3.2 Formulas and calculations

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Formulas and calculations

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Conversions

Knowing common conversions matters when you’re using formulas and doing calculations in pharmacy compounding. Here are key conversions to remember.

$1$ kilogram = $1, 000$ grams

$1$ kilogram = $2.2$ pounds

$1$ liter = $1, 000$ milliliters

$1$ gram = $1, 000$ milligrams

$1$ milligram = $1, 000$ micrograms

$1$ grain = $64.8$ milligrams

$1$ ounce = $28.35$ grams

$1$ ounce = $30 ml$

$1$ teaspoon = $5 ml$

$1$ tablespoon = $15$ milliliters

$1$ pound = $454$ grams

Roman numerals

Here are common Roman numerals and what they represent.

I $1$ - One

V $5$ - Five

X $10$ - Ten

L $50$ - Fifty

C $100$ - One hundred

D $500$ - Five hundred

M $1, 000$ - One thousand

SS $\frac{1}{2}$ - Half

Calculations

Concentration

Concentration tells you how much active ingredient is present in a total amount of product.

Liquids are often expressed as weight/volume ( $w / v$ ): the numerator is the drug’s weight, and the denominator is a specific volume of the final mixture (drug + vehicle).
- Example: A suspension labeled $10 mg /5 ml$ contains $10$ mg of active drug in each $5 ml$ of suspension.
Solid and semisolid topical products are often expressed as weight/weight ( $w / w$ ): the numerator is the drug’s weight, and the denominator is the total weight of the drug plus vehicle.
- Example: Nystatin cream labeled $100, 000 units / g$ contains $100, 000$ units of nystatin in each gram of cream.
Some liquid products are expressed as volume/volume ( $v / v$ ): the numerator is the volume of active ingredient, and the denominator is the total volume of the final liquid.
- Example: $10 ml /100 ml$ means $10 ml$ of active drug in $100 ml$ of total solution.

Specific gravity is the ratio of the weight of a substance to the weight of an equal volume of water (measured at the same time). If you know the specific gravity, you can convert between a substance’s weight and volume.

Specific gravity $= \frac{number of grams of a substance}{number of milliliters of a substance}$

Number of tablets $= \frac{desired dose}{stock strength}$

Use this when the prescribed tablet strength isn’t available.

Example: A prescription requires $1000 mg$ of drug A, but you only have $250 mg$ tablets.

Number of tablets of drug A $= \frac{1000 mg}{250 mg} = 4$ tablets.

So, you would dispense four tablets of $250 mg$ each to provide a total dose of $1000 mg$ .

Amount of solution to be given $= (\frac{desired dose}{stock strength}) \times stock volume$

Use this when you need to calculate the volume of a liquid medication to dispense and you know the stock concentration.

Example: The stock solution is $100 mg /10 ml$ . You need to dispense $500 mg$ . What volume should you dispense?

(spoiler)

Using the formula above:

Amount of solution to be given $= (\frac{500 mg}{100 mg}) \times 10 ml = 5 \times 10 = 50 ml$

So, dispense $50 ml$ of the stock solution.

Ratios

Ratios are another way to express the strength of a solution or preparation.

In ratio strength, the first number is always $1$ , followed by a colon and a second number (for example, $1 : 100$ ). The units are grams or milliliters, depending on whether the preparation is $w / w$ or $w / v$ .

A $1 : 100$ ratio strength can mean:
- $1$ g in $100$ ml (a liquid preparation), or
- $1$ g of drug in $100$ g of ointment (a solid/semisolid preparation).

For epinephrine:

A $1 : 1, 000$ ratio means $1$ gram in $1, 000$ ml, which is $1 mg / ml$ .
A $1 : 10, 000$ ratio means $1$ gram in $10, 000$ ml, which is $0.1 mg / ml$ .

Percent strength: Percent strength represents the number of grams contained in $100 mL$ of product.

