Pharmacy Calculations

Learning Outcomes

• Explain importance of standardized approach for math

• Convert between fractions, decimals, percentages

• Convert between different systems of measurement

• Perform & check key pharmacy calculations:

– to interpret prescriptions

– involving patient-specific information

Key Terms

• Alligation method

• Apothecary system

• Avoirdupois system

• Body mass index (BMI)

• Body surface area (BSA)

• Days supply

Key Terms

• Denominator • Fraction • Household system • Ideal body weight (IBW) • Metric system • Numerator • Proportion • Ratio • Ratio strengths

Review of Basic Math

• Arabic numerals (0,1,2,3)

• Roman numerals – ss = 1/2

– L or l = 50

– I or i = 1

– C or c = 100

– V or v = 5

– M or m = 1000

– X or x = 10

Roman Numeral Basics

• More than 1 numeral of same quantityadd them

• Locate smaller numerals – smaller numerals on right of largest numeral(s)

• add small numerals to largest numeral

– smaller numerals on left of largest numeral(s) • subtract smaller numerals from largest numeral

• Example: XXI = 10 + 10 + 1 = 21

• Example: XIX = 10 + 10 – 1 = 19

Numbers

• Whole numbers (0, 1, 2)

• Fractions (1/4, 2/3, 7/8

• Mixed numbers (1 ¼ , 2 ½ )

• Decimals (0.5, 1.5, 2.25)

Fractions

• Fraction represents part of whole number

– less than one

– quantities between two whole numbers

• Numerator=number of parts present

• Denominator=total number of parts

• Compound fractions or mixed numbers

– whole number in addition to fraction ( 3 ½)

Fractions in Pharmacy

• IV fluids include

– 1/2 NS (one-half normal saline)

– 1/4 NS (one-quarter normal saline)

• 3/4 teaspoon

• Med errors may occur if someone mistakes the / for a 1

Simplify or Reduce Fractions

• Find greatest number that can divide into numerator and denominator evenly

• Fractions should be represented in simplest form

• Example: Simplify the fraction 66/100

• 66 divided by 2 ⇒ 33

• 100 divided by 2 ⇒ 50

• This fraction cannot be reduced further because no single number can be divided into both 33 and 50 evenly

• Answer: 33/50

Adding Fractions

1. Make sure all fractions have common denominators

Example: 3/4 + 2/3 • 3/4 * 3/3 = 9/12

• 2/3 * 4/4 = 8/12

2. Add the numerators • 9/12 + 8/12 = 17/12

3. Reduce to simplest fraction or mixed number • 17/12 = 1 5/12

Subtracting Fractions

1. Make sure all fractions have common denominators

– Example: 1 7/8 – ½

• 1 7/8=1 + 7/8=8/8 + 7/8=15/8

• 1/2 * 4/4 = 4/8

2. Subtract the numerators • 15/8 – 4/8 = 11/8

3. Simplify the fraction

– subtract 8 from the numerator to represent one whole number

– 11/8 = 1 3/8

Multiplication

1. Multiply numerators

– Example: 9/10 * 4/5

• 9 * 4 = 36

2. Multiply denominators. • 10 * 5 = 50

3. Express answer as fraction

9/10 * 4/5 = 36/50

4. Simplify fraction

• 36 divided by 2 = 18 50 divided by 2 = 25

• Final answer = 18/25

Division

• Convert 2nd fraction to its reciprocal & multiply

• Example: 2/3 ÷ 1/3

1. 1/3 is converted to 3/1.

