Pharmacy for Technicians: Seventh Edition

6.8 Calculating Dosages Using Equal Proportion Equations

As has been explained, multiplying of extremes and means in equal proportion equations can be used to discover missing pharmaceutical information and answer questions. Now that you have a firm sense of metrics and pharmaceutical measurement systems and conversions for time, temperature, weight, and volume, you can apply these skills on even more pharmaceutical examples (see Table 6.10).

Table 6.10 Rules to Remember for Equal Proportion Equations

Three of the four values must be known.
The numerators must have the same unit of measurement.
The denominators must also have the same unit of measurement.

One of the most common calculations in pharmacy practice is that of determining the correct dosages to provide when using the specific drug concentrations of available stock. Setting up the equation correctly is essential to getting the correct answer. The term ratio strength in pharmacy refers to the concentration level of an active ingredient to the completed product or to the substance that holds it. In concentration ratios, the first number is the number of parts (numerator) of active ingredient compared to the whole total, which is the second number (denominator). The concentrations come in different formulations:

A liquid solution or suspension with a dissolved ingredient (solute) is usually expressed as a concentration of solute weight to volume (w/v) of the dissolving liquid (solvent) of the final product. The w/v is usually expressed in grams/per milliliters (g/mL) or milligrams/per milliters (mg/mL).
Liquid solutes are expressed as volume to volume (v/v)—the volume of the drug ingredient to the volume of the final solution with all ingredients. The v/v is usually expressed in milliliters/per milliliters (mL/mL) or microliters/per milliliters (mcL/mL).
Solid or semisolid concentrations are expressed as weight to weight (w/w)—the weight of the drug ingredient compared to the weight of the total solid with all ingredients added. You will usually see grams/per grams (g/g).
Solid drug products may be also divided into “units,” which are like units of dosage or like “servings” of food.

Remembering these ratio rules of thumb will help you when you are looking at a prescription or medical order and are trying to decide how to set up your calculation ratios and fractions. Also, follow the drug directions for use. The available stock is usually labeled with the concentration ratio of an active ingredient in the w/v, v/v, or w/w ratios described earlier, which refers to the following:

$\frac{active ingredient (weight or volume)}{final product volume (weight or volume)}$

Calculating Dosages with Weight-in-Volume Concentration Ratios

In prescriptions with weight-in-volume concentrations, the final weight of the active ingredient to be administered in the dosage will be provided. The unknown quantity to be calculated is the amount of the correct concentration of stock drug product to achieve this desired dose.

Practice Tip

If you are unsure how to set up an equation for filling a prescription or medical order, or for compounding, the directions on the label or package insert (PI) usually offer directions.

$\frac{active ingredient (prescribed)}{final product volume (needed for administration)}$

To calculate the amount of stock solution/suspension needed, you must set the two ratios into a proportion equation:

$\frac{active ingredient (prescribed)}{y product volume (needed)} = \frac{active ingredient weight (stock)}{solution volume (stock)}$

Or you can flip both sides to put the unknown on top, but it can be confusing, as it does not represent the units as stated in common pharmacy problems. It is better to have mg/mL than mL/mg even if the unknown y is in the denominator.

Example 20

A prescription for a sick child is received from the emergency room on Saturday night.

images Amoxicillin/clavulanate 600 mg/5 mL for 4 mL po twice daily

No other pharmacies are open in town. The pharmacy technician discovers that the pharmacy is out of stock on the 600 mg/5 mL concentration of the prescribed suspension, but the 400 mg/5 mL concentration is in stock. The prescribed concentration could be ordered, but the child would be without medication until Monday at noon. Since the child needs 4 mL of the 600 mg/5 mL concentration, what is the amount of amoxicillin/clavulanate per dose?

Safety Alert

Whenever you are calculating dosages with ratios, always make sure that the numerators have similar units of measurement, and the denominators have similar units. If not, first do the conversions, and then do your calculations.

How much of the 400 mg/5 mL stock concentration would be needed to make this individual dose?

