PHARMACY MATH

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ROMAN NUMERALS, TEMPERATURE

• I, i=1; V, v=5; X,x=10; L=50; C=100; D=500; M=1000 • Rules of Roman Numerals:

• - V,L and D are never repeated (ex. VV does not=10)

• - If the smaller numeral value is placed before the larger, the smaller is subtracted from the larger (ex. IV= 5-1=4)

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MEASUREMENT SYSTEMS

• Metric System: based on 1000

• Kilo, Hecto, Deca, Base unit, Deci, Centi, Milli, __,___, Micro

• Ex. 500 g= 0.5 Kg (move decimal point 3 places to the right from the base unit to kg)

• Ex. 5 mg= 0.005 g

• 8 oz= 1 cup; 16 oz= 1 pound; 16 oz= 1 pint

• 12 in= 1 foot; 36 inches= 3 feet= 1 yard; 1 inch= 2.54 cm

• 60 drops (gtts)= 1 tsp; 3 tsp= 1 TBSP

• 1 tsp= 5 mL; 1 TBSP= 15 mL

• 1 oz= 30 mL= 30 cc

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RATIO AND PROPORTION

• Ratio is the relationship between 1 quantity to another

• Proportion is the relationship between between 2 equal ratios

• The unknown is solved for using “X” as the unknown

• Ex) A physician orders an antibiotic available as 250 mg/5 mL. The household measurement to be used is a teaspoon. How many tsp?

• Known::Unknown

• 5 mL: 1 tsp= 5 mL: X

• 5x=5 (multiply the 2 outside numbers (5x) and then multiply the 2 inside numbers (5)

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DIMENSIONAL ANALYSIS

•

Dimensional analysis is ratio and proportion

expressed as fractional forms

•

Ex) A physician orders 250 mg of an antibiotic. The

medication is available as 125 mg/5 mL. How

many tsp should be administered?

•

x tsp= 250

mg

/1 dose X 5

mL

/125

mg

X 1 tsp/5

mL

•

x tsp= 250 X 5 X1/ 125 X 5

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•

Ex) 4 oz= ? Tsp

Ratio and Proportion

•

1 oz: 6 tsp= 4 oz: X

•

X= 24 tsp OR

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CALCULATION OF SOLID ORAL

DOSES

• Ratio and Proportion method to solve:

• Dose strength available (DA)/Dose form (DF)= Dose ordered

(DO)/Dose Given (DG)

• DA/DF= DO/DG

• Ex) An order is written for 500 mg of medication and the available

medication strength is 250 mg/capsule.

• 250 mg:1 capsule= 500 mg: x

• 250 x= 500

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EXAMPLE PROBLEMS

• Dimensional Analysis to solve: • 250 mg/ 1 capsule= 500 mg/ x • 250 x= 500

• X= 2 capsules • Formula Method:

• DD (dose desired)/ DH (dose on hand) X Qty (dose form)= dose given (DG)

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SOLID MEDICATIONS IN DIFFERENT

MEASUREMENT SYSTEMS

• Apothecary: 1 gr (grain)= 60 mg; gr V= 5 grains; gr X= 10 grains

• Ex) A physician orders codeine sulfate 60 mg q4-6 h prn for pain. The medication is available is gr ¼.

• R & P: 1 gr: 60 mg= 0.25 gr: X • X= 60 x 0.25

• X= 15 mg divided by 4x/day= 3.75 mg= 4 tablets • DA: 1gr/60 mg= 0.25 gr/x

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CALCULATE TOTAL DOSAGE OF

MEDICATION FROM MD ORDER

• Must find the number of doses in a specific day and multiply by the days the medication is necessary

• Ex) An MD writes an order for Amoxicillin 500 mg tid x10 days. The medication is available in 500 mg dose on the shelf.

• 3 times a day x 10 days= 30 capsules

• Ex) MD writes an order for Flagyl 250 mg po bid x7 days. Available dose is 500 mg.

• 250 mg x2= 500 mg (would need to take 2 tablets each time to reach dose prescribed)

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CALCULATION OF ORAL AND

PARENTERAL LIQUID MEDICATION

• Parenteral medications are given by injection into the body tissue, intravenously, or directly into the bloodstream

• Some parenteral medications come in powdered form and have to be reconstituted before administration

• Liquid medications are given via oral syringes, by a medication cup, dropper, medicine spoons, or by a teaspoon or tablespoon • Liquid medications can be tinctures, elixirs, suspensions and

syrups

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CALCULATIONS WITH LIQUID

MEDICATION

• Ex) An MD orders amoxicillin 500 mg oral suspension. The medication available is 250 mg/5 mL. How much medication will be administered?