2. Multiply 1st fraction by 2nd fraction’s reciprocal • 2/3 * 3/1 = 6/3

3. Simplify fraction • 6 divided by 3 = 2

• 3 divided by 3 = 1

• 6/3=2/1=2

• Final answer = 2

Decimals

• Decimals are also used to represent quantities less than one or quantities between two whole numbers

• Numbers to left of decimal point represent whole numbers

• Numbers to right of decimal point represent quantities less than one

1 0 0 . 0 0 0

hundreds, tens, ones, tenths, hundredths, thousandths

Decimal Errors

• Medication errors can occur

– decimals are used incorrectly or misinterpreted

– sloppy handwriting, stray pen marks, poor quality faxes

– copies can lead to misinterpretation

• To avoid errors

– use decimals appropriately

– never use trailing zero- not needed ( 5 mg, not 5.0 mg)

– always use leading zero (0.5 mg not .5mg)

Convert Fractions to Decimals

• If whole number present, that number is placed to left of decimal, then divide fraction

• Example:

• 1 2/3 → place 1 to left of decimal: 1.xx

• To determine numbers to right of decimal

– divide: 2/3 = 0.6667

• Final answer = 1.6667

• In most pharmacy calculations, decimals are rounded to tenths (most common) or other as determined

Rounding Decimals

• To round to hundredths

– look at number in thousandths place

– if it is 5 or larger increase hundredths value by 1

– if it is less than 5, number in hundredths place stays the same

– in either case, number in thousandths place is dropped

• Example: Round 1.6667 to hundreths

– look at number in thousandth place 1.6667

– final answer is 1.67

• Pharmacy numbers must be measureable/practical

Percentages

• Percentages are blend of fractions & decimals

• Percentage means “per 100”

• Percentages can be converted to fractions by placing them over 100

– Example:

• 78% =78/100

• Percentages convert to decimals

• Remove % sign & move decimal point two places to the left

• Example: 78% = 0.78

Ratios and Proportions

• A ratio shows relationship between two items

– number of milligrams in dose required for each kilogram of patient weight (mg/kg)

– read as “milligrams per kilogram”

• Proportion is statement of equality between two ratios

• Units must line up correctly

– (same units appear on top of equation & same units appear on bottom of equation)

• May need to convert units to make them match

Proportion Example

• Standard dose of a medication is 4 mg per kg of patient weight

• If patient weighs 70 kg, what is correct dose for this patient?

• Set up proportion:

4mg/kg=x mg/70kg

• x represents unknown value

(in this case, number of mg of drug in dose)

Solve the Proportion

• Using algebraic property

– if a/b=c/d then ad=bc

• Solve for x: 4mg/kg=x mg/70kg

4mg*70kg=1kg*xmg

isolate x by dividing both sides by 1kg:

4mg*70kg = 1kg*xmg

1kg 1kg

Completing the Problem

4mg*70kg = 1kg*xmg

1kg 1kg

Units cancel (kg) to give this equation:

4mg*70=x mg

Therefore: 280mg=x mg

A patient weighing 70kg receiving 4mg/kg should receive 280mg

Metric System

• Most widely used system of measurement in world

• Based on multiples of ten

• Standard units used in healthcare are:

– meter (distance)

– liter (volume)

– gram (mass)

• Relationship among these units is:

• 1 mL of water occupies 1 cubic centimeter & weighs 1 gram

Metric Prefixes

• “Milli” means one thousandth

– 1 milliliter is 1/1000 of a liter

• Oral solid medications are usually mg or g

• Liquid medications are usually mL or L

Metric Conversions

• Stem of unit represents type of measure

• Note relationship & decimal placement

0.001 kg = 1 gram = 1000 mg = 1000000 mcg

• 1 kilogram is 1000 times as big as 1 gram

• 1 gram is 1000 times as big as 1 milligram

• 1 milligram is 1000 times as big as 1 microgram

• Converting can be as simple as moving decimal point

Other Systems in Pharmacy

• Apothecary System

– developed in Greece for use by physicians/pharmacists

– has historical significance & has largely been replaced

– The Joint Commission (TJC) recommends

• avoid using apothecary units (institutional pharmacy)

– Apothecary units still used in community pharmacy

• Common apothecary measures still used

– grain is approximately 60-65 mg

– dram is approximately 5 mL

Other Systems in Pharmacy

• Avoirdupois System

– French system of mass: includes ounces & pounds

– 1 pound equals 16 ounces

• Household System

– familiar to people who like to cook

– teaspoons, tablespoons, etc.

