Home » 02. Calculation of drug dosages Dosage Calculation

02. Calculation of drug dosages Dosage Calculation

June 16, 2024

CALCULATION OF DRUG DOSAGES:

DOSAGE CALCULATION

There are several ways to determine how much of a medication you are supposed to administer to a patient. No matter what method you choose to use, if performed properly, they should all come up with the same answer. Following are three methods for determining the appropriate dose based on information that you have available to you.

Method 1

The ﬁrst method is based on the following formula:

An Example: Medical control orders you to administer 5 mg of morphine sulfate IV to your 84-year-old female patient who has signs and symptoms of a hip fracture. The morphine in your formulary contains 10 mg in 1 mL. How many milliliters of morphine sulfate do you need to administer to this patient in order to deliver 5 mg?

You have the following information:

Order: 5 mg morphine sulfate IV

On hand: 10 mg/1 mL

Fill in the formula:

Cancel any common values (volumes or concentrations) that exist on the top and on the bottom, and multiply across the top.

You need to administer 0.5 mL of morphine sulfate to your patient.

Method 2

This second method involves ratio and proportion. The symbol for proportion is and the symbol for ratio is using the same problem as in method 1, start with the known ratio on the left side of the proportion:

Place the unknown ratio on the right side of the proportion in the same sequence as the ratio on the left side of the proportion. This ratio is usually the physician order or the dosage that you are permitted to administer based on standing orders:

First, multiply the extremes ( the far outside values: 10 mg and X mL) and place the result on the left side of the equation. Second, multiply the means ( the numbers on either side of the proportion symbol: 1 mL and 5 mg) and place this value on the right side of the expression:

Multiply:

Divide both sides by the number in front of the X

You need to administer 0.5 mL of morphine sulfate to your patient.

Method 3

The third method is referred to as the cross multiplication method. This method sets the problem up using fractions. The ﬁ rst fraction is the concentration, and the second fraction is the physician’s order over the volume of medication being administered.

Cross multiply the fractions by multiplying numerators by the denominator on the opposite side. Express the results as an algebraic equation the same as used in the proportion method.

You need to administer 0.5 mL of morphine sulfate to your patient.

Many nurses are weak with drug calculations of all sorts. This article will help to review the major concepts related to drug calculations, help walk you through a few exercises, and provide a few exercises you can perform on your own to check your skills. There are many reference books available to review basic math skills, if you find that you have difficulty with even the basic conversion exercises.

Common Conversions:

1 Liter = 1000 Milliliters

1 Gram = 1000 Milligrams

1 Milligram = 1000 Micrograms

1 Kilogram = 2.2 pounds

Methods of Calculation

Any of the following three methods can be used to perform drug calculations. Please review all three methods and select the one that works for you. It is important to practice the method that you prefer to become proficient in calculating drug dosages.

Remember: Before doing the calculation, convert units of measurement to one system.

Basic Formula:

Frequently used to calculate drug dosages.

D (Desired dose)

H (Dose on hand)

V (Vehicle-tablet or liquid)

D H	x V = Amount to Give

D = dose ordered or desired dose

H = dose on container label or dose on hand

V = form and amount in which drug comes (tablet, capsule, liquid)

Example:

Order-Dilantin 50 mg p.o. TID
Drug available-Dilantin 125 mg/5ml

D=50 mg

H=125 mg

V=5 ml

50
125

x 5 =

250
125

= 2 ml

Ratio & Proportion:

Oldest method used in calculating dosage.

Known

Desired

Means

Extremes

Left side are known quantities

Right side is desired dose and amount to give

Multiply the means and the extremes

HX = DV

X =

DV
H

Example:

Order-Keflex 1 gm p.o. BID
Drug available-Keflex 250 mg per capsule

D=1 gm (note: need to convert to milligrams)

1 gm = 1000 mg

H=250 mg

V=1 capsule

250

1000

250X = 1000

X =

1000
250

X = 4 capsules

FRACTIONAL EQUATION

H
V

D
X

Cross multiply and solve for X.

H
V

D
X

HX = DV

X =

DV
H

Example:

Order – Digoxin 0.25 mg p.o. QD
Drug Available – 0.125 mg per tablet

D=0.25 mg

H=0.125 mg

V=1 tablet

0.125
1

0.25
X

0.125X = 0.25

X =	0.25 0.125

X = 2 tablets

INTRAVENOUS FLOW RATE CALCULATION (TWO METHODS)

Two Steps

Step 1 – Amount of fluid divided by hours to administer = ml/hr

Step 2 –

ml/hr x gtts/ml(IV set)
60 min

= gtts/min

One Step

amount of fluid x drops/milliliter (IV set)
hours to administer x minutes/hour (60)

Example:

1000 ml over 8 hrs
IV set = 15 gtts/ml

Two Steps

Step 1 –

1000
8

= 125

Step 2 –

125 x 15
60

= 31.25 (31 gtts/min)

One Step

1000 x 15
8 hrs x 60

15,000
480

= 31.25 (31gtts/min)

HOW TO CALCULATE CONTINUOUS INFUSIONS

mg/min (For example – Lidocaine, Pronestyl)

Solution cc x 60 min/hr x mg/min
Drug mg

= cc/hr

Drug mg x cc/hr
Solution cc x 60 min/hr

= mg/hr

Rule of Thumb

Lidocaine, Pronestyl
2 gms/250 cc D₅W

1 mg = 7 cc/hr
2 mg = 15 cc/hr
3 mg = 22 cc/hr
4 mg = 30 cc/hr

mcg/min (For example – Nitroglycerin)

Solution cc x 60 min/hr x mcg/min
Drug mcg

= cc/hr

Drug mcg x cc/hr
Solution cc x 60 min/hr

= mcg/hr

Rule of Thumb

NTG 100 mg/250 cc

1 cc/hr = 6.6 mcg/min

NTG 50 mg/250 cc

1 cc/hr = 3.3 mcg/min

mcg/kg/min (For example – Dopamine, Dobutamine, Nipride, etc.)

To calculate cc/hr (gtts/min)

Solution cc
Drug mcg

x 60 min/hr x kg x mcg/kg/min = cc/hr

Example:

Dopamine 400 mg/250 cc D₅W to start at 5 mcg/kg/min.
Patient’s weight is 190 lbs.

250 cc
400,000 mcg

x 60 min x 86.4 x 5 mcg/kg/min = 16.2 cc/hr

TO CALCULATE MCG/KG/MIN

Drug mcg/ x cc/hr
Solution cc x 60 min/hr x kg

= mcg/kg/min

Example:

Nipride 100 mg/250 cc D₅W was ordered to decrease your patient’s blood pressure.
The patient’s weight is 143 lbs, and the IV pump is set at 25 cc/hr. How many mcg/kg/min of Nipride is the patient receiving?

100,000 mcg x 25 cc/hr
250 cc x 60 min x 65 kg

2,500,000
975,000

= 2.5 mcg/kg/min

How to calculate mcg/kg/min if you know the rate of the infusion

Dosage (in mcg/cc/min) x rate on pump
Patient’s weight in kg

= mcg/kg/min

For example:

400mg of Dopamine in 250 cc D₅W =

1600 mcg/cc
60 min/hr

26.6 mcg/cc/min

26.6 is the dosage concentration for Dopamine in mcg/cc/min based on having 400 mg in 250 cc of IV fluid. You need this to calculate this dosage concentration first for all drug calculations. Once you do this step, you can do anything!

NOW DO THE REST!

If you have a 75 kg patient for example…

26.6 mcg/cc/min x 10 cc on pump
Patients’s weight in kg (75 kg)

= 3.54 mcg/kg/min

= 3.5 mcg/kg/min (rounded down)

How to calculate drips in cc per hour when you know the mcg/kg/min that is ordered or desired

mcg/kg/min x patient’s weight in kg
dosage concentration in mcg/cc/min

= rate on pump

For example:

400 mg Dopamine in 250 cc D₅W = 26.6 mcg/cc/min

3.5 mcg/kg/min x 75 kg
26.6 mcg/cc/min

= 9.86 cc

= 10 cc rounded up

ALWAYS WORK THE EQUATION BACKWARDS AGAIN TO DOUBLE CHECK YOUR MATH!

For example:

10 cc x 26.6 mcg/cc/min
75 Kg

= 3.5 mcg/kg/min

Dosage (in mcg/cc/min) x rate on pump
Patient’s weight in kg

= mcg/kg/min

For example:

400mg of Dopamine in 250 cc D₅W = 1600 mcg/cc 60 min/hr = 26.6 mcg/cc/min

NOW DO THE REST!!

