1. Practice Basics 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 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

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 (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)

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 • body surface area • ideal body weight • body mass index

35. 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

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*xmgx=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 • dose is 50 mg-requires two 25-mg tablets • dose is given bid (twice daily) 2 tabs* 2 = 4 tabs/day • 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:

45. Alligation • Add 5% and 10% to obtain 9%

46. Alligation • Add 5% and 10% to obtain 9% • Subtract crosswise to get # of parts of each

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

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

49. 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

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)