– good practice to dispense dosing spoon or oral syringe

• with both metric & household system units

Common Conversions

2.54 cm = 1 inch

1 kg = 2.2 pounds (lb)

454 g = 1 lb

28.4 g= 1 ounce (oz) but may be rounded to 30 g = 1 oz

5 mL = 1 teaspoon (tsp)

15 mL = 1 tablespoon (T)

30 mL = 1 fluid ounce (fl oz)

473 mL = 1 pint (usually rounded to 480 mL)

Household Measures

• 1 cup = 8 fluid ounces

• 2 cups = 1 pint

• 2 pints = 1 quart

• 4 quarts = 1 gallon

Conversions

• Formula for converting Fahrenheit temp (TF) to Celsius temp (TC): TC=(5/9)*(TF-32)

• Formula for converting Celsius temp ((TC ) to Fahrenheit temp (TF): TF=(9/5)*(TC +32)

Common Temps Celsius ° Fahrenheit°

Normal Body Temp 37° 98.6°

Freezing 0° 32°

Boiling 100° 212°

Military Time

• Institutions use 24-hour clock

• 24-hour clock=military time

– does not include a.m. or p.m.

– does not use colon to separate hours & minutes

• Examples:

0100=1 AM

1300=1 PM

2130 = 9:3o PM

Conversions • Example: How many mL in 2.5 teaspoons?

• Set up proportion, starting with the conversion you know: 5 mL per 1 tsp or 5mL/tsp

• Match up units on both sides of =

5mL/tsp= __ mL/__ tsp

• Fill in what you are given & put x in correct area

5mL/tsp= x mL/2.5tsp

• Now solve for x by cross multiplying and dividing:

5mL*2.5tsp=1tsp*x mL so 12.5mL=x mL

• Answer: There are 12.5mL in 2.5 tsp

Patient-Specific Calculations

• Three examples of patient-specific calculations

1. body surface area

2. ideal body weight

3. body mass index

Body Surface Area (BSA) • Value uses patient’s weight/height &

expressed as m2

• Example:

man weighs 150 lb (68.2 kg), stands 5’10” (177.8 cm) tall

BSA=1.8 m2

• BSA used to calculate chemotherapy doses

• Several BSA equations available

– find out which equation is used at your institution

• Hospital computer systems will usually calculate the BSA value

Ideal Body Weight (IBW)

• Ideal weight is based on height & gender

• Expressed as kg

• Common formula for determining IBW:

– IBW (kg) for males = 50 kg + 2.3(inches over 5’)

– IBW (kg) for females = 45.5 kg + 2.3(inches over 5’)

IBW Example

• Calculate IBW for 72-year-old male 6’2” tall

– Formula: IBW (kg) for males = 50 kg + 2.3(inches over 5’) – IBW (kg) = 50 kg + 2.3(14)

– IBW = 82.2 kg

• Example:

– calculate IBW for 52-year-old female 5’9” tall.

– IBW (kg) = 45.5 kg + 2.3(9)

– IBW = 66.2 kg

Body Mass Index (BMI)

• Measure of body fat based on height & weight

• Determines if patient is

– underweight

– normal weight

– overweight

– obese

• BMI is not generally used in medication calculations

Key Pharmacy Calculations

• Pediatric dosing determined by child’s weight

• Example: diphenhydramine syrup: 5 mg/kg per day

– if child weighs 43 lb, how many mg per day?

