspoons (tsp) this equals.
For a problem involving two category of units in this case apothecary and household, it is easier to convert back to mL. to solve we would set up two proportions: one going from gtt to mL, and the second going from mL to teaspoons.
\(\frac{1 \ gtt}{0.05 \ mL} = \frac{60 \ gtt}{x \ mL}\)
This yields an answer of 3 milliliters. Now that we have drops to milliliters, let’s find milliliters to teaspoons.
\(\frac{1\ tsp}{5\ mL} = \frac{x\ tsp}{3\ mL}\)
Applying the proportion leads to a dose of 0.6 tsp, which is the final answer.
Understanding and accurately converting between metric, household, and apothecary units fundamental for safe and effective pharmacy practice. The metric system is universal standard for drug dosing and compounding, household is often used when writing prescription instructions to the patients, and apothecary is a historical system that is outdated but still may appear in older prescriptions, particularly grains and drops. A proficiency and advancement in unit conversions allows pharmacists to interpret prescriptions accurately, prepare medications accordingly, and clearly communicate instructions to the patients. With continued practice, unit conversions become second nature and form a foundation for more advanced calculations in future chapters.
Weight, Volume, and Length Conversions
Weight, volume, and length conversions have for the most part been taught from previous chapters. Length, however, has not been. Length conversions are mandatory to know in pharmacy school as it is a vital piece to many conversions like Body surface area, ideal body weight, and BMI. Length is also good for checking diagnostic criteria for some disease states such as cellulitis as a a larger diameter might indicate a more severe of fast progressing infection.
Here is a table listing notable volume, weight, and length units:
| Unit | Measurement in Compounding | Measurements by Patients |
|---|---|---|
| Liquid (Volume) | ||
| 1 drop (gtt) | 0.05 mL | 0.05 mL |
| 1 teaspoon (tsp) | 5 mL | 5 mL |
| 1 tablespoon (tbsp) | 15 mL | 15 mL |
| 1 fluid ounce (fl oz) | 29.57 mL | 30 mL (approx.) |
| 1 cup = 8 oz | 236.56 mL | 240 mL (approx.) |
| 1 pint = 16 oz | 473 mL | 480 mL (approx.) |
| 1 quart = 4 cups | 946 mL | 960 mL (approx.) |
| 1 gallon = 4 quarts | 3785 mL | 3840 mL (approx.) |
| Solid (Weight) | ||
| 2.2 lb | 1 kg | 1 kg |
| 1 ounce (oz) | 28.35 g | 28.4 g |
| 1 grain | 64.8 mg | 65 mg |
| Length (Height) | ||
| 1 inch (in) | 2.54 cm | 2.5 cm |
For some examples below we are going to focus mainly on height conversions.
Example 1
A patient is listed at 5 ft 9”, what is the patient’s height in cm?
First what you want to do is convert the patients height into inches. 1 foot equals 12 inches and the patient is 5 foot 9 inches. you can set up a proportion if needed, but simple math is easy so it is not necessary. You should get an answer of 69 inches. now we convert inches to cm. You could set up a proportion like this:
\(\frac{1 \ in}{2.54 \ cm} = \frac{69 \ in}{x \ centimeters}\)
Or you could just multiply 69 by 2.54 to get the same answer which is 170.18 cm. it is important to make sure write centimeters to the hundredth decimal for the final answer if the number goes farther out.
Example 2
You are calculating a BSA formula for a patient. the patient is 172 cm tall. What is 172 cm in m?
By using the metric conversion system from the last topic, we know 100 cm = 1 m. you can set a proportion, or you can just divide 172 cm by 100 cm get 1.72 m as the answer.
Dimensional Analysis (unit factor method)
Dimensional analysis, also called unit factor method or factor label method, is a systematic approach to problem solving that focuses on unit conversion and unit cancelation. Dimensional analysis is one of the most used strategies to solve more advanced calculations in pharmaceutics. Dimensional analysis is used mainly when dealing with multi-step problems requiring 2 different type of units such as a dosing interval 20 mL/hr. Dimensional analysis is a little different from setting up a proportion, but similar in regards to finding the same answer. Here’s an example on how a dimensional analysis is set up. To less complicate the analysis, the example will only use units and not numbers to focus on the main point.
A patient is getting 5 mcg/kg/min dose of dopamine. The patient weighs 70 kg, and the concentration of dopamine is 400 mg in 250 mL. What is the rate in mL/hr should the infusion pump be set?
So we know we want the pump set to mL/hr and we have mcg/kg/min or mcg/kg * min. Dimensional analysis like stated before is about unit conversion and unit canceling. In this example we need to cancel the weight units and add volume while converting min to hr. This might sound complicated, but implementing dimensional analysis can make setting up the problem easy. Here is how a pharmacist might approach the problem using dimensional analysis:
\(\frac{mcg}{kg * min} \times \frac{kg}{1} \times \frac{mg}{mcg} \times \frac{mL}{mg} \times \frac{min}{hr}\)
To a first-time learner, this might look confusing. The first thing you do is write down what you have which is the dose of dopamine right now which is mcg/kg * min. the second fraction is the weight of the patient as because we are multiplying by kg, we can the kg of the dose of dopamine gets canceled out. The next 2 fractions are trying to get rid of the weight. The first part is unit converting mcg into mg so that the second part we can cancel out mg while also adding mL at the same time. the last part is converting min to hours as that is what the problem is asking. you can also cancel min. At the end you should get the units of mL/hr. You can arrange conversion factors in any order, but it’s good practice to start with the given value and also place related unit conversions such as mcg to mg next to each other. This helps maintain clarity and prevents confusion as you work through the problem. Most students like to cross out the units that get canceled for they can backtrack later if needed and it makes it easier to know what units are left.. This is what canceling each unit looks like:
\(\frac{\cancel{mcg}}{\cancel{kg} * \cancel{min}}\times \frac{\cancel{kg}}{1} \times \frac{\cancel{mg}}{\cancel{mcg}} \times \frac{mL}{\cancel{mg}} \times \frac{\cancel{min}}{hr}\)
As you can see mL and hr are the only units that don’t cancel out and when solving creates the dose of mL/hr. Of course this is half the battle as numbers were not used. This is what a full dimensional analysis would look like:
\(\frac{5\ \cancel{mcg}}{\cancel{kg}\ *\ \cancel{min}}\ \times\ \frac{70\ \cancel{kg}}{1}\ \times\ \frac{1\ \cancel{mg}}{1000\ \cancel{mcg}}\ \times\ \frac{250\ mL}{400\ \cancel{mg}}\ \times\ \frac{60\ \cancel{min}}{1\ hr}\)
The pump should be set to 13.125 mL/hr. Remember the metric unit conversions when conducting a dimensional analysis. When calculating everything, it is important to not round until the very end to avoid any error in accuracy. Let’s try some examples.
