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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