Percent weight in volume (w/v): The number of grams in $100 mL$ of solution is expressed as $% w / v$ . Powdered substances suspended in a liquid vehicle are typically calculated as $% w / v$ .

Example: A $1%$ $% w / v$ solution contains 1 gram of powder in $100 mL$ of solution.

Practice problem A prescription calls for $150 mL$ of $2%$ $% w / v$ gentamicin solution. How much gentamicin powder is needed to make the desired solution? Let’s assume the unknown quantity is “G”.

(spoiler)

$\frac{2}{100} = \frac{G}{150}$

$G = \frac{2 \times 150}{100} = \frac{300}{100} = 3$

Hence, 3 grams of gentamicin powder are needed to make $150 mL$ of a $2%$ $% w / v$ solution. Dissolve 3 grams in $150 mL$ to get the desired strength.

:::

Percent volume in volume (v/v): The number of milliliters in $100 mL$ of solution is expressed as $% v / v$ . A liquid component in a liquid preparation is typically calculated on a $% v / v$ basis.

Practice problem How much alcohol is in $100 mL$ of a $10%$ $% v / v$ solution?

(spoiler)

A $10%$ solution of alcohol contains $10 mL$ of alcohol in $100 mL$ of solution. Hence, the answer is $10 mL$ .

:::

Percent weight in weight (w/w): The number of grams in $100$ grams of total dosage form is expressed as $% w / w$ . Powdered substances mixed with a solid or semisolid base (for example, ointments) are typically calculated as $% w / w$ .

Practice problem How much betamethasone is in $150 mg$ of a $0.1%$ $% w / w$ betamethasone cream? Assuming the unknown quantity of betamethasone is “B”,

(spoiler)

$\frac{1}{1000} = \frac{B}{150}$

$B = \frac{1 \times 150}{1000} = 0.15$ gram

The steps are the same as calculating $% w / v$ , but here the strength is $0.1%$ $% w / w$ . Converting $0.1%$ to a fraction gives $\frac{1}{1000}$ .

:::

If a solution requires a dilution, the amount of active drug stays constant while the total volume increases.

When two solutions have equal osmotic pressure and salt concentration, they are isotonic. Normal saline contains $0.90%$ $% w / v$ NaCl in sterile water and is therefore an isotonic crystalloid.

Calculating drug dosage based on body surface area: In some conditions, drug dosage is calculated using body surface area (BSA).

$BSA (m^{2}) = \frac{1}{6} (W H)^{0.5}$ , where $W$ is body weight in kg, $H$ is body height in meters

Pediatric dosing: Clark’s and Young’s rules are used to calculate drug dosages in the pediatric age group (from birth to about 18 years of age).

According to Clark’s rule, the pediatric dose is obtained by dividing the patient’s weight in pounds by the average standard weight of $150$ pounds, then multiplying by the adult dose.

$Pediatric dose = [\frac{weight of child (lbs)}{150}] \times adult dose$

Young’s rule uses the patient’s age to calculate the pediatric dosage.

$Pediatric dose = adult dose \times [\frac{age}{age + 12}]$

Fried’s rule also uses age to calculate the pediatric dosage.

$Pediatric dose = (\frac{age in months}{150}) \times adult dose$

Pediatric doses are often stated per body weight, for example, $X mg / kg / day$ .

Practice problem If the prescription calls for Amoxicillin $30 mg / kg / day$ for ten days, then how much amoxicillin will you dispense? Assume body weight is 22 pounds.

(spoiler)

First, convert pounds to kg. $1 kg = 2.2$ pounds.

$\frac{22}{2.2} = 10 kg$

Now apply the dosing:

$30 \times 10 = 300 mg / day$

Total dosage $= 300 \times 10 = 3000$ mg

Hence, you will dispense a total of $3000$ mg.

Typically, this is then mixed to form a solution of a particular concentration (for example, $400 mg /5 mL$ ). To find the volume to dispense, use:

$Amount of Solution to be given = (\frac{Desired Dose}{Stock Strength}) \times Stock Volume$ .