If you have a 75 kg patient for example

26.6 mcg/cc/min x 10 cc on pump
Patients’s weight in kg (75 kg)

= 3.54 mcg/kg/min

SUMMARY

Many nurses have difficulty with drug calculations. Mostly because they don’t enjoy or understand math. Practicing drug calculations will help nurses develop stronger and more confident math skills. Many drugs require some type of calculation prior to administration. The drug calculations range in complexity from requiring a simple conversion calculation to a more complex calculation for drugs administered by mcg/kg/min. Regardless of the drug to be administered, careful and accurate calculations are important to help prevent medication errors. Many nurses become overwhelmed when performing the drug calculations, when they require multiple steps or involve life-threatening drugs. The main principle is to remain focused on what you are doing and try to not let outside distractions cause you to make a error in calculations. It is always a good idea to have another nurse double check your calculations. Sometimes nurses have difficulty calculating dosages on drugs that are potentially life threatening. This is often because they become focused on the actual drug and the possible consequences of an error in calculation. The best way to prevent this is to remember that the drug calculations are performed the same way regardless of what the drug is. For example, whether the infusion is a big bag of vitamins or a life threatening vasoactive cardiac drug, the calculation is done exactly the same way.

Many facilities use monitors to calculate the infusion rates, by plugging the numbers in the computer or monitor with a keypad and getting the exact infusion titration chart specifically for that patient. If you use this method for beginning your infusions and titrating the infusion rates, be very careful that you have entered the correct data to obtain the chart. Many errors take place because erroneous data is first entered and not identified. The nurses then titrate the drugs or administer the drugs based on an incorrect chart. A method to help prevent errors with this type of system is to have another nurse double check the data and the chart, or to do a hand calculation for comparison. The use of computers for drug calculations also causes nurses to get “rusty” in their abilities to perform drug calculations. It is suggested that the nurse perform the hand calculations from time to time, to maintain her/his math skills.

Drugs and Dosage Formulas and Conversions

Volume

60 minims = 1 dram = 5cc = 1tsp

4 drams = 0.5 ounces = 1tbsp

8 drams = 1 ounce

16 ounces = 1pt.

32 ounces = 1qt.

Weight

60 grains = 1dram 1/60 grain=1mg

8 drams = 1 ounce 15 grains=1g

12 ounces = 1 lb. (apothecaries’) 2.2 lbs.=1kg

Household Apothecary

1tsp = 1 dram

1tsp = 60 gtts (drops)

3tsp = 0.5 ounce

1tbsp = 0.5 ounce

Household Apothecary Metric

1tsp=5cc 1fl.dram=4cc 5cc=1tsp

3tsp=1tbsp 4drams=0.5oz 15cc=1tbsp

1tbsp=0.5oz or 15cc 8drams=2tbsp(1oz) 30cc=2tbsp(1oz)

2tbsp=1oz or 30cc 16minims=1cc 1cc=16minims

1pt.=16oz or 480cc 500cc=0.5L or 1pt.

1qt=32oz or 960cc 1000cc=1L or 1qt.

Temp. Conversion

C= F-32/1.8

F= 1.8*C-32

METHODS OF CALCULATION

Remember: Before doing the calculation, convert units of measurement to one system.

I. Basic Formula: Frequently used to calculate drug dosages.

D (Desired dose)

H (Dose on hand)

V (Vehicle-tablet or liquid)

D H	x V = Amount to Give

D = dose ordered or desired dose

H = dose on container label or dose on hand

V = form and amount in which drug comes (tablet, capsule, liquid)

Example:

Order-Dilantin 50 mg p.o. TID
Drug available-Dilantin 125 mg/5ml

D=50 mg

H=125 mg

V=5 ml

50
125

x 5 =

250
125

= 2 ml

II. Ratio & Proportion: Oldest method used in calculating dosage.

Known

Desired

Means

Extremes

III. Left side are known quantities

IV. Right side is desired dose and amount to give

V. Multiply the means and the extremes

HX = DV

X =

Example:

Order-Keflex 1 gm p.o. BID Drug available-Keflex 250 mg per capsule

VII. D=1 gm (note: need to convert to milligrams)

VIII. 1 gm = 1000 mg

IX. H=250 mg

X. V=1 capsule

250

1000

XI. 250X = 1000

X =

1000
250

XII. X = 4 capsules

XIII. Fractional Equation

H
V

D
X

XIV. Cross multiply and solve for X.

HX = DV

X =

DV
H

Example:

Order – Digoxin 0.25 mg p.o. QD Drug Available – 0.125 mg per tablet

D=0.25 mg

H=0.125 mg

V=1 tablet

0.125
1

0.25
X

XX. 0.125X = 0.25

X =

0.25
0.125

XXI. X = 2 tablets

XXII. How to Calculate Continuous Infusions

A. mg/min (For example – Lidocaine, Pronestyl)

Solution cc x 60 min/hr x mg/min
Drug mg

= cc/hr

Drug mg x cc/hr
Solution cc x 60 min/hr

= mg/hr

Rule of Thumb

Lidocaine, Pronestyl
2 gms/250 cc D₅W

1 mg = 7 cc/hr

2 mg = 15 cc/hr

3 mg = 22 cc/hr

4 mg = 30 cc/hr

D. mcg/min (For example – Nitroglycerin)

Solution cc x 60 min/hr x mcg/min
Drug mcg

= cc/hr

Drug mcg x cc/hr
Solution cc x 60 min/hr

= mcg/hr

Rule of Thumb

NTG 100 mg/250 cc

1 cc/hr = 6.6 mcg/min

NTG 50 mg/250 cc

1 cc/hr = 3.3 mcg/min

G. mcg/kg/min (For example – Dopamine, Dobutamine, Nipride, etc.)

1. To calculate cc/hr (gtts/min)

Solution cc

Drug mcg

x 60 min/hr x kg x mcg/kg/min = cc/hr

Example:

Dopamine 400 mg/250 cc D₅W to start at 5 mcg/kg/min.
Patient’s weight is 190 lbs.

250 cc
400,000 mcg

x 60 min x 86.4 x 5 mcg/kg/min = 16.2 cc/hr

4. To calculate mcg/kg/min

Drug mcg/ x cc/hr
Solution cc x 60 min/hr x kg

= mcg/kg/min

Example:

100,000 mcg x 25 cc/hr
250 cc x 60 min x 65 kg

2,500,000
975,000

= 2.5 mcg/kg/min

A. How to calculate mcg/kg/min if you know the rate of the infusion

Dosage (in mcg/cc/min) x rate on pump
Patient’s weight in kg

= mcg/kg/min

B. For example:

400mg of Dopamine in 250 cc D₅W =

1600 mcg/cc
60 min/hr

26.6 mcg/cc/min

C. 26.6 is the dosage concentration for Dopamine in mcg/cc/min based on having 400 mg in 250 cc of IV fluid. You need this to calculate this dosage concentration first for all drug calculations. Once you do this step, you can do anything!

D. NOW DO THE REST!

E. If you have a 75 kg patient for example…

26.6 mcg/cc/min x 10 cc on pump
Patients’s weight in kg (75 kg)

= 3.54 mcg/kg/min

= 3.5 mcg/kg/min (rounded down)

F. How to calculate drips in cc per hour when you know the mcg/kg/min that is ordered or desired

mcg/kg/min x patient’s weight in kg
dosage concentration in mcg/cc/min

= rate on pump

G. For example:

H. 400 mg Dopamine in 250 cc D₅W = 26.6 mcg/cc/min

3.5 mcg/kg/min x 75 kg
26.6 mcg/cc/min

= 9.86 cc

= 10 cc rounded up

I. ALWAYS WORK THE EQUATION BACKWARDS AGAIN TO DOUBLE CHECK YOUR MATH!

J. For example:

10 cc x 26.6 mcg/cc/min
75 Kg

= 3.5 mcg/kg/min

Dosage (in mcg/cc/min) x rate on pump
Patient’s weight in kg

= mcg/kg/min

L. For example:

M. 400mg of Dopamine in 250 cc D₅W = 1600 mcg/cc 60 min/hr = 26.6 mcg/cc/min

N. 26.6 is the dosage concentration for Dopamine in mcg/cc/min based on having 400 mg in 250 cc of IV fluid. You need this to calculate this dosage concentration first for all drug calculations. Once you do this step, you can do anything!

O. NOW DO THE REST!!

P. If you have a 75 kg patient for example

26.6 mcg/cc/min x 10 cc on pump
Patients’s weight in kg (75 kg)

= 3.54 mcg/kg/min

CALCULATING IV DRUG DOSES

Principles

1. For drugs already in solution, check the amount of drug in each ml and the total amount of drug in the container.

2.Make sure you are clear about the dose units used.Most commonly prescribed are milligrams (mg) or micrograms.

