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# Chapter 6 pharmacy calculation

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### Chapter 6 pharmacy calculation

1. 1. The Pharmacy Technician 4E Chapter 6 Basic PharmaceuticalMeasurement Calculation
2. 2. Topic Outline Numbers  Percents & Solutions Fractions  Alligation Decimal Numbers  Powder Volume Significant Figures  Children’s Doses Measurement  Calculations for Equations & Variables Business Ratio & Proportion
3. 3. Comparison of Roman and Arabic Numerals
4. 4. Example: xxx = 30 or 10 plus 10 plus 10 DC = 600 or 500 plus 100 LXVI = 66 or 50 plus 10 plus 5 plus 1 When the second of two letters has a value greater than that of the first, the smaller is to be subtracted from the larger.
5. 5. Fractions• When something is divided into parts, each part is considered a fraction of the whole.• If a pie is cut into 8 slices, one slice can be expressed as 1/8, or one piece (1) of the whole (8). If we have a 1000 mg tablet, • ½ tablet = 500 mg • ¼ tablet = 250 mg
6. 6. FractionsFractions have two parts: 1• Numerator (the top part) 8• Denominator (the bottom part) 1 8
7. 7. FractionsA fraction with the same numerator and the samedenominator has a value equivalent to 1.In other words, if you have 8 pieces of a pie that hasbeen cut into 8 pieces, you have 1 pie. 8 =1 8
8. 8. TerminologyProper fraction• A fraction with a value of less than 1. 1• A fraction with a numerator value smaller than the <1 denominator’s value. 4Improper fraction• A fraction with a value larger than 1.• A fraction with a numerator value larger than the 6 denominator’s value. >1 5
9. 9. Adding or Subtracting Fractions• When adding or subtracting fractions with unlikedenominators, it is necessary to create a commondenominator. •This is like making both fractions into the same kind of “pie.”• Common denominator is a number that each of theunlike denominators of two or more fractions can bedivided evenly.
10. 10. RememberMultiplying a number by 1 does not changethe value of the number. 5 ×1 = 5Therefore, if you multiply a fraction by afraction that equals 1 (such as 5/5), you donot change the value of a fraction. 5× 5 = 5 5
11. 11. Guidelines for Finding a Common Denominator1. Examine each denominator in the given fractions for its divisors, or factors.2. See what factors any of the denominators have in common.3. Form a common denominator by multiplying all the factors that occur in all of the denominators. If a factor occurs more than once, use it the largest number of times it occurs in any denominator.
12. 12. Example 1Find the least common denominator of the following fractions.Step 1.) Find the prime factors (numbers divisible only by 1 andthemselves) of each denominator. Make a list of all the differentprime factors that you find. Include in the list each different factoras many times as the factor occurs for any one of thedenominators of the given fractions. • The prime factors of 28 are 2, 2, and 7 (because 2 3 2 3 7 5 28). •The prime factors of 6 are 2 and 3 (because 2 3 3 5 6).The number 2 occurs twice in one of the denominators, so it mustoccur twice in the list. The list will also include the unique factors 3and 7; so the final list is 2, 2, 3, and 7.
13. 13. Example 1Find the least common denominator of thefollowing fractions.Step 2. Multiply all the prime factors on your list.The result of this multiplication is the least commondenominator.
14. 14. Example 1Find the least common denominator of the following fractions.Step 3. To convert a fraction to an equivalent fraction with thecommon denominator, first divide the least commondenominator by the denominator of the fraction, then multiplyboth the numerator and denominator by the result (thequotient). •The least common denominator of 9⁄28 and 1⁄6 is 84. In the first fraction, 84 divided by 28 is 3, so multiply both the numerator and the denominator by 3.
15. 15. Example 1Find the least common denominator of the followingfractions.In the second fraction, 84 divided by 6 is 14, somultiply both the numerator and the denominator by14.
16. 16. Example 1Find the least common denominator of the followingfractions.The following are two equivalent fractions:
17. 17. Example 1Find the least common denominator of the followingfractions.Step 4. Once the fractions are converted to containequal denominators, adding or subtracting them isstraightforward. Simply add or subtract the numerators.
18. 18. Multiplying Fractions• When multiplying fractions, multiply the numerators by numerators and denominatorsbydenominators.• In other words, multiply all numbers above the line; then multiply all numbers below the line.• Cancel if possible and reduce to lowest terms.