:::

Estimating intravenous (IV) flow rates: Intravenous infusions administer fluids directly into the veins. The infusion rate can be calculated in milliliters or in drops.

Drops per minute are abbreviated as gtts/min. The drop factor is the number of drops in $1 mL$ . The tubing used determines the drop size:

Macrodrip tubing may be $10 gtts / mL$ , $15 gtts / mL$ , or $20 gtts / mL$ . Macrodrips are used when rapidly infusing large amounts of fluid.
Microdrip tubing administers $60 gtts / mL$ and is typically used in children.

All IV packages clearly label the gtts/mL for that set.

The following formulae can be used to set the IV infusion rate:

Total IV volume/time (hour or minute) $= mL per hour or minute$

$[\frac{Total IV volume}{time (minute)}] \times drop factor = drops per minute$

Practice problem The physician has ordered D5W $1200$ milliliters in $12$ hours using a $15$ -drop per milliliter infusion rate. Calculate the number of drops per minute.

(spoiler)

Convert $12$ hours to minutes because the infusion rate is in drops per minute.

$12 hours = 12 \times 60 = 720 minutes$

Using the formula:

$Drops per minute = (\frac{1200 mL}{720 min}) \times 15 = 1.6667 \times 15 = 25$

:::

International units (IU): International units are used to denote doses for hormones like insulin, vaccines, vitamins, blood products, etc. They measure the drug’s effect or biological activity, not its weight or mass.

Milliequivalent (mEq): The milliequivalent (mEq) is a unit of measure often used for electrolytes. It indicates the chemical activity (combining power) of an element relative to the activity of $1 mg$ of hydrogen.

$m Eq / l = \frac{( mg / l ) \times valence}{molecular weight}$

Alligation: Alligation is a method of mixing two liquids or solids of different concentrations to produce a mixture with a desired concentration. The final concentration differs from either starting concentration, so you must mix specific proportions of each ingredient. Those proportions are given by the alligation ratio.

$Alligation ratio = \frac{H}{L}$

Let “m” be the desired concentration, “l” the lower concentration, and “h” the higher concentration.

$H = m - l$

$L = h - m$

Once the alligation ratio is known (for example, $2 : 3$ ), it means two parts of the higher concentration to three parts of the lower concentration. Next, convert “parts” into actual amounts.

Example: If you’re compounding $500 ml$ total and the ratio is $2 : 3$ , then:
- Higher concentration: $200 ml$
- Lower concentration: $300 ml$

Example: An order comes in for $10$ grams of $2%$ hydrocortisone cream. You only have $1%$ and $3%$ hydrocortisone available. Using the alligation method:

$m = 2$

$h = 3$

$l = 1$

$H = 2 - 1 = 1$

$L = 3 - 2 = 1$

Hence, alligation ratio = $\frac{1}{1} = 1$ .

You need equal parts of $1%$ and $3%$ to make $2%$ . Since you need $10$ grams total, mix $5$ grams of $1%$ hydrocortisone with $5$ grams of $3%$ hydrocortisone to get $10$ grams of $2%$ hydrocortisone.

In another example, suppose you get an order for $100 ml$ of a solution with a $50 mg/ml$ concentration. You have $30 mg/ml$ and $100 mg/ml$ concentrations available. How will you use the alligation method to calculate the volume of each concentration required?

$m = 50$

$h = 100$

$l = 30$

$H = 50 - 30 = 20$

$L = 100 - 50 = 50$

Alligation ratio = $\frac{H}{L} = \frac{20}{50} = \frac{2}{5}$

So you need two parts of the higher concentration ( $100 mg/ml$ ) and five parts of the lower concentration ( $30 mg/ml$ ). One way to convert parts into volume is:

$2 x + 5 x = 100$

$7 x = 100$

$x = \frac{100}{7} = 14.286$

$2 x = 2 \times 14.286 = 28.572 ml$

$5 x = 5 \times 14.286 = 71.43 ml$

Hence, mix $28.572 ml$ of $100 mg/ml$ and $71.43 ml$ of $30 mg/ml$ to get $100 ml$ of $50 mg/ml$ .