3. Beware of drugs such as insulin and heparin, for which doses are prescribed in international units (which is sometimes, but should never be abbreviated to i.u. which can be misread as 10).

4. Check the dose on the prescription and that it is expressed in the same units as on the medicine label.

5. If the prescription and the medicines label use different units of strength, refer to the conversion table and calculation examples on page 4 and 5.

6. Once you are sure that the units are the same, divide the required dose by the amount of the drug in the ampoule and multiply by the volume of solution in the vial or ampoule.

7. The answer is the volume needed for each dose.

CALCULATING DRIP RATES FOR GRAVITY FLOW INFUSIONS

PRINCIPLES

1. Without a flow control device such as a pump, infusion rates depend entirely on gravity. Rate of flow is measured by counting drops per minute.

2. Administration sets deliver controlled amounts of fluid at a predetermined fixed rate, measured in drops per minute.

3. It is also important to check the number of drops per ml delivered by the administration set (which is printed on the outer packaging). This may vary between sets, between manufacturers and between different infusion fluids or drug solutions.

4. A (drug) solution administration set will usually deliver 20 drops per ml of clear infusion fluid such as 0.9% sodium chloride injection.

5. A blood administration set will deliver 15 drops per ml of blood.

6. A burette administration set will usually deliver 60 drops per ml of infusion fluid or drug solution.

CALCULATING INFUSION RATES FOR INFUSION DEVICES

1. All infusions require rate control. This can be achieved using a roller clamp (gravity flow), an infusion pump, a syringe driver, a syringe pump or a disposable device.

2. When using any sort of rate control device, check at least the following parameters at regular intervals in accordance with local policy:

3. • Volume given

4. • Volume remaining

5. • Administration rate

6. • Condition of the patient including the administration site.

7. Before and after transfer of care between units or teams, make sure you repeat the above checks.

8. You should always check the manufacturer’s instructions or refer to local policy to ensure you use the correct administration set for the device and that the device is programmed correctly.

9. An administration device should only be used by practitioners who have been trained and are competent in the use of the particular device.

10. The rate may be prescribed in terms of:

11. Volume: For example ml per hour or ml per min OR amount of drug:

For example mg per min or international units per hour.

CALCULATING RATES FOR SYRINGE DRIVERS

A syringe driver pushes the plunger of a syringe forward at an accurately controlled rate.

• For most syringe pumps the rate is set according to the volume of solution injected per hour i.e. in ml per hour.

• For some syringe drivers the rate is set according to the distance travelled by the plunger in mm per hour or mm per 24 hour.

If the rate is to be set in mm, the volume to be adminstered by a syringe driver depends on the diameter of the syringe barrel as well as on the rate setting. Different makes of syringe may have different barrel sizes. It is essential that the brand of syringe to be used is specified and the stroke length is measured.

Serious errors have occurred when settings in mm per hour and ml per hour have been confused.

1. Prepare prescribed infusion.

2. Prime the extension set with fluid.

3. If using a syringe driver, measure the stroke length (the distance the plunger has to travel) in mm.

4. Check carefully the units of time in which the syringe driver operates:

Is the rate set in mm per hour or mm per day (24 hours)?

See diagram for example.

CALCULATING IV DRUG DOSES FOR CHILDREN AND NEONATES

1. Remember that many injections are made for adults. For children’s doses, you may need as little as one tenth (1/10) or even one hundredth (1/100) of the contents of one ampoule or vial.

2.When calculating infusions, consider the child’s total daily fluid allowance.More concentrated individual infusions may be required. Discuss with the pharmacist or prescriber.

3. Ensure the prescribed infusion fluid/diluent is appropriate for the child e.g. the sodium content of an infusion contributes to the child’s total daily sodium requirement.

4. Displacement Values.

For many injections presented as powders for reconstitution, the powder adds to the volume of the final solution after the diluent has been added. This ‘displacement value’ must be taken into account when the dose needed is less than the full contents of the vial or ampoule.

The displacement value can be found on the package insert. It may vary with brands, so it is crucial to check the package insert for the product you are actually using.

5. For the above reasons, the calculations involved in preparing and administering infusions for children are often particularly complex. It is most important that these calculations are independently checked

DRUGS CALCULATION.

Method 1

The ﬁrst method is based on the following formula:

You have the following information:

Order: 5 mg morphine sulfate IV

On hand: 10 mg/1 mL

Fill in the formula:

Cancel any common values (volumes or concentrations) that exist on the top and on the bottom, and multiply across the top.

You need to administer 0.5 mL of morphine sulfate to your patient.

Method 2

Multiply:

Divide both sides by the number in front of the X

You need to administer 0.5 mL of morphine sulfate to your patient.

Method 3

Cross multiply the fractions by multiplying numerators by the denominator on the opposite side. Express the results as an algebraic equation the same as used in the proportion method.

You need to administer 0.5 mL of morphine sulfate to your patient.

TOP CALCULATION TIPS

Drugs are formulated into medicines in such a way that most adult doses are easily calculated and predictable, e.g.1 or 2 tablets, 1 or 2 capsules, 1 vial or ampoule of an injectable medicine, 1 suppository.

Before doing a calculation, it is sensible to estimate the dose you are likely to require so that you know whether your calculated answer seems reasonable i.e. roughly what you expected.

To check doses use a reliable reference source, such as the BNF or BNF for Children.

For recommended administration methods, see local drug policies or national guides such as The IV Guide.

Dose volumes of oral liquid medicines are typically 5-20mls for adults and 5mls or less for children.

Crushing tablets should be avoided wherever possible. Some tablets, such as ‘modified release’ products should never be crushed.Always ask your pharmacists’ advice before crushing tablets. If itmust be done, a pestle and mortar or tablet crusher should be used and the tablet ground to as fine a powder as possible.

Always check childrens’ and babies’ weights carefully.Make sure they are weighed in kg and that their weight is recorded in kg.

If a calculation using weight or surface area gives an answer equivalent to or greater than the normal adult dose, reconfirm that it is what is really required.

If you are in any doubt about a calculation, stop, and contact the ward pharmacist, an on-call pharmacist or the prescriber.

Principles

The way the strength of a drug in a solution is described will affect the way a dose calculation is carried out

Doses may be expressed in a number of different ways:

1.Mass (weight) per volume of solution e.g. mg in 10ml, mMol/L.

2. Units of activity per volume of solution e.g. units per ml.

3. Percentage This is the weight of the drug in grams that is contained in each 100ml of the solution. Common examples are 0.9% sodium chloride; 5% glucose

If you know the number of grams in 1000ml, divide by 10 to convert to % strength.

If you know the % strength, multiply by 10 to give the number of grams of drug in 1000ml.

If you know the % strength, divide by 100 to calculate the amount of drug in 1ml.

4. Ratios Strengths of some drugs such as adrenaline (epinephrine) are commonly expressed in ratios

CALCULATING ORAL DOSES IN TABLETS

Principles

1. Check the strength of (amount of drug in) each tablet or capsule.

2.Make sure you are clear about the dose units used, most commonly prescribed are milligrams or micrograms.

3. Check the dose on the prescription and that it is expressed in the same units as on the medicine label.

4. If the prescription and the medicines label use different units of strength, refer to the conversion table and calculation examples on page 4 and 5.

5. Once you are sure that the units are the same, divide the required dose by the strength of the tablet or capsule.

6. The answer is the number of tablets/capsules needed for each dose.

Extra safety tip

If your first calculation gives a dose of more than two tablets, double check the calculation and confirm thatthe dose doesn’t exceed the manufacturer’s recommended maximum. If it does, or if you are still unsure that the dose is correct, check with the prescriber or pharmacist.

CALCULATING ORAL DOSES FOR CHILDREN AND NEONATES

Principles

1. Always use the smallest oral syringe that will hold the volume you need to measure.

2. If the dose prescribed means that less than a whole tablet or capsule is required, check with the pharmacy that it is appropriate to break a tablet or split a capsule before doing so.

3. If it is essential, dissolve or disperse the powder/crushed tablet in an accurately measured amount of water (e.g. 5ml). Stir and draw up the required volume immediately

4. If the result cannot be accurately measured e.g. 0.33ml, it is generally acceptable to round the dose up or down. However, the actual dose given must be within 10% of the calculated dose. If this cannot be achieved, discuss with the prescriber and pharmacist.

Dimensional analysis (also known as the factor-label method or unit-factor method) is by far the most useful math trick you’ll ever learn. Maybe you’ve learned some algebra, but will you use it? For many people the answer is, “not after the final exam.”

