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FUNDAMENTALS OF PHARMACEUTICAL CALCULATIONS John Alechxander M. Advincula, RPh  Pharmaceutical Calculations The area of study that applies the basic principles of mathematics to the preparation and safe and effective use of pharmaceuticals. It includes calculations from initial product f...

FUNDAMENTALS OF PHARMACEUTICAL CALCULATIONS John Alechxander M. Advincula, RPh  Pharmaceutical Calculations The area of study that applies the basic principles of mathematics to the preparation and safe and effective use of pharmaceuticals. It includes calculations from initial product formulations through clinical administration and outcomes. Roman Numerals -are used with the apothecary’s system of measurement to designate quantites on prescription. ROMAN ARABIC ROMAN ARABIC I 1 VIII 8 II 2 IX 9 III 3 X 10 IV 4 L 50 V 5 C 100 VI 6 D 500 VII 7 M 1000 (1) when roman numerals of lesser value follows one of a greater value, they are added. EXAMPLES: 1. II = 1 + 1 2. VI = 5 + 1 3. XV = 10 + 5 4. XXII = 10 + 10 + 1 +1 5. VVII =5 + 5 +1 +1 (2) When Roman Numerals of lesser value preceeds one of a greater value, they are subtracted from the greater value numeral. EXAMPLES: 1. IV = 5 - 1 = 4 2. IX = 10 - 1 = 9 3. XL= 50 - 10 = 40 4. LD = 500 - 50 = 450 5. IL = 50 - 1 = 49 Additional ruling for Rule no.2 The proposed additional rules are as follows: 1. ACCEPTED SUBTRACTIVE NOTATIONS The following are the numerical combinations that use a subtractive notation: IV (4) IX (9) XL (40) XC (90) CD (400) CM (900) Additional ruling for Rule no.2 Other combinations that use subtractive notations not found in the above list are NOT ALLOWED. ARABIC ALLOWED SUBTRACTIVE NOT ALLOWED NOTATION USED 45 XLV XL VL 49 XLIX XL,IX IL 95 XCV XC VC 99 XCIX XC,IX IC 145 CXLV XL CVL 495 CDXCV CD,XC VD 950 CML CM LM 990 CMXC CM,XC XM 999 CMXCIX CM,XC,IX IM 1999 MCMXCIX CM,XC,IX MIM Additional ruling for Rule no.2 2. Least Possible Letters and Operations In accordance with Rule 1 and 3, the Roman numeral with the lowest possible letter count should always be used. The well-known “up to three consecutively” rule of repeatable Roman numerals do not apply to V, L, and D. These cannot be used more than once. VV (10) is not allowed because it can be written as X (10), which contains less characters compared to the former. The same applies to LL (100) and DD (1000) because they can be written as C (100) and M (1000), respectively. Regarding operations, IVIV (8) is not allowed because it can be written as VIII (8). Both contain 4 letters, but VIII only requires one addition process (5 + 1 + 1 + 1). IVIV requires subtraction operations (two counts of 5 - 1 = 4) followed by one addition process (4 + 4 = 8). In this case, the one with less operations involved is the correct choice. IXL (39) contains less letters than the correct version XXXIX (39), but IXL is unclear on its operations. It can be viewed as XL (40) – I (1) = 39, or L (50) – IX (9) = 41. Roman numerals that do not exhibit singularly clear and linear results should not be used. (3) When Roman Numeral of a lesser value is placed between two greater values, it is first subtracted from the greater numeral placed after it, and then the value is added to the other numerals. EXAMPLES: 1. XXIX = 10 + 10 + (10 - 1) = 29 2. XIV = 10 + (5 - 1) = 14 (4) Roman numerals may not be repeated more than three times in succession. example: 15 = VVV ? , IT SHOULD BE = XV  Common Fractions portions of a whole, expressed at 1⁄3, 7⁄8, and so forth. It is recalled, that when adding or subtracting fractions, the use of a common denominator is required. Example # 1: If the adult dose of a medication is 2 teaspoonful (tsp.), calculate the dose for a child if it is 1⁄4 of the adult dose.  Common Fractions Example # 2: If a child’s dose of a cough syrup is 3⁄4 teaspoonful and represents 1⁄4 of the adult dose, calculate the corresponding adult dose. NOTE: When common fractions appear in a calculations problem, it is often best to convert them to decimal fractions before solving.  Decimal Fraction a fraction with a denominator of 10 or any power of 10 and is expressed decimally rather than as a common fraction 1 is expressed as 0.10 10 45 as 0.45 100 Note: It is important to include the zero before the decimal point.  Decimal Fraction To convert a common fraction to a decimal fraction, divide the denominator into the numerator. 1 = 1 ÷ 8 = 0.125 8 To convert a decimal fraction to a common fraction, express the decimal fraction as a ratio and reduce. 0.25 = 25 = 1 100 4  Percent its corresponding sign, %, mean ‘‘in a hundred.’’ 50 percent (50%) → 50 parts in each one hundred  Common Fraction to Percent Common fractions may be converted to percent by dividing the numerator by the denominator and multiplying by 100. EXAMPLE: Convert 3/8 to percent  Decimal Fraction to Percent Decimal fractions may be converted to percent by multiplying by 100. EXAMPLE: Convert 0.125 to percent  Test your Brains! SEATWORK # 1 1. How many 0.000065-gram doses can be made from 0.130 gram of a drug? 2. Give the decimal fraction and percent equivalents for each of the following common fractions: 1 1 a. c. 35 250 3 1 b. d. 7 400 3. If a clinical study of a new drug demonstrated that the drug met the effectiveness criteria in 646 patients of the 942 patients enrolled in the study, express these results as a decimal fraction and as a percent. 