Step 1	Calculate the original prescribed dose for the child. $\begin{matrix} y mg / 4 mL = 600 mg / 5 mL or \frac{y m g}{4 m L} = \frac{600 m g}{5 m L} \\ y mg = \frac{600 mg \times 4 mL}{5 mL} = 2, 400 mg \div 5 = 480 mg \end{matrix}$
The amount of prescribed amoxicillin needed to be taken two times each day is 480 mg. How much would that be of a 400 mg/5 mL suspension?
Step 2	Calculate the volume (mL) of the available concentration of 400 mg/5 mL. $\begin{matrix} \frac{480 mg (prescribed)}{y mL} = \frac{400 m g (stock)}{5 m L} \\ 400 y m L = 480 m g \times 5 m L \end{matrix}$
Step 3	Find the value of y by dividing each side by 400. $\begin{matrix} \frac{400 y mg}{400 mg} = \frac{480 g \times 5 mL}{400 mg} \\ y mL = \frac{2, 400 mL}{400} = 24 mL \div 4 = 6 mL \end{matrix}$

You can also solve this problem using dimensional analysis.

The calculated 6 mL dosage of the stock 400 mg/5 mL concentration of amoxicillin/clavulanate suspension is equal to the prescribed 4 mL dosage of the 600 mg/5 mL suspension. Consequently, the pharmacy fills the prescription and allows the sick child to begin treatment right away. The child will take this 6 mL twice a day.

Safety Alert

Remember, if you are calculating the drug volume for an injectable or IV medication, do not include the volume (in milliliters [mL] or liters [L]) or strength (in percent [%]) of the stock base in equations. Select the correct base solution from the medical order but do not add its numbers to your equations.

Calculating Dosages with Volume-in-Volume Percent Ratios

Volume-in-volume (v/v) concentrations are usually expressed as a percentage of the active ingredient in milliliters (mL) per 100 mL. For example, the common disinfectant used in the hospital is 70% isopropyl alcohol (IPA), which is 70 mL of IPA for every 100 mL of sterile water solution—as seen below, which is a v/v equation.

Example 21

How many milliliters of isopropyl alcohol (IPA) are contained in an 8 fl oz bottle of 70% IPA?

Step 1	Rewrite 70% IPA as a volume-in-volume equivalent. $70 % IPA = \frac{70 mL alcohol}{100 mL solution}$ This means that 70% IPA is 70 mL alcohol in 100 mL of solution (70 mL alcohol/100 mL).
Step 2	Set up a ratio and proportion to calculate the unknown amount of milliliters of IPA (y) in 8 fl oz of the 70% solution by using the conversion rate from Table 6.9 of 8 fl oz to 240 mL. $\frac{y m L a l c o h o l}{240 m L s o l u t i o n} = \frac{70 m L a l c o h o l}{100 m L s o l u t i o n}$
Step 3	Use the butterfly technique of multiplying extremes and means. 100y mL = 70 mL × 240 100y mL = 16,800 mL
Step 4	Solve for the value of y by transferring the multiplier of y to the other side of the equation as a denominator (to divide that side proportionally).

$y mL = \frac{16, 800}{100} = 168 mL$ alcohol, the volume of IPA in 8 fl oz of solution.

You can also solve this problem using dimensional analysis.

Math Morsel

Remember that if a multiplier is on one side of an equation, you can keep the same values by transferring it to the other side by dividing and vice versa. You transfer a denominator from one side of an equation and turn it into a multiplier on the other.

Calculating Dosages with Weight-in-Weight Concentration Ratios

Ointments and creams commonly use a percentage weight-in-weight (w/w) concentration, which is defined as the weight of the active ingredient per 100 grams of the final drug formulation (cream or ointment). For example, a 2.5% hydrocortisone cream is equal to 2.5 grams of active ingredient (hydrocortisone) per 100 grams of cream.

Math Morsel

Many neonatal physicians and pharmacists are deeply concerned about adult therapeutic dosages calculated proportionally down directly for premature babies, newborns, and children without more adaptation and studies. To improve safety and reduce the number of calculations needed, a table that lists recommended neonatal doses for common drugs was published in the Journal of Pediatric Pharmacology and Therapeutics in 2014. It can be accessed at https://PharmPractice7e.ParadigmEducation.com.

Example 22

What would be the weight (in grams) of zinc oxide in 240 g of 40% zinc oxide ointment?

Step 1

Rewrite 40% zinc oxide as a weight-in-weight ratio.