For a fraction of the effort needed to learn algebra, you too can learn “dimensional analysis.” First off, however, let’s get rid of the big words. What this is all about is just conversion—converting one thing to another. This is something you will have occasion to do in real life. This is seriously useful stuff.

This trick is about applied math, not about numbers in the abstract. We’re talking about measurable stuff, stuff you can count. Anything you measure will have a number with some sort of “unit of measure” (the dimension) attached. A unit could be miles, gallons, miles per second, peas per pod, or pizza slices per person.

Example 1

How many seconds are in a day?

First, don’t panic. If you have no idea what the answer is or how to come up with an answer, that’s fine—you’re not supposed to know. You’re not going to solve THE PROBLEM. What you are going to do is break the problem down into several small problems that you can solve.

Here’s your first problem:

1. Ask yourself, “What units of measure do I want to know or have in the answer?” In this problem you want to know “seconds in a day.” After you figure out what units you want to know, translate the English into Math. Math is a sort of shorthand language for writing about numbers of things. If you can rephrase what you want to know using the word “per,” which means “divided by,” then that’s a step in the right direction, so rephrase “seconds in a day” to “seconds per day.” In math terms, what you want to know is:

2. Ask, “What do I know?” What do you know about how “seconds” or “days” relate to other units of time measure? You know that there are 60 seconds in a minute. You also know that in 1 minute there are 60 seconds. These are two ways of saying the same thing. You know that there are 24 hours in a day (and in one day there are 24 hours). If you could now connect “hours” and “minutes” together you would have a sort of bridge that would connect “seconds” to “days” (seconds to minutes to hours to days). The connection you need, of course, is that there are 60 minutes in an hour (and in one hour there are 60 minutes). When you have this kind of connection between units, then you know enough to solve the problem–but first translate what you know into math terms that you can use when solving the problem. If in doubt, write it out:

All of these statements, or conversion factors, are true or equivalent (60 seconds = 1 minute). All you need to do now is pick from these statements the ones that you actually need for this problem, so….

3. Ask, “From all the factors I know, what do I need to know?”

Remember that you want to know:

So pick from the things you know a factor that has seconds on top or day(s) on the bottom. You could pick either of the following two factors as your “starting factor:”

Write down your starting factor (say you pick 60 seconds per 1 minute):

Now the trick is to pick from the other things you know another factor that will cancel out the unit you don’t want. You start with “seconds” on top. You want “seconds” on top in your answer, so forget about the seconds—they’re okay. The problem is you have “minutes” on the bottom but you want “days.” You need to get rid of the minutes. You cancel minutes out by picking a factor that has minutes on top. With minutes on top and bottom, the minutes will cancel out. So you need to pick 60 minutes per 1 hour as the next factor because it has minutes on top:

You now have seconds per hour, since the minutes have cancelled out, but you want seconds per day, so you need to pick a factor that cancels out hours:

4. Solve it. When you have cancelled out the units you don’t want and are left only with the units you do want, then you know it’s time to multiply all the top numbers together, and divide by all the bottom numbers.

In this case you just need to multiple 60x60x24 to get the answer: There are 86,400 seconds in a day.

Here’s how this problem might look if it were written on a chalkboard:

Remember that you don’t need to worry about the actual numbers until the very end. Just focus on the units. Plug in conversion factors that cancel out the units you don’t want until you end up with the units you do want. Only then do you need to worry about doing the arithmetic. If you set up the bridge so the units work out, then, unless you push the wrong button on your calculator, you WILL get the right answer every time.

Example 2

How many hours are in a year?
Let’s go through the steps:
Don’t panic
What do you want to know? This one’s easy: hours in a year, or better, hours per year

Translate into math terms:

What do you know that you might need to use? You know that there are 24 hours in a day, 7 days in a week, 4 weeks in a month, 12 months in a year, and about 365 days in a year. The converse, of course, is also true: In one day there are 24 hours, in one week there are 7 days, in one month there are 4 weeks, in one year there are 12 months, and in one year there is about 365 days (actually closer to 365.25 days).

Translate into math terms—writing them all down can’t hurt:

Pick a starting factor. Since you want hours on top (or years on the bottom), you could start with 24 hours per day:

Now pick whatever factors you need to cancel out day(s):

Keep picking factors that cancel out what you don’t want until you end up with the units you do want:

Do the math. You have all the units cancelled out except hours per year, which is what you want, so when you do the math, you know you’ll get the right answer:

But wait, why not go from “hours” to “year” using the fact that there are 365 days per year?

Oh no! There is a 696-hour difference between the two answers! How can this be? Exact answers can only be obtained if you use conversion factors that are exactly correct. In this case there are actually more than 4 weeks in a month (about 4.35 weeks per month). Since there is a bit more than 365 days in a year (365.25 days), a more accurate answer still, to the nearest hour, would be that there are 8766 hours in a year. The point of this example is that your answer can only be as accurate as the conversion factors you use.

Example 3

Sometimes both the top and bottom units need to be converted:
If you are going 50 miles per hour, how many feet per second are you traveling?
If you were to do this one on the blackboard, it might look something like this:

You want your answer to be in feet per second. You are given 50 miles per hour. Notice that both are in distance per time. Normally you can use any value given by the problem as your starting factor. One thing you know, then, is given. The other things you just know or have to look up in a conversion table. Although every conversion factor can be written two ways, you really only need to write each one way. That’s because you know you can always just flip it over and then use it. If you have written 60 min/1 hr, then to solve this problem you would just flip the 60 min/hour factor over. With practice you won’t eveeed to write down what you know, you’ll just pull it out of your head and write down the last part, do the math, and get the right answer.

Example 4

How much bleach would you need to make a quart of 5 percent bleach solution?
You’re not told what answer unit to use, but ounces would work since there are 32 oz in a quart. If the only thing you had to measure bleach with was in milliliters, then you would pick “mL” as your answer unit.

When you are given something like “5 percent” or “5%” by a problem, you need to translate it into math terms you can use. “5%” means 5 per 100 or 5/100, but 5 what? per 100 what? You need to label the numbers appropriately. In this example you write down “5 oz bleach/100 oz bleach solution” or just “5 oz B/100 oz BS” as a factor you are given. If you were going for milliliters, then you would use “5 mL B/100 mL BS.” The top and bottom units must be the same or equivalent, but otherwise can be any units you may need. If you had 5 gal. bleach/100 gal. bleach solution, then you would still have a 5% bleach solution.

Setting up and solving this problem is now easy:

So you would add 1.6 oz bleach to a quart measuring cup, then add water (30.4 oz) to make 32 ounces of 5% bleach solution.

Example 5
Your little sister’s hamster is slowly dying a horrable death from multiple tumors and has stopped eatting. The vet wants $75 to put it to sleep. It is Friday night and even if Mom was willing to spend $75 the hamster and your sister would have to suffer until Monday. You look online and find information about small animal euthanasia. It says you need some vinegar or other 5% acid solution. There is no vinegar in the house, only some muraitic swimming pool acid in the garage. Instead of waiting until tomorrow to go to town and buy vinegar, you want to do something now. According to the label it is 31.45% hydrochloric acid (HCl). Realizing it is dangerous to mess with hydrodhloric acid, you take precautions you learned in chemistry class. Also from chemisty class you recall there is a formula, something about V2 times C2 over C1 but this is not a test, this is for real, and getting the wrong answer is not an option. Fortunately, also in chemistry, you learned dimentional analysis and know that if the formula (and answer) is correct, the units of measure will cancel out leaving you with only the ones appropriate for the answer. So you stop trying to recall the formula and decide to use DA. You have told your sister you can help Fuzzy go to sleep and stop thrashing about. She is crying and won’t go to bed. She stares hopefully at you. She trusts you. You realize it is up to you now to not mess up.

So what you want to know is the number of ounces of concentrated (32%) HCl you need to use to make, say, a quart of 5% dilute acid solution. In the previous example involving bleach, the assumption was that the initial bleach solution was 100% bleach. But HCl doesn’t come in a 100% acid form and what you have is about 32% acid. This means that in 100 oz. of concentrated HCl solution there are 31.45 oz. of actual HCl. You also know that in a 5% dilute solution there are 5 oz. actual pure HCl in 100 oz. of dilute solution. Since you want to end up with a quart of solution, you’ll need to know that there are 32 oz/qt.