19. 19. Multiplying FractionsDividing the denominator by a number is thesame as multiplying the numerator by thatnumber. 3 × 5 15 3 = = 20 20 4
20. 20. Multiplying FractionsDividing the numerator by a number is the sameas multiplying the denominator by that number. 6 6 1 = = 4 × 3 12 2
21. 21. Dividing FractionsTo divide by a fraction, multiply by its reciprocal,and then reduce it if necessary. 1 1× 3 3 = = =3 1/ 3 1 1
22. 22. Reciprocals Reciprocals are two different fractions that equal 1 when multiplied together. Every fraction has a reciprocal (except those fractions with zero in the numerator). The easiest way to find the reciprocal of a fraction is to switch the numerator and denominator, or just turn the fraction over. To find the reciprocal of a whole number, just put 1 over the whole number. EXAMPLE:  The reciprocal of 2 is 1/2.
23. 23. Example 2Multiply the two given fractions
24. 24. Decimal Places 1000 500 50 5 0.5 mg mg mg mg mgWhole 0.5 0.05 0.005 0.0005 tenths hundredths thousands ten thousands (1 place to (2 places to (3 places to (4 places to the right) the right) the right) the right)
25. 25. DecimalsAdding or Subtracting Decimals• Place the numbers in columns so that the decimal points are aligned directly under each other.• Add or subtract from the right column to the left column.Multiplying Decimals• Multiply two decimals as whole numbers.• Add the total number of decimal places that are in the two numbers being multiplied.• Count that number of places from right to left in the answer, and insert a decimal point.
26. 26. DecimalsDividing Decimals1. Change both the divisor and dividend to whole numbers by moving their decimal points the same number of places to the right. • divisor: number doing the dividing, the denominator • dividend: number being divided, the numerator2. If the divisor and the dividend have a different number of digits after the decimal point, choose the one that has more digits and move its decimal point a sufficient number of places to make it a whole number.
27. 27. DecimalsDividing Decimals3. Move the decimal point in the other number the same number of places, adding zeros at the end if necessary.4. Move the decimal point in the dividend the same number of places, adding a zero at the end. 1.45 ÷ 3.625 = 0.4 1.45 1450 = = 0. 4 3.625 3625
28. 28. DecimalsRounding to the Nearest Tenth1. Carry the division out to the hundredth place2. If the hundredth place number ≥ 5, + 1 to the tenth place3. If the hundredth place number ≤ 5, round the number down by omitting the digit in the hundredth place: 5.65 becomes 5.7 4.24 becomes 4.2
29. 29. DecimalsRounding to the Nearest Hundredth or Thousandth Place 3.8421 = 3.84 41.2674 = 41.27 0.3928 = 0.393 4.1111 = 4.111
30. 30. System International Prefixes Micro - One millionth (basic unit × 10–6 or unit × 0.000,001) Milli - One thousandth (basic unit × 10–3or unit × 0.001) Centi - One hundredth (basic unit × 10–2 or unit × 0.01) Deci - One tenth (basic unit × 10–1 or unit × 0.1) Hecto - One hundred times (basic unit × 102 or unit × 100) Kilo - One thousand times (basic unit × 103 or unit × 1000)
31. 31. Common Metric Units: WeightBasic Unit Equivalent1 gram (g) 1000 milligrams (mg)1 milligram (mg) 1000 micrograms (mcg)1 kilogram (kg) 1000 grams (g)
32. 32. Common Metric Conversions kilograms (kg) to grams (g)  Multiply by 1000 (move decimal point three places to the right).  Example: 6.25 kg = 6250 g grams (g) to milligrams (mg)  Multiply by 1000 (move decimal point three places to the right).  Example: 3.56 g = 3560 mg milligrams (mg) to grams (g)  Multiply by 0.001 (move decimal point three places to the left).  Example: 120 mg = 0.120 g
33. 33. Common Metric Conversions Liters (L) to milliliters (mL)  Multiply by 1000 (move decimal point three places to the right).  Exmaple: 2.5 L = 2500 mL Milliliters (mL) to liters (L)  Multiply by 0.001 (move decimal point three places to the left).  Example: 238 mL = 0.238 L
34. 34. Avoirdupois System 1 gr (grain) - 65 mg 1 oz (ounce) - 437.5 gr or 30 g (28.35 g) 1 lb (pound) - 16 oz or 7000 gr or 1.3 g
35. 35. Household Measure: Volume 1 tsp (teaspoonful) - 5 mL 1 tbsp (tablespoonful) - 3 tsp (15 mL) 1 fl oz (fluid ounce) - 2 tbsp (30 mL (29.57 mL) 1 cup - 8 fl oz (240 mL) 1 pt (pint) - 2 cups (480 mL) 1 qt (quart) - 2 pt (960 mL) 1 gal (gallon) - 4 qt (3840 mL)
36. 36. Household Measure: Weight 1 oz (ounce) - 30 g 1 lb (pound) - 16 oz (454 g) 2.2 lb - 1 kg
37. 37. Numerical RatiosRatios represent the relationship between: • two parts of the whole • one part to the wholeWritten as follows: 1:2 “1 part to 2 parts” ½ 3:4 “3 parts to 4 parts” ¾Can use “per,” “in,” or “of,” instead of “to”• Proportions are frequently used to calculate drug doses in the pharmacy.• Use the ratio-proportion method any time one ratio is complete and the other is missing a component.
