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Calculations Chapter 6 Numbers Knowing how to work with numbers is essential to the proper handling of drugs and preparation of prescriptions. The amount of a drug in its manufactured or prescribed form is always stated numerically (with numbers). Roman Numerals The Roman numerals are letters that represent numbers. They can be capital or lower case letters. (see handout) ss = ½ I or i = 1 V or v = 5 X or x = 10 L = 50 C = 100 D = 500 M = 1000 Roman Numerals Two Rules: Rule 1: When the second of the two letters has a value equal to or smaller than the first, their values are added together. xx = 20 DC = 600 lxvi = 66 10 + 10 = 20 500 + 100 = 600 50 + 10 + 5 + 1 = 66 Roman Numerals Two Rules: Rule 2: When the second of the two letters has a value greater than the first, the value is subtracted from the larger value. iv = 4 xxxix = 39 xc = 90 1 subtracted from 5 = 4 30 + ( 1 – 10 ) = 39 10 subtracted from 100 = 90 Common Roman Numerals on RX’s i=1 ii = 2 V=5 X = 10 C = 100 Fractions A fraction is a numerical representative indicating that there is part of a whole. Fractions have numerators and denominators. The denominator is the bottom number of the fraction. It tells us how many pieces the whole is divided into. The numerator is the top number of the fraction. It tells us how many pieces exist. Fractions 2 ─ 5 Example: Numerator we have 2 parts Denominator out of 5 total parts Converting Fractions to Decimals Fractions can be converted to decimals by dividing. 2 ─ 5 = 2 ÷ 5 = 0.4 Decimals A decimal point is used to represent an amount less than one (fraction). 0.1 0.01 0.001 0.0001 = = = = one tenth one hundredth one thousandth one ten-thousandth Reciprocals Reciprocals are two different fractions that equal 1 when multiplied together. The reciprocal of 2 ─ 3 is 3 ─ 2 Reciprocals When multiplied together, reciprocals equal 1 2 ─ 3 x x 3 ─ 2 = = 6 ─ 6 or 1 Adding & Subtracting Fractions In order to add & subtract fractions, the fraction must have the same common denominator. 1 ─ 4 2 ─ 5 + 1 ─ 4 - 1 ─ 5 = 2 ─ 4 = 1 ─ 5 or 1 ─ 2 Multiplying Fractions Multiply each numerator, then multiply each denominator 2 ─ 3 x 24 ─ 1 = 48 ─ 3 or 16 Dividing Fractions To divide a fraction, find the reciprocal of the fraction (or invert the fraction), then multiply. Example: 1/3 ÷ 1/2 = 1 ─ 3 x 2 ─ 1 = 2 ─ 3 Working with Decimals The key to adding and subtracting with decimals is to line up the decimal points, then work the problem as you would a whole number equation. 45.6 + 3.2 48.8 45.6 - 3.2 42.4 Multiplying with Decimals To multiply with decimals, first multiply as if using whole numbers. Count the total number of decimal places in the equation. Insert the decimal point the total number of decimal places, starting from the right. 47.2 (1 decimal place) x 5.5 (1 decimal place) 2360 2360 25960 259.60 (2 decimal places) Percentages Percentages are fractions in which the denominator is always 100. Percentages are expressed (using the % symbol) and mean parts out of 100 units. Example: 25% = 25 ── 100 1 or ─ 4 Basic Percentages 50% = 50 ── 100 2% = 2 ── 100 or 1 ─ 2 2% represents 2 parts out of 100 parts Significant Figures 4 Rules for significant figures: 1. Digits other than zero are always significant. (1,2,3,4) 2. Final zeros after a decimal point are always significant. (1.20) 3. Zeros between two other significant digits are always significant. (1.05) 4. Zeros used only to space the decimal are never significant. (0.1) Measurements There are different systems of measurement used in pharmacy: metric, English, apothecary & avoirdupois. The major system of weights & measures in medicine is the metric system. The different measurement units are related by measures of ten. Metric measures apply to both liquids and solids. Liquids Liquids (including lotions) are measures by metric volume. The most common being liters (L) or milliliters (ml). Unit liter milliliter Symbol L ml Liquid Conversion 1L = 1000 ml 