So how much of the concentrated acid do you need to use to make a quart of dilute 5% acid? In your answer you want concentrated on top and dilute on bottom, so start with dilute on bottom or concentrated on top:

If the volume of concentrated acid is V1 and its concentration is C1, and V2 is the dilute volume and C2 is the concentration of the dilute acid, then V1 = (V2xC2)/C1. You might remember this formula for a test, but don’t expect to remember it when you need it. With dimensional analysis you can always think your way to the right answer. But knowing the right answer is not enough. You should also know not to ever add water to concentrated acid. So to make a quart of dilute acid you measure out 27 oz of water and add 5 oz concentrated acid to it. Following the instructions of the euthanasia site, Fuzzy goes quietly to sleep. This example is a real world one. Knowing some math can make all the difference.

Example 6
Your car’s gas tank holds 18.6 gallons and is one quarter full. Your car gets 16 miles/gal. You see a sign saying, “Next gas 73 miles.” Your often-wrong brother, who is driving, is sure you’ll make it without running out of gas. You’re not so sure and do some quick figuring:

“Ah! I knew I was right,” declares your brother on glancing at your calculator,
“your calculations prove it!” Is this a good time to be assertive and demand that he turn around to get gas, or do you conclude that your brother is right for once?

Unless the next stop is Las Vegas and you’re feeling really lucky, you should turn around and get some more gas. Your calculator reads “74.4” exactly, but if you believe what it says, then you believe that when the car runs out of gas, it will have gone some where between 74.35 miles and 74.45 miles or 74.4 + 0.05 miles. Obviously the “point four” is meaningless, so you round to 74. Is 74 the right answer? If you think it is, then you think your car will go some where between 73.5 miles and 74.5 miles before running out of gas. When you run out of gas, what chance do you think you’ll have of having gone 74 + 0.5 miles? A proverbial “fat chance” would be a good guess.

When you’re given something like a “quarter tank” you should wonder just how accurate such a measurement is. Can you really divide 18.6 by 4 and conclude that there really is 4.65 gallons of gas in the tank? The last time you figured mileage you came up with 16 miles/gallon, but is the engine still operating as efficiently? Are the tires still properly inflated? Will the next 73 miles be uphill? Do you have a head wind? Surely a calculated estimate of 74 miles is overly precise. A realistic answer, then, might be 74 + 10 miles. You realize you could run out of gas anywhere between 66 and 84 miles, so you finally and correctly conclude that you have a slightly less than a 50/50 chance of running out of gas before reaching the next gas station.

The point of this example is to remind you that dimensional analysis is applied math, not abstract math. The numbers used should describe the real world in so far as possible and indicate no more accuracy than is appropriate. If you overlook this point, you might have a five-mile walk to the next gas station.

Example 7
You’re throwing a pizza party for 15 and figure each person might eat 4 slices. How much is the pizza going to cost you? You call up the pizza place and learn that each pizza will cost you $14.78 and will be cut into 12 slices. You tell them you’ll call back. Do you have enough money? Here’s how you figure it out, step by step.

1. Ask yourself, “What do I want to know?” In this case, how much money is the pizza going to cost you, which in math terms is: cost (in dollars) per party, or just $/party. This is your “answer unit.” This is what you are looking for.

2. Ask, “What do I know?” Write it all down, everything you know: one pizza will cost you $14.78 (in math terms 1 pizza/$14.78). You also know that for $14.78 you can buy one pizza ($14.78/1 pizza). It can be important to realize that every conversion factor you know can be written two ways. One of these ways may be needed to solve the problem and the other won’t, but in the beginning you don’t know which, so just write them both ways. Continue writing down other things you know. You know, or hope, that only 15 people will be eating pizza (15 persons/1 party), or for this one party, 15 people will come (1 party/15 persons). You also know there will be 12 slices per pizza (12 slices/1 pizza), or that each pizza has 12 slices (1 pizza/12 slices). The last thing you know is that each person gets 4 slices (1 person/4 slices), or that you are buying 4 slices per person (4 slices/person). Math is a language that is much briefer and clearer than English, so writing every thing you know in math terms, here’s what you might have written down:

3. Ask, “From all the things above I know, what do I actually need to know to figure out the problem?”

Remember that you want to know $/party, so pick one of the things you know that has either dollars on top, or “party” on the bottom. Let’s start with $14.78/pizza as the starting factor. Great, you got dollars on top, but “pizza” on the bottom where you want “party.” To get rid of “pizza” pick one of the things you know that has “pizza” on the top. “Pizza” over “pizza” cancels out, so you get rid of the “pizza:”

Okay, you now have dollars per slice, but you want dollars per party, so now what? Easy, just keep picking from the things you know whatever cancels out the units you don’t want. The numbers go with the units, but don’t worry about numbers, just pay attention to the units. So you pick 4 slices/1 person to get rid of “slices,” then 15 persons/1 party to get rid of “person(s):”

Now multiply all the top numbers, and then divide by any bottom numbers to get the right number. Finally add the units that are left over to the number to get the answer you wanted. Using this method, you can hardly go wrong unless you push the wrong button on your calculator.

By the way, how many pizzas should you order? Figuring this out should be as easy as….

Example 8

Chemists often use dimensional analysis. Here’s a chemistry problem. To solve it you need to know that, as always, there are 6.02 x 10²³ molecules (or atoms) of whatever in a mole.
A sample of calcium nitrate, Ca(NO₃)₂, with a formula weight of 164 g/mol, has 5.00 x 10²⁵ atoms of oxygen. How many kilograms of Ca(NO₃)₂ are present?
This may look like a some sort of horribly complex nightmare problem, but don’t panic, just take it step by step. Since you want kilograms (kg) in your answer, pick a starting factor with weight (g=grams) on top. Note that in each molecule of calcium nitrate there are two nitrate ions each having three atoms of oxygen (for a total of 6).

Example 9

You have come down with a bad case of the geebies, but fortunately your grandmother knows how to cure the geebies. She sends you an eyedropper bottle labeled:

Take 1 drop per 10 lbs. of body weight per day divided into 4 doses until the geebies are gone.

This problem is a bit more challenging, but don’t panic. Break the problem down into a bunch of small problems, and tackle each one by one.

What do you want to know? In order to take one dose 4 times a day you need to know how many drops to take per dose.

Translated into math terms you want the answer to be in:

What do you know? Well, you know you weigh 160 pounds. You know that you need to take 4 doses per day (implied). You know that you need to take 1 drop per 10-lbs. body weight per day.

Translate this into math terms:

The problem now is the last factor. What can you do with such an odd factor? You rewrite it so it is in a different form that you can use. What you do is multiply the middle term by the bottom term:

So whenever you have a triple-decker, use this trick to rewrite the factor.

Now you should be able to solve the problem. The one thing you know that isn’t a conversion factor is that you weigh 160 lbs., so use that as your starting factor:

If you had wanted to know how many drops per day to take, you would have just left off the last conversion factor, which would give you an answer of 16 drops/day.

Example 10
Okay, enough easy problems, let’s try something harder. Well, not really harder, just longer. The point of this example is that no matter how ridiculously long your conversion might be, long problems are not really more difficult. If you get the point, then skip this example; otherwise read on.

At the pizza party you and two friends decide to go to Mexico City from El Paso, TX where y’all live. You volunteer your car if everyone chips in for gas. Someone asks how much the gas will cost per person on a round trip. Your first step is to call your smarter brother to see if he’ll figure it out for you. Naturally he’s too busy to bother, but he does tell you that it is 2015 km to Mexico City, there’s 11 cents to the peso, and gas costs 5.8 pesos per liter in Mexico. You know your car gets 21 miles to the gallon, but we still don’t have a clue as to how much the trip is going to cost (in dollars) each person in gas ($/person).

1. What do you want to know? $/person round trip—the answer unit(s).
2. What do you know so far? There will be 3 persons going on a round trip (3 persons/1 round trip), or in the planned round trip 3 persons will be going (1 round trip/3 persons), it will be a 2015 km trip one-way (2015 km/one-way trip), or one-way is 2015 km (one-way trip/2015 km), there are 2 one-way trips per round trip (2 one-way/round trip), there is 11 cents per peso (11 cents/1 peso), or one peso is worth 11 cents (1 peso/11 cents). Finally you know that one liter of gas costs 5.8 pesos (1 liter/ 5.8 pesos), or 5.8 pesos will buy you 1 liter (5.8 pesos/1 liter).

You know a lot, but still not enough. Knowing the number of miles in a kilometer, or liters in a gallon would be nice, but one of your friends recalls that there is 39.37 inches in a meter and the other is sure that there is 4.9 ml in a teaspoon. This still isn’t enough. You might need to know that there are 1000 meters in a kilometer, 1000 ml in a liter, 100 cents per dollar, 12 inches to the foot, and 5,280 feet to the mile, but then you already knew that.