Convert values to the appropriate units

x=19.5 kg

Determine dose

5mg/kg=xmg/19.5kg

5mg*19.5kg=1kg*xmg x=97.5mg of diphenhydramine

Days Supply

• Evaluate dosing regimen to determine

– how much medication per dose

– how many times dose is given each day

– how many days medication will be given

• Example: Metoprolol 50 mg po bid for 30 days

only 25 mg tablets available

1. dose is 50 mg-requires two 25-mg tablets

2. dose is given bid (twice daily) 2 tabs* 2 = 4 tabs/day

3. given for 30 days, so 4 tabs/day*30 days = 120 tablets

Concentration & Dilution

• Mixtures may be 2 solids added together

– percentage strength is measured as weight in weight (w/w) or grams of drug/100 grams of mixture

• Mixtures may be 2 liquids added together

– Percentage strength measured as volume in volume (v/v) or mL of drug/100mL of mixture

• Mixtures may be solid in liquid

– percentage strength is measured as weight in volume (w/v) or grams of drug per 100mL of mixture

Standard Solutions

• To determine how much dextrose is in 1 liter of D5W

– weight (dextrose) in volume (water) mixture (w/v)

• Set up proportion-start with concentration you know & then solve for x

• Make sure you have matching units in the numerators & denominators

• D5W means 5% dextrose in water=5 g/100 mL

• Start with 5 g/100 mL

• Convert 1 liter to mL so that denominator units are mL on both sides of equation

Standard Solutions

• How much dextrose is in 1 liter of D5W?

• Steps to solve the problem

– 5g/100mL=xg/1000mL

– 5g*1000mL=100mL*xg

– divide each side by 100mL to isolate x

– perform calculations & double check your work

– 50g=x There are 50 grams of Dextrose in l liter of D5W

Alligation Method

• It may be necessary to mix concentrations above and below desired concentration to obtain desired concentration

• Visualize alligation as a tic-tac-toe board:

Conc you have Conc you want Parts of each

Alligation

• Add 5% and 10% to obtain 9%

%Conc you have %Conc you want # of parts of each

5%

9%

10%

Alligation

• Add 5% and 10% to obtain 9%

• Subtract crosswise to get # of parts of each

%Conc you have %Conc you want # of parts of each

5% 10-9=1 Part

9%

10%

Alligation

• Add 5% and 10% to obtain 9%

• Subtract crosswise to get # of parts of each

• Need 1 part of 5% solution & 4 parts of 10% solution

• Total parts=5 parts

%Conc you have %Conc you want # of parts of each

5% 10-9=1 Part

9%

10% 9-5=4 Parts

Alligation

• Determine how much you need to mix by using proportions relating to parts

• If you want a total of 1 L or 1000 mL set up like this:

1 part/5 parts=x mL/1000 mL x=200mL of 5%

Since total is 1000 mL, 1000mL-200mL=800mL of 10% solution

Y

%Conc you have %Conc you want # of parts of each

5% 10-9=1 Part

9%

10% 9-5=4 Parts

Another Solution

• Another method to solve similar problems mixing 2 concentrations to obtain a 3rd concentration somewhere between original 2 concentrations:

• C1V1 = C2V2

• You need to know 3 of these values to solve for the 4th

Specific Gravity

• Specific gravity is ratio of weight of compound to weight of same amount of water

• Specific gravity of milk is 1.035

• Specific gravity of ethanol is 0.787

• Generally, units do not appear with specific gravity

• In pharmacy calculations, specific gravity & density are used interchangeably

• specific gravity = weight (g)

volume(mL)

Chemotherapy Calculations

• System of checks & rechecks important in chemotherapy

• Example: medication order is received for amifostine

200 mg/m2 over 3 minutes once daily 15–30 minutes prior to radiation therapy

patient is 79-year-old man weighing 157 lb & standing 6’ tall

BSA is 1.9 m2

What is the dose of amifostine for this patient?

Solution

• Set up equation

• Ordered dose of amifostine 200mg/m2

• BSA is 1.9 m2

• 200mg/m2=xmg/1.9m2 Note how units match up

200mg*1.9m2 =1m2 *xmg Now divide both sides by 1m2

380mg=xmg

The correct dose of amifostine is 380mg