1ml = 0.001 L **cc’s (cubic centimeters) are often used in place of ml Solids Solids (pills, granules, ointments) are measured by weight. Units Symbol kilogram kg gram g milligrams mg microgram mcg Solid Conversion_ 1kg = 1000 g 1g = 1000 mg =0.001 kg 1mg = 1000 mcg =0.001 g 1mcg =0.001 mg Ounces, Pounds & Grains Units Symbol Conversion pound ounce grain lb oz gr 1 lb = 16 oz. 1 oz = 437.5 gr 1 gr = 64.8 mg **grains are often rounded up to 65mg or down to 60mg in the pharmacy practice. Apothecary Measurements Unit gallon quart pint ounce Symbol gal qt pt fl oz Conversion 1 gal = 4 qt 1 qt = 2 pt 1 pt = 16 fl oz 1 oz = 30 ml Household Measures Teaspoons, tablespoons & cups are common household measures. Unit Symbol 15 drops gtts Teaspoon tsp Tablespoon tbsp Cup cup Conversion 15gtts = 1ml 1 tsp = 5ml 1 tbsp = 15 ml = 3 tsp 1 cup = 8 oz Conversions 1L = 1 pt = 1 fl oz = 1 kg = 1 lb = 1 oz = 1g = 1gr = 33.8 fl oz 473.167 ml (473 or 480ml) 29.57 ml (30ml) 2.2 lbs 453.59 g (454g) 28.35 g (30g) 15.43 gr (15gr) 64.8 mg (65mg) Dose Measurements Not all doses doctor’s write prescriptions for are available from the manufacturer…so what do you do? Tablets can be doubled up: for example, the RX is written for 250mg, and only 125mg is made, the result is the dose taken by the patient is 2 tablets together to equal 250mg. A tablet can also be cut in half, or even a quarter to provide the adequate dose for the patient. Dose Measurements Example 1: Rx: Flagyl 125mg po bid x 7 days On hand is 500mg tablets, how would you accommodate the patients needs? 500mg/125mg = 4 Take ¼ tablet (=125mg) by mouth two times a day for 7 days. Dose Measurements How many tablets would you dispense to last the patient the duration of the treatment? ¼ + ¼ = 2/4 or ½ tablet daily ½ X 7 days = 3.5 **always round up on quantity to dispense, ***not dose 4 tablets would be dispensed; and that would equal a 7 day supply. Dose Measurements Example 2: Rx: Amoxicillin 500mg po tid X 10 days On hand Amox 250mg/5ml 250mg = 5ml Take 2 teaspoonfuls (=500mg) by mouth three times a day for 10 days What is the quantity you will dispense to cover the duration of the therapy? Dose Measurements 2 tsp = 10 ml / dose 10ml X 3 = 30ml / day 30 ml x 10 days = 300 ml / 10 days Equations & Variables In pharmacy calculations, there is often an unknown variable that needs to be determined. We solve for the unknown value by setting up mathematical equations. In equations, this is usually represented by the letter “x”. Equations & Variables Example: How many ounces is equal to 120ml? total ml x (oz) = ──────────── ml to oz conversion 120ml x (oz) = ──────── 30ml x (oz) = 4 Equations & Variables Example: Calculate the quantity for an Rx with a sig of 1 tid x 7. x (qty) = (1 cap per dose) x (3 times a day) x (7 days) x (qty) = 1 x 3 x 7 x (qty) = 21 Dose Equation D/A X Q = dose quantity D = desired A = Available on hand Q = Quantity (dose) on hand Dose Equation Example: A prescription calls for 200mg of a drug that you have in a 10mg/15ml concentration. How many ml of the liquid do you need? Desired = 200mg Available = 10mg Quantity (dose) on hand = 15ml Dose Equation 200mg / 10mg X 15ml 200 / 10 = 20 (the mg are canceled) 20 X 15ml = 300 ml Ratio & Proportion Understanding ratios & proportions is important for pharmacy technicians so they can perform the calculations necessary for the job. Ratio: A ratio states a relationship between two quantities. a It can be stated as: a : b or ─ b Ratio & Proportion Proportion: Two equal ratios form a proportion. a c ─ = ─ b d 1 ─ 2 = 2 ─ 4 1/2 & 2/4 are equivalent ratios, therefore the equation is a proportion. Ratio & Proportions Proportion Example: If one person has 1 bottle containing 5 tablets, and another has 3 bottles containing a total of 15 tablets, it is still an equivalent ratio. (5:1) 5 ─ = 1 15 ─ 3 It is not the quantity, but the relationship between the quantity that we are looking for. Solving Ratio & Proportion Equations There are 3 conditions for using ratio & proportion equations: 1. Three of the four values must be known. 