Almost enough, but how can you get from teaspoons to gallons? Simple, call your mom. She knows that there are 3 teaspoons in a tablespoon, 16 tablespoons in a cup, 2 cups in a pint, 2 pints in a quart, and 4 quarts in a gallon. Wow, that’s a lot of things to know, but it should be enough. Write it all down in math terms and see what you have:

3. What do you need to know from the above? If any of the above are upside down from what you end up needing, just turn them over, then use them. This problem looks harder than it is. Since we want to end up with $/person, let’s start with:

Even with 18 factors to plug in to get your answer, it’s still pretty much a no-brainer whether you have two or 30 factors. If you know enough conversion factors and set the “bridge” up correctly to cancel out all unwanted units, then you get the correct answer. You may have to look up a few conversion factors you don’t know, but once you do, you’re home free. Looking up the number of liters per gallon or miles in a kilometer would have saved quite a few steps, but if you can remember any relationship you can still figure out your answer. It takes a little longer, but really adds nothing to the inherent difficulty.

Example 11

Now this one is a bit hard if you haven’t paid close attention to the previous examples.
You have come down with a bad case of the geebies, but fortunately your grandmother has a sure cure. She gives you an eyedropper bottle labeled:

Take 1 drop per 15 lb of body weight per dose four times a day until the geebies are gone. Contains gr 8 heebie bark per dr 100 solvent. 60 drops=1 tsp.

You weigh 128 lb, and the 4-oz bottle is half-full. You test the eyedropper and find there are actually 64 drops in a teaspoon. You are going on a three-week trip and are deeply concerned that you might run out of granny’s geebie tonic. Do you need to see her before leaving to get a refill?

Try working this one out before reading further.

First, what do you want to know? You want to know how long the bottle will last. You could figure out days/bottle or weeks/bottle and see if the bottle will last longer than 3 weeks or 21 days. So you write down “days/bottle” as the units you want in your answer.

What do you know to start off with that you might need to know? You write down the following:

You realize that if a 4-oz bottle is half-full, then there is 2 oz of tonic in it, but you could figure it out dimensionally if you wanted to:

You would then end up with “days/half-bottle” in your answer, but it’s easier to just go with 2 oz/bottle as you’re given.

What should you use as a starting factor? You pick 128 lb because it’s something you’re given and it seems lonely. You set the problem up:

Houston, we have a problem. You ended up with units reversed from what you wanted. You figured out how much of the bottle you would use in one day. What to do? You could hit the 1/x button on your calculator if it had one, or invert the answer by dividing 1 by 0.044, or start over with 128 lb on the bottom. What? Can you do that? Sure you can. You could even put 128 lb on the end and on the bottom, or put it in the middle somewhere. You decide to start over, this time picking a starting factor that already has “day” or “bottle” in the right place.

So, it looks like you’ll have enough. At some point you need to know how many drops per dose you will need to take, so you figure it out:

As a practical matter, you can’t take 8.533 drops per dose; you have to round off. At this point you realize that when you calculated 22.5 days/bottle, you were not figuring on 9 drops/dose. You decide to recalculate to see if rounding up to 9 makes a significant difference.

You note a small difference, but conclude that you have just enough geebie tonic. Concluding that you have enough, however, and having enough may not be the same thing. The story continues:

You leave on your trip and on the 19th day you run out of geebie juice. You didn’t spill any, and no one took any. You sit in a stunned stupor trying to figure out where you went wrong in your calculations.

You finally realize there might not have been 2.0 oz of tonic in the bottle to begin with. A measurement like “half a bottle” should not inspire great certainty. You wish you had measured the amount and found that the bottle contained 2.0 + 0.05 oz of tonic, but what you were given, more or less, was that you had 2 + 0.5 oz of tonic. There could be anything from 1.5 to 2.5 oz in the bottle. Recalculating using the low and high values, you find you had enough tonic to last somewhere between 16 and 26 days. If you had figured out the correct answer of 21 + 5 days the first time, you would have realized you had only slightly less than a 50/50 chance of running out, and would have gone to see Granny for a refill.

Summary

Don’t panic.

Figure out what answer unit(s) you want to end up with. This is usually easy.

Write down, in math terms, everything you know that relates to the problem. You may need to read the problem several times, rephrasing parts of it, so you can translate everything into math terms. You may need to look up a few conversion factors, but that’s inconvenient, not difficult.

Pick a starting factor. If possible pick one that already has one of the units you want in the right place. Otherwise start with something you are given that is not a conversion factor.

Plug in conversion factors that allow you to cancel out any units you don’t want until you are left with only the units you do want (your answer units).

If you can’t solve the problem, pick a different starting factor and start over.

Do the math. You may be less apt to make an error if you first multiply all the top numbers, then divide by all the bottom numbers. Now double-check your calculations. If you make a mistake it will probably be in hitting the wrong key on the calculator.

Ask yourself if the answer seems right or reasonable. If not, recheck everything.

A Step by Step Guide to Dimensional Analysis

The following summary can be used as a guide for doing DA. Some familiarity with DA is assumed. See above for an introduction to DA. While not all steps listed below will be needed to solve all problems, I have found that any problem that can be solved using DA will yield its answer if the following steps are followed. I would not suggest memorizing the sequence of steps, but rather understanding and practicing them. Understanding is more durable than memory.

1. Determine what you want to know. Read the problem and identify what you’re being asked to figure
     out, e.g. “how many milligrams are in a liter of solution.”
     a. Rephrase if necessary using “per.” Example: You want to know “milligrams per liter.”
     b. Translate into “math terms” using appropriate abbreviations to end up with “mg/L” as your answer
          unit (AU). Write this down, e.g. “AU= mg/L”

2. Determine what you already know.
     a. What are you given by the problem, if anything? Example: “In one minute, you counted 45 drops.”
          • Rephrase if necessary. Think: “Drip rate is 45 drops per minute.”
          • Translate into math terms using abbreviations, e.g. “45 gtt/min”
               — If a given is in the form mg/kg/day, rewrite as mg/kg x day (see example 8)
               — If a percentage is given, e.g. 25%, rewrite as 25/100 with appropriate labels (see example 4)
     b. Determine conversion factors that may be needed and write them in a form you can use, such as
         “60 min/1 hour.” You will need enough to form a “bridge” to your answer unit(s). See example 1.
          • Factors known from memory: You may know that 1 kg = 2.2 lb, so write down “1 kg/2.2 lb”
               and/or “2.2 lb/1 kg” as conversion factors you may need.
          • Factors from a conversion table: If the table says “to convert from lb to kg multiply by 2.2,” then
               write down “2.2 lb/1 kg”

3. Setup the problem using only what you need to know.
     a. Pick a starting factor.
          • If possible, pick from what you know a factor having one of the units that’s also in your answer
               unit and that’s in the right place. See example 1.
          • Or pick a factor that is given, such as what the physician ordered.
          • Note that the starting factor will always have at least one unit not in the desired answer unit(s) that
               will need to be changed by canceling it out.
     b. Pick from what you know a conversion factor that cancels out a unit in the starting factor that you
          don’t want. See example 1.
     c. Keep picking from what you know factors that cancel out units you don’t want until you end up with
          only the units (answer units) you do want.
     d. If you can’t get to what you want, try picking a different starting factor, or checking for a needed
          conversion factor.
    e. If an intermediate result must be rounded to a whole number, such as drops/dose which can only be
          administered in whole drops, setup as a separate sub-problem, solve, then use the rounded off
          answer as a new starting factor. See example 10.

4. Solve: Make sure all the units other than the answer units cancel out, then do the math.
     a. Simplify the numbers by cancellation. If the same number is on the top and bottom, cancel them out.
     b. Multiply all the top numbers together, then divide into that number all the bottom numbers.
     c. Double check to make sure you didn’t press a wrong calculator key by dividing the first top number
          by the first bottom number, alternating until finished, then comparing the answer to the first one.
          Miskeying is a significant source of error, so always double check.
     d. Round off the calculated answer.
          • Be realistic. If you round off 74.733333 to 74.73 mL that implies that all measurements were of
               an extreme accuracy and that the answer is known to fall between 74.725 and 74.735, or 74.73
               + 0.005 mL. A more realistic answer would probably be 74.7 mL or 75 mL. See example 6.
          • If you round to a whole number that implies a greater accuracy than is appropriate, write your
               answer to indicate a range, such as 75 + 5 mL. See example 10.
     e. Add labels (the answer unit) to the appropriately rounded number to get your answer. Compare
          units in answer to answer units recorded from first step.

5. Take a few seconds and ask yourself if the answer you came up with makes sense. If it doesn’t,
start over.

This is a fairly bare outline. The steps are best taught, rather than read, and so would serve better as a guide to tutoring students than as a self-teaching guide.