4. A pharmacist had 3 ounces of hydromorphone hydrochloride. He used the following: 1 ounce 8 How many ounces of hydromorphone hydrochloride were 1 ounce left? 4 1 1 ounces 2  Ratio relative magnitude of two quantities it resembles a common fraction except in the way in which it is presented 1/2 represent a ratio of 1 : 2 Note: If the two terms of a ratio are multiplied or are divided by the same number, the value is unchanged 20 The ratio 20:4 or 4 has a value of 5 10 if divided by 2, 10 : 2 or 2 has a value of 5  Proportion A proportion is the expression of the equality of two ratios. It may be written in any one of three standard forms:  Proportion Each of these expressions is read: a is to b as c is to d a and d are called the extremes ‘‘outer members’’ b and c are the means ‘‘middle members’’ Note: The product of the extremes is equal to the product of the means.  Proportion “The product of the extremes is equal to the product of the means” allows us to find the missing term of any proportion when the other three terms are known.  If the missing term is a mean, it will be the product of the extremes divided by the given mean  If it is an extreme, it will be the product of the means divided by the given extreme  Proportion NOTE: It important to label the units in each position (e.g., mL, mg) to ensure the proper relationship between the ratios of a proportion. EXAMPLE: 1. If 3 tablets contain 975 milligrams of aspirin, how many milligrams should be contained in 12 tablets? 2. If 12 tablets contain 3900 milligrams of aspirin, how many tablets should contain 975 milligrams? Problem #1. If an insulin injection contains 100 units of insulin in each milliliter, how many milliliters should be injected to receive 40 units of insulin? Problem #2. An injection contains 2 mg of medication in each milliliter (mL). If a physician prescribes a dose of 0.5 mg to be administered to a hospital patient three times daily, how many milliliters of injection will be required over a 5-day period? Problem #3. In a clinical study, a drug produced drowsiness in 30 of the 1500 patients studied. How many patients of a certain pharmacy could expect similar effects, based on a patient count of 100? Problem #4. A formula for 1250 tablets contains 6.25 grams of diazepam. H ow many grams of diazepam should be used in preparing 550 tablets?  Dimensional Analysis This method involves the logical sequencing and placement of a series of ratios (termed factors) into an equation. The ratios are prepared from the given data as well as from selected conversion factors and contain both arithmetic quantities and their units of measurement. Some terms are inverted (to their reciprocals) to permit the cancellation of like units in the numerator(s) and denominator(s) and leave only the desired terms of the answer. Step 1. Identify the given quantity and its unit of measurement. Step 2. Identify the wanted unit of the answer. Step 3. Establish the unit path, and identify the conversion factors needed. This might include: (a) a conversion factor for the given quantity and unit, and/or (b) a conversion factor to arrive at the wanted unit of the answer. Step 4. Set up the ratios in the unit path such that cancellation of units of measurement in the numerators and denominators will retain only the desired unit of the answer. Step 5. Perform the computation by multiplying the numerators, multiplying the denominators, and dividing the product of the numerators by the product of the denominators.  DIMENSIONAL ANALYSIS using Conversion Factors in a Unit Path How many fluidounces (fl. oz.) are there in 2.5 liters (L)? Step 1. The given quantity is 2.5 L. Step 2. The wanted unit for the answer is fluidounces. Step 3. The conversion factors needed are those that will take us from liters to fluidounces. 1 liter = 1000 mL (to convert the given 2.5 L to milliliters), and 1 fluidounce= 29.57 mL (to convert milliliters to fluidounces) Step 4. The unit path setup: Step 5. Perform the computation:  Compute the same problem by Direct Method (Inverting the Units) 1. How many tablets will be taken in seven days if a prescription order reads furosemide 20 mg/tab, one tab twice a day? 2. How many tablets are needed to fill a prescription for seven days for alprazolam 0.5mg/tab, one tablet three times a day?  ANSWER 1. How many tablets will be taken in four days if a prescription order reads sucralfate 1g/tab, one tablet four times a day? 2. How many tablets will be taken in 10 days if a prescription order reads warfarin 5mg/tab, one tab daily at bed time? 3. How many tablets will be taken in 30 days if a prescription order reads metoprolol tartrate 50mg/tab, one tablet two times a day  Compute using the 4. A medication order calls for 1000 milliliters of a dextrose intravenous infusion to be administered over an 8-hour period. Using an intravenous administration set that delivers 10 drops/milliliter, how many drops per minute should be delivered to the patient?  Significant Figures Consecutive figures that express the value of a denominate number accurately enough for a given purpose. Any of the digits in a valid denominate number must be regarded as significant. Whether, zero is significant, however, depends on its position or on known facts about a given number. EXAMPLES: RULE EXAMPLE All nonzero digits are considered 98.513 has five significant SIGNIFICANT numbers Leading zeros are NOT 0.00361 has three significant SIGNIFICANT numbers Trailing zeros in a number 998.100 has six significant containing a decimal point are numbers SIGNIFICANT All zeros between other significant 607.123 has six significant number figures are SIGNIFICANT ALLIGATION  is an arithmetic two types of method of solving alligation: problems relating alligation medial mixtures of alligation alternate componetns of different strengths. ALLIGATION MEDIAL  may be used to determine the strength ALLIGATION of a common ALTERNATE ingredient in a mixture of two or more  may be used to preparations. determine the proportion or quantities of two or more components to combine in order to prepare a mixture of a desired strength.