2. The numerators must have the same units. 3. The denominators must have the same units. Solving Ratio & Proportion Equations Example: You receive an Rx for Ktabs 1 bid x 30. How many tablets are needed to fill the Rx? Define the unknown variable: Establish the known ratio: Establish the unknown ratio: x tabs ───── 30 days = 2 tabs ───── 1 day We need 60 tablets to fill the Rx. x = total tablets needed. 2 tablets per day x tablets/30 days = x = 60 Percents & Solutions Percents are used to indicate the amount or concentration of something in a solution. Concentrations are indicated in terms of weight to volume or volume to volume. Weight to volume = grams per 100 milliliters ( g/ml ) Volume to volume = milliliters per 100 milliliters (ml/ml) Percents & Solutions Example: You have a 70% dextrose solution. How many grams in 20mls of solution? 70g ─ = 100ml xg ─ = 20ml 1400 = 100x(g) = 14g Calculations for Business Terms: Usual & customary price (U&C) is the lowest price for a customer paying cash on that day for that drug. Average wholesale price (AWP) is the average wholesale price for that drug. Professional fee or fee for service: the charge for service. Calculations for Business Prescription prices are determined by a variety of formulas. The easiest is AWP + professional fee = the selling price. To calculate the retail price, you must first calculate the AWP for the specific quantity of tablets then apply the appropriate professional fee. Calculations for Business Example: What is the retail price for #30 glyburide 5mg tablets (AWP 480.15/M) and a professional fee of $4.00? 1. Calculate AWP per tablet: 480.15/1000 = .48 per tab. 2. Multiply # of tablets by price per tablet = 30 x .48 = 14.40. 3. Add professional fee = 14.40 + 4.00 = 18.40 retail price. Discounts Pharmacies sometimes give a discount to certain groups of patients (such as senior citizens). To begin, calculate the retail price of a prescription as we did on the previous slide, then we calculate the discount. Discounts Example: The glyburide Rx is for a senior citizen, who qualifies for a 10% discount. (retail was 18.40). 18.40 x .10 = 1.84 18.40 – 1.84 = 16.56 The customer would pay 16.56 as the discounted price. Practice What would a customer pay for Verapamil SR #30 (AWP 135.85/C) with a $5.00 professional fee and a 10% senior discount? AWP per tab = 135.85/100 = 1.36 Cost per 30 tablet = 1.36 x 30 tabs = 40.80 Professional fee = 40.80 + 5.00 = 45.80 10 % Discount = 45.80 x .10 = 4.58 Retail price = 45.80 – 4.58 = 41.22 Answer: $ 41.22 Gross Profit & Net Profit Gross profit is the difference between the selling price and the acquisition cost. To calculate gross profit, there is no consideration for any of the other expenses associated with filling a prescription. In its simplest form, cash prescriptions, it is the difference between the selling price and the cost paid for the item Gross Profit Gross profit = selling price – acquisition cost Example: An Rx for Amoxil 250mg # 30 has a U & C of 8.49. The acquisition cost is 2.02. What is the gross profit? GP = 8.49 – 2.02 GP = 6.47 Answer: $ 6.47 Net Profit The net profit is the difference between the selling price and the sum total of all the costs associated with filling the prescription. Costs associated with filling an Rx: Cost of the medication Cost of the vials, lids, labels, caution labels Cost of labor: pharmacy tech, pharmacist, clerk, etc. Costs of operations: rent, utilities These are often grouped together and called a dispensing fee. Net Profit Net profit = selling price – acquisition cost – dispensing fee or Net profit = Gross profit – dispensing fee Example: Amoxil 250mg #30 (U&C 8.49) with an acquisition cost of 2.02 and a dispensing fee of 5.50. What is the net profit? selling price acquisition dispensing fee NP = 8.49 – 2.02 – 5.50 NP = .97 The End Read Chapter 6 Do Practice Problems Throughout Chapter Take Self Test Study Review Pages