A copy of the above guide in Word format is available, click here. Also see Medication Math for the Nursing Student which contains this page and is also available as a Word document

The steps for doing dimensional analysis are:

1. Determine the starting factor and answer unit.
2. Formulate a conversion equation.
3. Solve the conversion equation.

Determining the answer unit or units is crucial; they are not always obvious and can be challenging to determine. For some problems, reading the problem correctly is the only challenge. Students need to be able to translate sometimes convoluted English descriptions of a problem into clear, properly labeled factors they can later use to solve the problem. This skill is not emphasized in the textbook. If the answer unit is always given in the examples used, then this is because the examples have been contrived to be more simple and consistent than actual problems tend to be.

In some real-world problems no starting factor is given, or several possible starting factors are given with no way to decide, initially, which to use. It is preferable, in such cases, to determine everything you know that might be relevant to solving the problem, then decide, after the answer unit is determined, which of the factors you know would make an appropriate starting factor.

All examples used throughout the text use only numbers having a single unit attached for starting factors. Apparently “1 hour” is an acceptable starting unit, but “250 mL/hour” is not. This is not correct as starting factors are often in the form of “something per something.” Indeed, some problems cannot be solved if they have a single unit starting factor (see example 3 in Appendix A).

While many conversion factors are approximations, and fraction of a percent errors are unimportant in medication math, 10 percent errors are a bit worrisome. Equating 1 grain with 60 milligrams when the actual equivalency is closer to 64.8 mg, is questionable, as is equating liters and quarts, or 1 mL to 15 minims (actually 1 mL = 16.23 minims). It is possible to solve a problem and come up with answers that differ by as much as 10% depending on which approximate conversion factors you decide to use. If + 5% errors are acceptable, then, as an aside, any answer to a test question that is within 5% of the correct answer should be counted as correct. It is oddly inconsistent to insist on carrying out calculations to two decimals, rounding to the nearest tenth, when far greater errors can be introduced by using loose approximations.

When, in chapter 6, a problem involving amount/body weigh/day comes up, the solution is presented in an unorthodox way. The problem (p. 49) gives 25 mg/kg/24 hr. When doing dimensional analysis it is essential that all the units given should be used and accounted for. Ignoring a given unit, then pulling it out of thin air at the end is poor technique, yet this is what the textbook does. The solution is given as:

The problem is that the correct answer units should be how many mL should be administered per day, or “mL/day.” Omitting the “per day” part doesn’t alter the fact that that is what you want to know—not per hour, not per dose, but per day. There is actually a simple rule that applies here. For example, when acceleration is measured in feet per second per second, it is not written as ft/sec/sec, but as ft/sec ²because ft/sec/sec is equal to ft/sec x sec. So if you’re given mg/kg/day, the preferred way to deal with such a “triple decker” is to rewrite it as mg/kg x day. In this form it can be used, all undesired units cancel, and you end up with the desired answer with the right units attached:

If the problem called for “mL/dose” given 4 doses per day, then the solution is straightforward:

If “day” were omitted, however, this problem would become more difficult to solve. The textbook method is to calculate “mL,” then divide by 4 to get “mL/dose.” Students must remember to perform this final “critically important” step which would not exist if better technique were used. As the text acknowledges, “it is easy to forget to divide the total daily dose into the prescribed number of doses, thus greatly increasing the risk of administering an overdosage (sic).”

Problems of this sort are common, and it is unfortunate that the authors neglect to show students how to logically deal with them. The risk of confusing some students by introducing a new rule can hardly be worth the risk of error introduced by teaching a flawed technique.

In Chapter 10, page 184, an example is shown, as a model for students to follow, to determine how many mcg/min must be administered to a 215 lb patient at 3 mcg/kg/min:

In this example, at least, minutes are not omitted then added at the end, and the technique is not even erroneous, but merely confusing to many students and visually awkward. A student might try to logically extend this technique to determine mL/hr:

The student who notices that the answer doesn’t make sense might wonder what went wrong. Would they realize that when “mcg” was cancelled that “3 1/min” was left requiring the use of 60 min/1 hr instead of 1 hr/60 min? Trying to explain how to work around the poor technique employed by this example only digs a deeper hole. The better response to student confusion would be to have them put a big X mark over this section of the textbook and show them a sensible way to set it up:

Another case of flawed technique arises in Chapter 10. Students are given problems that require converting from mL/hr to gtt/min, and are shown conversion equations like the following:

The problem, again, is that the correct answer unit is “gtt/min” and not “gtt” as it appears. The correct answer is just pulled out of nowhere and declared to be “33 gtt/min.” The initial starting factor of “1 min” is spurious. It is not a given, and it means absolutely nothing to say that you know “1 min” or “1 hour” or “1 cabbage.” If such meaningless starting factors are simply omitted from such examples, the problems are perfectly setup to yield the correct answers with the correct answer units. It seems that the pseudo-starting factor is used to avoid having a starting factor with more than one unit attached. As mentioned, however, there is no such requirement when doing dimension analysis. In the above example “90 mL/1 hr” would make a logical and perfectly good starting factor.

Students should be told to just ignore the nonsensical “1 min” and “1 hour” starting factors. If you were to introduce “1 hour” as a starting factor in example 3 in Appendix A, you would be committing mathematical suicide as the problem would be rendered unsolvable once “hour” is cancelled out.

Here’s an actual example from chapter 10:

Calculation of IV Flow Rate When Total Infusion Time is Specified

Order: 1000 mL of D5W (5% Dextrose in water) IV to infuse over a period of 5 hr

Drop Factor: 10 gtt/mL

Starting Factor Answer Unit

1 min gtt (drops)

Equivalents: 1000 mL = 5 hr, 10 gtt = 1 mL, 60 min = 1 hr

Conversion Equation:

1 min x 1 hr x 1000 mL x 10 gtt = 33.3 = 33 gtt
60 min 5 hr 1 mL

Flow Rate: 33 gtt/min

For review, let’s go over this problem.

1. There are two errors relating to the starting factor. One is procedural—there is no logical way to pick a starting factor as the first step. The other is that “1 min” is a meaningless factor. I can meaningfully say that I know there are 10 drops per mL, but it means nothing to say that I know “1 min” in the context of this problem.

2. The answer unit is wrong. I want to know a rate of flow in drops per some unit of time. Just “gtt” doesn’t cut it.

3. Factors are expressed as equalities. It should read “something per something” and not “something equals something” which leads to absurd statements like “25 mg = 1 kg”

4. By introducing a spurious starting factor the setup is in error, as is the resultant answer. The number is correct, but the answer unit is not.

5. The final statement, that the flow rate is 33 gtt/min, is the only part of the example that is correct, but it is logically disconnected from everything that precedes it.

So, let’s see, the text manages to state an incorrect answer unit, then introduces a spurious starting factor, which makes the setup wrong, which yields 33 gtt for an answer, which is also wrong. But through some sort of mental slight-of-mind, they finally come up with the correct answer, which they simply declare to be 33 gtt/min.

Is there a better way to do this problem? First ask, what do I want to know? The flow rate in gtt/min, which is my answer unit, not just gtt (drops). What do I know? I’m given that there are 10 gtt/mL and that the infusion rate is 1000 mL/5 hr. Since I want gtt on top and 10 gtt/mL has gtt in the right place, 10 gtt/mL makes a perfectly good starting factor—I just need to get from mL to min. My set up then:

10 gtt x 1000 mL x 1 hr = 33 gtt
1 mL 5 hr 60 min min

Just omitting the “1 min” from the textbook’s setup would also work.

As to what the authors might be thinking, the only clue to their reasoning was given in the following paragraph that preceded this example:

“In calculating the flow rate for drops per minute , one minute becomes the labeled value that must be converted to an equivalent value: number of drops. One minute , therefore, is the starting factor and drops is the answer unit and these, as in all dimensional analysis conversions, form an equivalent relationship.”

On page 9 is the following table:

Table 1-2 Conversion Equation

This table reveals how the authors think about dimensional analysis. They see the starting factor as something given; there can be only one starting factor; it has only one unit associated with it, and it forms a special “equivalent relationship” with the answer unit, which, being equivalent, must also have only a single unit associated with it. In between are conversion factors that are fundamentally different from the starting factor.

All of these assumptions are incorrect as generalizations about dimensional analysis. The only equivalent relationship is between what is on the left side of the equal sign and what is on the right side. One could speak of an equivalent relationship between the “numerator” and “denominator” of a conversion factor (2.2 lb/1 kg means 2.2 lb = 1 kg), but otherwise there is no necessary “equivalent relationship” implied.