38. 38. Proportions• An expression of equality between two ratios.• Noted by :: or = 3:4 = 15:20 or 3:4 :: 15:20
39. 39. Rules for Ratio-Proportion Method • Three of the four amounts must be known • The numerators must have the same unit of measure • The denominators must have the same unit of measure
40. 40. Steps for Solving for X1. Calculate the proportion by placing the ratios in fraction form so that the x is in the upper-left corner.2. Check that the unit of measurement in the numerators is the same and the unit of measurement in the denominators is the same.3. Solve for x by multiplying both sides of the proportion by the denominator of the ratio containing the unknown, and cancel.4. Check your answer by seeing if the product of the means equals the product of the extremes.
41. 41. RememberWhen setting up a proportion to solve aconversion, the units in the numerators mustmatch, and the units in the denominators mustmatch.
42. 42. Example 3 Solve for X
43. 43. Percents• The number of parts per 100 can be written as a fraction, a decimal, or a ratio.• Percent means “per 100” or hundredths.• Represented by symbol %. 30% = 30 parts in total of 100 parts 30:100, 0.30, or 30 100
44. 44. Percents in the Pharmacy• Percent strengths are used to describe IV solutions and topically applied drugs.• The higher the % of dissolved substances, the greater the strength.• A 1% solution contains • 1 g of drug per 100 mL of fluid • Expressed as 1:100, 1/100, or 0.01
45. 45. Equivalent Values 45 100 0.5 100
46. 46. Converting a Ratio to a Percent1. Designate the first number of the ratio as the numerator and the second number as the denominator.2. Multiply the fraction by 100%, and simplify as needed.3. Multiplying a number or a fraction by 100% does not change the value.
47. 47. Converting a Ratio to a Percent5:1 = 5/1 × 100% = 5 × 100% = 500%1:5 = 1/5 × 100% = 100%/5 = 20%1:2 = 1/2 × 100% = 100%/2 = 50%
48. 48. Converting a Percent to a Ratio1. Change the percent to a fraction by dividing it by 100.2. Reduce the fraction to its lowest terms.3. Express this as a ratio by making the numerator the first number of the ratio and the denominator the second number.
49. 49. Converting a Percent to a Ratio 2% = 2 ÷ 100 = 2/100 = 1/50 = 1:50 10% = 10 ÷ 100 = 10/100 = 1/10 = 1:10 75% = 75 ÷ 100 = 75/100 = 3/4 = 3:4
50. 50. Converting a Percent to a Decimal1. Divide by 100% or insert a decimal point two places to the left of the last number, inserting zeros if necessary.2. Drop the % symbol.
51. 51. Converting a Decimal to a Percent1. Multiply by 100% or insert a decimal point two places to the right of the last number, inserting zeros if necessary.2. Add the the % symbol. symbol
52. 52. Percent to Decimal4% = 0.04 4 ÷ 100% = 0.0415% = 0.15 15 ÷ 100% = 0.15200% = 2 200 ÷ 100% = 2 Decimal to Percent0.25 = 25% 0.25 × 100% = 25%1.35 = 135% 1.35 × 100% = 135%0.015 = 1.5% 0.015 × 100% = 1.5%
53. 53. Example 4How many milliliters are there in 1 gal, 12 fl oz?According to the values in Table 5.7, 3840 mL are found in1 gal. Because 1 fl oz contains 30 mL, you can use the ratio-proportion method to calculate the amount of milliliters in12 fl oz as follows:
54. 54. Example 4How many milliliters are there in 1 gal, 12 fl oz?
55. 55. Example A solution is to be used to fill hypodermic syringes, each containing 60 mL, and 3 L of the solution isavailable. How many hypodermic syringes can be filled with the 3 L of solution?1 L is 1000 mL. The available supply of solution is thereforeDetermine the number of syringes by using the ratio-proportionmethod:
56. 56. ExampleHow many hypodermic syringes can be filled with the 3 L of solution?
57. 57. ExampleYou are to dispense 300 mL of a liquid preparation. Ifthe dose is 2 tsp, how many doses will there be in the final preparation? Begin solving this problem by converting to a common unit of measure using conversion values.
58. 58. Example 6 If the dose is 2 tsp, how many doses will there be in the final preparation?Using these converted measurements, the solution can bedetermined one of two ways:Solution 1: Using the ratio proportion method and themetric system.