Dilution: Dilution is a method of preparing a solution of a desired concentration by diluting a concentrated stock solution (typically with water). A common dilution relationship is:

$M_{1} V_{1} = M_{2} V_{2}$

where $M_{1}$ and $M_{2}$ are the molarity of the solutions (mol/L or M), and $V_{1}$ and $V_{2}$ are the solution volumes. If concentration is given as $v / v$ or $w / v$ , substitute $M_{1}$ and $M_{2}$ with those concentrations.

How much water should you add to $150 ml$ of a $0.4%$ $v / v$ solution to reduce its strength to a $0.02%$ $v / v$ solution?

Using the formula above:

$0.4 \times 150 = 0.02 \times V_{2}$

$V_{2} = \frac{0.4 \times 150}{0.02} = 3000 ml$

Since you already have $150 ml$ of solution, the amount of water to add is:

$3000 - 150 = 2850 ml$

So, add $2850 ml$ of water to $150 ml$ of a $0.4%$ $v / v$ solution to obtain a $0.02%$ $v / v$ solution.

Make sure you keep units consistent on both sides of the equation (for example, use ml throughout, or convert L to ml as needed).

Aliquot: An aliquot is a dilution method used when only a very small quantity of drug is needed - smaller than the minimum measurable or weighable quantity (MMQ or MWQ) of the available equipment.

Example: If the MWQ is $100 mg$ but you need $25 mg$ , you can’t accurately weigh $25 mg$ directly.

In that case, weigh the MWQ ( $100 mg$ ) and dilute it to a convenient concentration. For example, dilute with $20 ml$ of water to make:

$\frac{100 mg}{20 ml} = 5 mg/ml$

Then measure $5 ml$ of the $5 mg/ml$ solution to obtain $25 mg$ of the drug.

Conversions

1 kg = 1,000 g; 1 kg = 2.2 lbs; 1 lb = 454 g
1 L = 1,000 mL; 1 oz = 28.35 g or 30 mL
1 g = 1,000 mg; 1 mg = 1,000 mcg; 1 grain = 64.8 mg
1 tsp = 5 mL; 1 tbsp = 15 mL

Roman numerals

I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1,000
SS = ½

Concentration

Liquids: weight/volume (w/v); Solids: weight/weight (w/w); Liquids (v/v) for volume/volume
Specific gravity = grams substance / mL substance
Number of tablets = desired dose / stock strength
Amount of solution = (desired dose / stock strength) × stock volume

Ratios and percent strength

Ratio strength: 1:x format (e.g., 1:1,000 = 1 mg/mL for epinephrine)
Percent strength: grams per 100 mL (w/v), mL per 100 mL (v/v), or grams per 100 g (w/w)
- w/v: g/100 mL; v/v: mL/100 mL; w/w: g/100 g
Isotonic solutions: equal osmotic pressure (e.g., 0.9% NaCl)

Pediatric and body surface area dosing

BSA (m²) = (1/6) × (weight × height)^0.5 (kg, m)
Clark’s rule: (weight in lbs / 150) × adult dose
Young’s rule: adult dose × (age / (age + 12))
Fried’s rule: (age in months / 150) × adult dose
Pediatric dosing: mg/kg/day

IV flow rates

Drop factor: macrodrip (10, 15, 20 gtts/mL), microdrip (60 gtts/mL)
Drops/min = (total IV volume / time in min) × drop factor

International units and milliequivalents

IU: measures drug effect/biological activity (not weight)
mEq/L = (mg/L × valence) / molecular weight

Alligation

Used to mix two concentrations to achieve a desired concentration
- Alligation ratio: H = m - l; L = h - m; ratio = H:L
- Convert ratio parts to actual amounts for compounding

Dilution

Formula: M₁V₁ = M₂V₂ (use consistent units)
Used to prepare lower concentration from a stock solution

Aliquot

Used when needed quantity is below minimum weighable/measurable quantity (MWQ/MMQ)
Dilute MWQ to a measurable volume, then measure aliquot to obtain desired dose

Formulas and calculations

Conversions

Knowing common conversions matters when you’re using formulas and doing calculations in pharmacy compounding. Here are key conversions to remember.