There is a particular type of DA problem, the simple conversion problem, that does involve going from one unit of measure to another equivalent measure (such as converting from feet to meters). In this subtype of problem you have only one logical starting factor, which can be said to be equivalent to your answer (10 inches x 2.54 cm/1 inch = 25.4 cm), but such problems should not be taken as a model for all DA problems, which appears to be what has happened.

By the Commutative Law of Multiplication, it doesn’t matter what order the factors on the left side are multiplied in. Therefore any factor could be first, and thus be the starting factor, although usually only one or two factors qualify to be thought of as logical starting factors. Both starting factors and answer units are often in the form of something per something. You could start with miles/hour and end up with seconds in your answer, for example, without any equivalence between starting factor and answer unit.

It appears that such fundamental misunderstandings underlie the errors in the textbook. Problems that do not conform to their notions are tortured into compliance by introducing spurious starting factors and using obviously incorrect answer units. I don’t think it is going too far to suggest that the poor technique exhibited by the textbook makes it difficult for students to master med-math. Indeed, those who do must do so in spite of the textbook and not because of it.

Recommended Corrections to:

Clinical Calculations: A unified approach (4th ed.)

A Google search shows that only this textbook and a few nursing sites associate “label factor” with dimensional analysis (DA). Likewise “unit conversion” is not a synonym for DA. The only synonym commonly used is “factor-label method.” While this point is nit-picky, I would expect the authors to use the same terminology as everyone else by the 4th edition.

At the bottom, “Step I: Determining the Starting Factor and Answer Unit,” should read, “Step I: Determining the Answer Unit.” Determining the starting factor should come after Step II, since the starting factor is not always given, there can be more than one possible starting factor, and the best starting factor to use may be one of the factors determined in Step II. Picking a starting factor from what you are given or know is the first step of Step III—setting up/solving the conversion equation.

In the box is the statement: “When the conversion equation is solved, it will be seen that the starting factor and the labeled answer have formed an equivalent relationship.” The belief that this is true leads to serious error and confusion in Chapter 10. If true, the collorilary would be that if the starting factor has one unit of measure associated with it, then the answer unit can have only one unit of measure associated with it and vice versa .

Emphasize that several of the equivalents in the table are fairly rough approximations. Give the actual equivalents—some students will want to know. Also, if the value of an equivalent can be 5% off, then, to be consistent, any test answer that is within + 5% of the correct value should be counted correct. In some (unlikely) cases answers could be as much as 10% off when several approximate equivalents are used to compound the error.

In the example at the bottom of the page you are given 25 mg/kg/24-hr (or day). The third unit given should not be dropped. There is a way to deal with problems of this type (25 mg/kg/day = 25 mg/kg-day) that can be consistently applied to all problems of this type. Triple unit factors are common and the difficulty they pose should be dealt with head on. All the various ad hoc attempts to get around these problems result in endless trouble in the long run. In this example the answer unit is given as “mL,” whereas the correct answer unit is “mL/day.” The problem should be setup as:

Whatever initial difficulty this technique may present for students not already familiar with it, it is still the technique of choice and will save a lot of grief later on. Some of the techniques contrived to deal with these problems work on some problems, but not others. The technique used above has the virtue of working with all problems involving triple unit factors.

In the two examples on this page the Answer Unit is incorrectly given as “cap” whereas “cap/dose” is what is really desired. In the first example, you are given 50 mg/kg/day and 4 doses/day, but not knowing what to do with “mg/kg/day” the problem is broken into two problems. The “day” is initially ignored, then brought back in the second part of the problem, thus paving the way for confusion and error. The logically consistent one-step setup would be:

For the second example the setup should be:

In the box at the bottom on the page are several warnings (“critically important,” “easy to forget”) that do not apply when the problems are done in a single step.

Avoid the two-step technique, and ignore the two examples at the bottom of the page. Work out as above.

Cross out the second paragraph: “In calculating the flow rate for drops per minute , one minute becomes the labeled value that must be converted to an equivalent value: number of drops. One minute , therefore, is the starting factor and drops is the answer unit and these, as in all dimensional analysis conversions, form an equivalent relationship.”

Ignore examples. Omit the spurious “1 min” Starting Factors. Note that Answer Units are also wrong (should be “gtt/min,” not just “gtt”). All that needs to be done is to cross out the “1 min” at the beginning of each example and add “/min” to “gtt” (to get the correct answer unit).

Another ad hoc variation in technique is introduced without comment in step 1 of the first example. Students will get into trouble if they try to extend this example to other problems. Also, what if the desired answer units were “mcg/hr?” Would students have trouble canceling out “min” with “min” apparently on top? Putting “mcg/min” on top invites confusion. A better setup for step 1 would be:

For step 2:

For steps 3 and 4, just omit the “1 min.” and “1 gtt”

Cross out the meaningless Starting Factors in examples 1, 3, 4, 5, and 6. In example 2, change “mcg/min” over “kg” to “mcg” over “kg x min.”

In Example a., the setup is in error due to a failure to fully label units. The 10 mL is “10 mL water.” You have to ask, “10 mL of what?” Your answer unit is “mL Chloromycetin sol” and not just “mL.” You can’t use “mL water” and end up with “mL Chlor. sol.” When you add 10 mL water to reconstitute you will end up with somewhat more than 10 mL Chlor. solution. Since you want “mL Chlor. sol” in your answer, pick a factor that has “mL Chlor. sol” in it and in the right place. You are given “100 mg/mL” which should be more completely written as “100 mg Chlor./mL Chlor. sol” and “10 mL/g” should be “10 mL water/1 g Chlor.” which is quite an unnecessary bit of information for solving this problem, though the text incorrectly uses it (and by luck gets away with it). The correct setup should be:

Omit spurious Starting Factors from example.

The first example asks, “How many mL should the child receive per dose?” The answer unit, therefore, should be “mL/dose” and not “mL.” You are given 15 mcg/kg/dose, so solve as shown above for examples on pages 49 and 50—likewise with the second example on page 221.

Page 225: Again, example gives 50,000 U/kg/day and 4 doses/day, so a one-step setup would be:

That’s about it. The other 96% of the text is okay.

Textbook Guide to Dimensional Analysis

(as compiled from various pages throughout the textbook)

Determine the starting factor* and answer unit.

Initially, it is essential to determine exactly what information is sought: the known quantity called the starting factor , and the desired unit to which the starting factor will be converted, the answer unit.

When the conversion equation is solved, it will be seen that the starting factor and the labeled answer have formed an equivalent relationship.

In calculating the flow rate for drops per minute (or mL per hour) one minute (or one hour) becomes the labeled value that must be converted to an equivalent value: number of drops (or mL). One minute , therefore, is the starting factor and drops is the answer unit and these, as in all dimensional analysis conversions, form an equivalent relationship.

Formulate a conversion equation consisting of a sequence of labeled factors, in which successive units can be cancelled until the desired answer unit is reached.

If a given is in the form mg/kg/day, ignore the third unit, do the conversion, then remember to factor the omitted unit back in. If in the form mcg/kg/min, change to mcg/min over kg if mcg/min is the answer unit.

If a percentage is given, e.g. 25%, rewrite as 25/100 with appropriate labels.

Determine conversion factors that may be needed. You will need enough to form a “bridge” to your answer unit(s).

Use only conversion factors that have a 1:1 relationship

It is desirable that conversion factors be arranged in a sequence so that identical units are placed diagonally.

In setting up the conversion factors, it is helpful to write the denominator first, as this contains the unit of the preceding numerator and facilitates cancellation of successive units.

Solve the conversion equation by use of cancellation and simple arithmetic.

Cancel units first

Reduce numbers to lowest terms.

Multiply/divide to solve the equation.

Reduce answer to lowest terms, convert to decimal, and/or round off.

Take a few seconds and ask yourself if the answer you came up with makes sense. If it doesn’t, start over.

* The text in red represents weak or erroneous technique. Errors of omission are not indicated.

Conclusions

This may be a case of a book being the worst textbook on dimensional analysis available—with the exception of all the others. I’ve heard that it is much better than its predecessor. Several medication math textbook titles are currently available, but not having reviewed them, I can’t assume any do a better job, but I think other titles should be looked into.

There are errors of omission where students are not given a complete enough understanding of dimensional analysis to do all problems that could crop up. There are errors of commission where students are taught flawed or even erroneous technique. Throughout the textbook, overly simplified examples are used that fail to show the range of problems that students may encounter. A wider range of problems, however, would have illustrated the shortcomings of the techniques as taught, and may have been omitted for that reason.

Overall, however, I would say that this book is quite useable provided its shortcomings and flaws are amended. A better rounded, more robust presentation of dimensional analysis is definitely needed. Students should not only do well solving test problems, but come away feeling confident in their ability to handle any problems that may come their way in the future.