59. 59. Example 6If the dose is 2 tsp, how many doses will there be in the final preparation?
60. 60. Example 7 How many grains of acetaminophen should be used in a Rx for 400 mg acetaminophen?Solve this problem by using the ratio-proportion method.The unknown number of grains and the requestednumber of milligrams go on the left side, and the ratio of1 gr 65 mg goes on the right side, per Table 5.5.
61. 61. Example 7How many grains of acetaminophenshould be used in the prescription?
62. 62. Example 8 A physician wants a patient to be given 0.8 mg of nitroglycerin. On hand are tablets containing nitroglycerin 1/150 gr. How many tablets should the patient be given?Begin solving this problem by determining the number ofgrains in a dose by setting up a proportion and solvingfor the unknown.
63. 63. Example 8How many tablets should the patient be given?
64. 64. Common Calculations in the Pharmacy• Calculations of DosesActive ingredient (to be administered)/solution(needed) =Active ingredient (available)/solution(available)
65. 65. Example 9You have a stock solution that contains 10 mg of activeingredient per 5 mL of solution. The physician orders adose of 4 mg. How many milliliters of the stock solution will have to be administered?
66. 66. Example 9How many milliliters of the stock solution will have to be administered?
67. 67. Example 10An order calls for Demerol 75 mg IM q4h prn pain. The supply available is in Demerol 100 mg/mL syringes. How many milliliters will the nurse give for one injection?
68. 68. Example 10How many milliliters will the nurse give for one injection?
69. 69. Example 11An average adult has a BSA of 1.72 m2 and requiresan adult dose of 12 mg of a given medication. A childhas a BSA of 0.60 m2.If the proper dose for pediatric and adult patients is alinear function of the BSA, what is the properpediatric dose? Round off the final answer.
70. 70. Example 11What is the proper pediatric dose?
71. 71. Example 11What is the proper pediatric dose?
72. 72. Example 12A dry powder antibiotic must be reconstituted foruse. The label states that the dry powder occupies0.5 mL. Using the formula for solving for powdervolume, determine the diluent volume (the amountof solvent added). You are given the final volume forthree different examples with the same powdervolume.
73. 73. Example 12Using the formula for solving for powder volume, determine the diluent volume.
74. 74. Example 12Using the formula for solving for powder volume, determine the diluent volume.
75. 75. Example 13You are to reconstitute 1 g of dry powder. The labelstates that you are to add 9.3 mL of diluent to makea final solution of 100 mg/mL. What is the powdervolume?
76. 76. Example 13 What is the powder volume?Step 1. Calculate the finalvolume. The strength of thefinal solution will be 100mg/mL.mg/mL
77. 77. Example 13What is the powder volume?
78. 78. Example 14Dexamethasone is available as a 4 mg/mLpreparation. An infant is to receive 0.35 mg. Preparea dilution so that the final concentration is 1 mg/mL.How much diluent will you need if the originalproduct is in a 1 mL vial and you dilute the entirevial?
79. 79. Example 14How much diluent will you need if the original product is in a 1 mL vial and you dilute the entire vial?
80. 80. Example 14How much diluent will you need if the original product is in a 1 mL vial and you dilute the entire vial?
81. 81. Example 15Prepare 250 mL of dextrose 7.5% weight in volume(w/v) using dextrose 5% (D5W) w/v and dextrose50% (D50W) w/v. How many milliliters of each willbe needed?
82. 82. Example 15 How many milliliters of each will be needed?Step 1. Set up a box arrangement and at the upper-leftcorner, write the percent of the highest concentration(50%) as a whole number.
83. 83. Example 15 How many milliliters of each will be needed?Step 2. Subtract the center number from the upper-leftnumber (i.e., the smaller from the larger) and put it atthe lower-right corner. Now subtract the lower-leftnumber from the center number (i.e., the smaller fromthe larger), and put it at the upper-right corner.
84. 84. Example 15 How many milliliters of each will be needed?50 2.5 mL parts D50W 7.55 42.5 mL parts D5W 45 mL total parts D7.5W
85. 85. Example 15How many milliliters of each will be needed?
86. 86. Example 15How many milliliters of each will be needed?
87. 87. Example 15How many milliliters of each will be needed?
88. 88. Example 15How many milliliters of each will be needed?
89. 89. Example 15How many milliliters of each will be needed?
90. 90. Terms to Remember1. Body surface area 8. Nomogram2. Concentration 9. Numerator3. Conversions 10. Positional notation4. Denominator 11. Qs ad5. Flow rate 12. Total parenteral nutrition6. Least common 13. Usual and customary (U&C) denominator 14. Valence7. Milliequivalent (meq) 15. Variable