Chapter 14: 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 quantity  add 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 2 nd fraction to its reciprocal & multiply  Example: 2/3 ÷ 1/3 1. 1/3 is converted to 3/1.

2. Multiply 1 st fraction by 2 nd 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 (T F ) to Celsius temp (T C ): T C =(5/9)*(T F -32)  Formula for converting Celsius temp ((T C ) to Fahrenheit temp (T F ): T F =(9/5)*(T C +32)

Common Temps

Normal Body Temp Freezing Boiling

Celsius °

37° 0° 100°

Fahrenheit°

98.6° 32° 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.

3.

ideal body weight body mass index

Body Surface Area (BSA)

 Value uses patient’s weight/height & expressed as m 2  Example: man weighs 150 lb (68.2 kg), stands 5’10” (177.8 cm) tall BSA=1.8 m 2  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 1.

2.

only 25 mg tablets available dose is 50 mg-requires two 25-mg tablets 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 5% %Conc you want # of parts of each 9% 10%

Alligation

 Add 5% and 10% to obtain 9%  Subtract crosswise to get # of parts of each

%Conc you have 5% %Conc you want # of parts of each 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 5% 10% %Conc you want # of parts of each 10-9=1 Part 9% 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

%Conc you have 5% 10% %Conc you want # of parts of each 10-9=1 Part 9% 9-5=4 Parts

Another Solution

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

C1V1 = C2V2



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

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/m 2 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 m 2 What is the dose of amifostine for this patient?

Solution

 Set up equation  Ordered dose of amifostine 200mg/m 2  BSA is 1.9 m 2  200mg/m 2 =xmg/1.9m

2 Note how units match up 200mg*1.9m

2 =1m 2 *xmg Now divide both sides by 1m 2 380mg=xmg The correct dose of amifostine is 380mg