1. Chapter 14: Pharmacy Calculations

2. 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

3. Key Terms Alligation method Apothecary system Avoirdupois system Body mass index (BMI) Body surface area (BSA) Days supply

4. Key Terms Denominator Fraction Household system Ideal body weight (IBW) Metric system Numerator Proportion Ratio Ratio strengths

5. 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

6. 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

7. 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)

8. 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)

9. 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

10. 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

11. 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

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

13. 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

14. 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

15. 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

16. 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)

17. 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

18. 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

19. 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

20. 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

21. 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)

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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)

30. Household Measures 1 cup = 8 fluid ounces 2 cups = 1 pint 2 pints = 1 quart 4 quarts = 1 gallon

31. 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 TempsCelsius °Fahrenheit° Normal Body Temp37°98.6° Freezing0°32° Boiling100°212°

32. 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

33. 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

34. Patient-Specific Calculations Three examples of patient-specific calculations 1. body surface area 2. ideal body weight 3. body mass index

35. 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

36. 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’)

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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 haveConc you wantParts of each

45. Alligation Add 5% and 10% to obtain 9% %Conc you have%Conc you want# of parts of each 5% 9% 10%

46. 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%

47. 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

48. 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

49. 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

50. 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)

51. 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?

52. 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