$1$ kilogram = $1, 000$ grams

$1$ kilogram = $2.2$ pounds

$1$ liter = $1, 000$ milliliters

$1$ gram = $1, 000$ milligrams

$1$ milligram = $1, 000$ micrograms

$1$ grain = $64.8$ milligrams

$1$ ounce = $28.35$ grams

$1$ ounce = $30 ml$

$1$ teaspoon = $5 ml$

$1$ tablespoon = $15$ milliliters

$1$ pound = $454$ grams

Roman numerals

Here are common Roman numerals and what they represent.

I $1$ - One

V $5$ - Five

X $10$ - Ten

L $50$ - Fifty

C $100$ - One hundred

D $500$ - Five hundred

M $1, 000$ - One thousand

SS $\frac{1}{2}$ - Half

Calculations

Concentration

Concentration tells you how much active ingredient is present in a total amount of product.

Liquids are often expressed as weight/volume ( $w / v$ ): the numerator is the drug’s weight, and the denominator is a specific volume of the final mixture (drug + vehicle).
- Example: A suspension labeled $10 mg /5 ml$ contains $10$ mg of active drug in each $5 ml$ of suspension.
Solid and semisolid topical products are often expressed as weight/weight ( $w / w$ ): the numerator is the drug’s weight, and the denominator is the total weight of the drug plus vehicle.
- Example: Nystatin cream labeled $100, 000 units / g$ contains $100, 000$ units of nystatin in each gram of cream.
Some liquid products are expressed as volume/volume ( $v / v$ ): the numerator is the volume of active ingredient, and the denominator is the total volume of the final liquid.
- Example: $10 ml /100 ml$ means $10 ml$ of active drug in $100 ml$ of total solution.

Specific gravity $= \frac{number of grams of a substance}{number of milliliters of a substance}$

Number of tablets $= \frac{desired dose}{stock strength}$

Use this when the prescribed tablet strength isn’t available.

Example: A prescription requires $1000 mg$ of drug A, but you only have $250 mg$ tablets.

Number of tablets of drug A $= \frac{1000 mg}{250 mg} = 4$ tablets.

So, you would dispense four tablets of $250 mg$ each to provide a total dose of $1000 mg$ .

Amount of solution to be given $= (\frac{desired dose}{stock strength}) \times stock volume$

Use this when you need to calculate the volume of a liquid medication to dispense and you know the stock concentration.

Example: The stock solution is $100 mg /10 ml$ . You need to dispense $500 mg$ . What volume should you dispense?

(spoiler)

Using the formula above:

Amount of solution to be given $= (\frac{500 mg}{100 mg}) \times 10 ml = 5 \times 10 = 50 ml$

So, dispense $50 ml$ of the stock solution.

Ratios

Ratios are another way to express the strength of a solution or preparation.

A $1 : 100$ ratio strength can mean:
- $1$ g in $100$ ml (a liquid preparation), or
- $1$ g of drug in $100$ g of ointment (a solid/semisolid preparation).

For epinephrine:

A $1 : 1, 000$ ratio means $1$ gram in $1, 000$ ml, which is $1 mg / ml$ .
A $1 : 10, 000$ ratio means $1$ gram in $10, 000$ ml, which is $0.1 mg / ml$ .

Percent strength: Percent strength represents the number of grams contained in $100 mL$ of product.

Percent weight in volume (w/v): The number of grams in $100 mL$ of solution is expressed as $% w / v$ . Powdered substances suspended in a liquid vehicle are typically calculated as $% w / v$ .

Example: A $1%$ $% w / v$ solution contains 1 gram of powder in $100 mL$ of solution.