$40 % zinc code = \frac{40 g zinc oxide}{100 g ointment}$

Step 2

Set up a ratio and proportion to calculate the unknown value of zinc oxide (y) in 240 grams of the ointment.

$\frac{y g z i n c o x i d e}{200 g o i n t m e n t} = \frac{40 g z i n c o x i d e}{100 g o i n t m e n t}$

Step 4

Use the butterfly technique of multiplying extremes and means.

$\begin{array}{l} 100 y g = 40 g \times 240 & 100 y g = 9, 600 g & y g = \frac{9600 g}{100} = 96 g \end{array}$

The weight of zinc oxide in 240 grams of ointment is 96 g.

You can also solve this problem using dimensional analysis.

Calculating Dosage Concentrations Based on Patient Weight

Dosages are often based on the patient’s body weight in kilograms. Before determining the correct dosage of the existing concentrations, you must first convert the patient’s weight from pounds to metric kilograms. As has been emphasized, most calculation dosage errors happen to neonates, infants, and children, so question the pharmacist or the prescriber about any prescription that would be in the normal range for adults but would not be in a safe range for children.

Example 23

For a serious infection in an 11-pound, 2-month-old infant, the hospital physician put in the following antibiotic order.

images Ceftazidime, 50 mg IV per 1 kg of patient weight every 8 hours

What is the total daily dose of IV ceftazidime for an 11-pound, 2-month-old infant?

Step 1

Convert pounds to kilograms. You can do this by dimensional analysis or ratio and proportion with butterfly multiplication, and both answers will be the same. The ratio and proportion method will be used in this example.

$\begin{matrix} \frac{y k g}{11 l b} = \frac{1 k g}{2 . 2 l b}; then 2 . 2 y k g = 11 k \\ y kg = 11 kg \div 2.2 = 5 kg \end{matrix}$

The patient weighs 5 kg.

Step 2

Calculate the dose of the antibiotic based on the patient’s weight using the ratio and proportion method. Please note this could also be solved using dimensional analysis.

$\begin{matrix} \frac{y m g}{5 k g} = \frac{50 m g}{1 k g} & y m g = 5 \times 50 m g = 250 mg per dose \end{matrix}$

Step 3

Calculate the total daily dose. Since ceftazidime is given every 8 hours, which makes 3 doses in a 24-hour period (24 ÷ 8 = 3), the total daily dose is as follows:

3 doses per day × 250 mg per dose = 750 mg per day

Safety Alert

Pharmacy personnel should remind parents or caregivers to use only the oral syringe provided with their child’s product to administer the medication.

Example 24

A parent visits your pharmacy on a Saturday night with a prescription for Infants’ Tylenol for their 3-month-old daughter, who weighs 15 lb. The dose recommended is 10 mg/kg. The concentration of Infants’ Tylenol is 160 mg/5 mL. What is the correct dose in milligrams (mg) and volume in milliliters (mL) that the mother should administer to their child?

This problem can be solved using the ratio and proportion method or dimensional analysis. Dimensional analysis will be used in this example. The one-step and multiple-step methods will be shown.

One-step method:

Solve for the correct dose in milligrams.

$1.5 lb \times \frac{1 kg}{2.2 lb} \times \frac{10 mg}{kg} = 68 mg$

Now solve for the correct dose in milliliters.

$1.5 lb \times \frac{1 kg}{2.2 lb} \times \frac{10 mg}{kg} \times \frac{5 mL}{160 mg} = 2.1 mL, rounded down to 2 mL$

Multiple-step method:

Step 1

Convert the patient’s weight from pounds to kilograms.

$1.5 lb \times \frac{1 kg}{2.2 lb} =6 .8 kg$

Step 2

Using the patient’s weight in kilograms, calculate the dose of Infants’ Tylenol needed in milligrams.

$6.8 kg \times \frac{10 mg}{1 kg} = 68 mg$

Step 3

Convert the dose in milligrams to milliliters.

$6.8 kg \times \frac{5 mL}{160 kg} = 2 .1 ml, rounded down to 2 mL$

The dose is 68 mg or 2 mL.

Safety Alert

Medication doses for premature infants, neonates, and pediatric patients may use two decimal places rather than just one if they are not rounded down.