Example 1
A pharmacist prepares 60 mL of a 2% lidocaine solution. When looking into options for lidocaine, they notice only 5% lidocaine available. How many mL of the 5% stock solution should be used, and how much diluent should be added to make 60 mL of a 2% solution?
Let’s define the variables, we know the final volume is 60 mL, the final concentration is 2%, and the stock concentration we have is 5%. First, we need to know how much drug is needed to be in a 60 mL of a 2% solution. we also know that % is grams of active ingredient over 100 milliliters. This is how you would set up a dimensional analysis:
\(\frac{2\ g}{100\ mL}\times\frac{60\ mL}{1}\)
the total active ingredient we need is 1.2 g. Now we need to find how much of the 5% stock has 1.2g of lidocaine.
\(\frac{1.2 \ g}{1} \times \frac{100 \ mL}{5 \ g}\)
This dimensional analysis will yield an answer of 24 mL. breaking down to find the active ingredient to find the mL required does not have to be a separate step. The problem is asking for mL and combining both dimensional analysis, will still yield you with the same answer, and you do not have to write down the extra step of finding the total g of the active ingredient. Here’s what a combined dimensional analysis would look like:
\(\frac{100 \ mL}{5 \ g} \times \frac{2 \ g}{100 \ mL} \times \frac{60 \ mL}{1}\)
Now the final part is to know how much diluent to add to the solution. Well, we need 60 mL and know that 24 mL is required of the 5% solution to get the same g of the active ingredient. If we subtract 24 from 60 we get 36 mL of diluent as the answer. The answer to the problem is 24 mL of the 5% lidocaine solution and 36 of a diluent to prepare 60 mL of a 2% lidocaine prescription.
Example 2
A 55-year-old female (5’3″, 130 lbs) is started on vancomycin 1.25 grams IV every 12 hours. Vancomycin comes in vials labeled 1 gram per 20 mL. The dose is added to 200 mL of normal saline and infused over 90 minutes.
- How manymilliliters of vancomycin solution are needed for each 1.25-gram dose?
- What is theinfusion rate in mL/hr for this dose? (Round to the nearest tenth)
To solve the problem, lets write out what we have and what we need
What we have:
– Dose: 1.25 g/ 12 hr
– Vials: 1g / 20 mL
What we need:
– How many milliliters needed for the 1.25 dose
– Infusion rate of when the vancomycin is given after the dose is put into a 200 mL saline bag and infused over 90 mins
To find how many milliliters you need for the dose you can set up a simple dimensional analysis:
\(\frac{1.25g}{1} \times \frac{20\ mL}{1\ g}\)
The milliliters required for 1.25 g of vancomycin is 25 mL. We don’t account for the dose interval which is every 12 hr because we are only calculating one dose and not multiple, so the dosing interval is not relevant to this case.
For finding the infusion rate we must first combine the mL needed for the dose of vancomycin and the 200 mL of the saline solution. the total milliliters we are infusing is 225 mL and now set up a dimensional analysis to find the infusion rate:
The milliliters required for 1.25 g of vancomycin is 25 mL. We don’t account for the dose interval which is every 12 hr because we are only calculating one dose and not multiple, so the dosing interval is not relevant to this case.
For finding the infusion rate we must first combine the mL needed for the dose of vancomycin and the 200 mL of the saline solution. the total milliliters we are infusing is 225 mL and now set up a dimensional analysis to find the infusion rate:
\(\frac{225 \ mL}{90 \ mins} \times \frac{60 \ mins}{1 \ hr}\)
This results in an infusion rate of 150 mL/hr. we again do not include the dosing interval as the infusion rate is the rate as the drug is being delivered to the body every dose.
Dimensional analysis is more than just a systematic method, it is a disciplined form of thinking that ensures accuracy by reinforcing attention to detail. As different calculations will require multiple variables like weight, volume, time, and concentration, dimensional analysis is a consistent method that can navigate through all of them. it trains students and pharmacists to approach a problem methodically by verifying units at each step. Building confidence in this topic will provide you with method that supports safe medication practice and sharpens a problem solving mindset that pharmacists rely on every day.
In conclusion, This chapter introduced students to building blocks for pharmaceutical calculations like exploring core measurement systems, metric, household, or apothecary, and methods for converting precisely between them. This chapter also preference the units for weight, volume, length, and time as they are vital for interpreting prescriptions, preparing compounds, and communicating effectively to patients. Units conversions and each measurement system prepared students for the concepts of dimensional analysis which leverage unit relationships to simplify even the most complex calculations. With continued practice, unit references, unit conversions, and dimensional analysis will become second nature and assist you in more advanced calculations in later chapters.