• Ratio/Proportion:

• 250 mg/ 5 mL= 500 mg/X • 250 X= 2500

• X= 10 mL

• Dimensional Analysis:

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LIQUID MEDICATION CALCULATION

• Ex) An MD orders Diflucan suspension 35 mg po daily. Available medication is 10 mg/mL. How much medication is

administered?

• 10 mg/1 mL x 35 mg/X • 10 x=35

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CALCULATIONS WITH PARENTERAL

MEDICATIONS

• Available in ampules and vials

• Strength usually stated in the metric system- mg, g, micrograms for weight and mL for volume

• Once the amount of parenteral medication has been decided, the appropriate syringe should be chosen according to that volume

• Ex) An MD orders prochlorperazine 10 mg IM q6h prn for N &V. Available medication is 5 mg/mL. What is the volume of medication to be given?

• 5 mg/1 mL X 10 mg/x

• 5x= 10

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RECONSTITUTION OF POWDERS

INTO LIQUID MEDICATIONS

• Medication can become unstable with lengthy storage; they are usually shipped in powder form to be stored and then reconstituted before administration

• Very common with pediatric medications and immunizations

• A diluent is required for reconstitution; usually sterile water or distilled water; directions for reconstitution are provided by manufacturer

• When powders are dissolved in the liquid, the weight or strength of the medication will always remain the same as the amount given on the label

• Total liquid volume of the medication will the amount of medication plus the amount of liquid

• Powder Volume= difference between final volume and diluent volume • Ex) Final volume after adding 78 mL diluent is 100 mL

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RECONSTITUTION CALCULATIONS

• **Read the labels for necessary information!!

• Ex) The label for Zithromax has a reconstituted strength of 100 mg/mL. The total dosage in the vial is 500 mg. Directions for reconstitution states to add 4.8 mL of sterile water. How many mL are there after reconstitution? What is the powder volume?

• 100 mg/1 mL X 500 mg/x

• 100x=500

• X= 5 mL

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RECONSTITUTION CONT’D

• Ex) The label for Oxacillin sodium for injection contains 250 mg/1.5 mL. Each vial is equivalent of 1 g (1000mg). What volume of medication is needed for an order of 750 mg q6h IM? What is the powder volume if amount of diluent is 5.7 mL? • 250 mg/1.5 mL X 750 mg/x

• 250x=1125

• X=4.5 mL at a dose of 750 mg

• Total volume after reconstitution: 250 mg/1.5 ml X 1000 mg/x • 250x=1500

• X=6 mL in the vial

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CALCUL ATIONS OF MEDICATIONS FOR SPECIAL

POPUL ATIONS BASED ON BODY WEIGHT OR AGE

• Main consideration in this category of patients is children • Classifications:

• Neonates/newborns: birth to 1 month

• Infants: 1 month to 1 year

• Early Childhood: 1 to 5 years old • Late Childhood: 6 to 12 years

• Adolescence: 13 to 17 years

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CALCULATIONS FOR SPECIAL

POPULATIONS

• Pediatric doses of medication may be calculated using weight….mg

(or g or mcg)/kg

• Conversion: 2.2 lbs= 1 kg

• If an infant is weighed in pounds and ounces, you must convert

ounces into pounds

• Ex) An infant weighs 18 lbs 4 oz. Convert to kg.

• 16 oz: 1 lb= 4 oz: x

• 16x=4

• X= 0.25 lb

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EXAMPLE PROBLEMS

• _{Ex) A child weighs 75 lbs. What is the weight in kg?}

• 75/2.2= 34.1 kg

• _{Remember that child’s dosage and weight is expressed as mg/kg}

• _{Ex) A child weighs 53 lbs. The MD orders Dilantin 30 mg capsules to be} given at a 2.5 mg/kg/dose. What dose should be given to the child.

• _{First convert lb to kg: 53/2.2= 24.1 kg}

• Second, consider the formula mg x kg x dose

• _{2.5 x 24.1 x 1= 60.25 mg/dose}

• _{If the dose available is in 30 mg capsules, how many should the child} take to meet the dosage prescribed?

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PEDIATRIC “RULES” FOR

CALCULATION

• Clark’s Rule: Based on body weight, assuming the average adult weights@150 lbs. and the manufacturer considers this in calculating correct dosage

• Child’s dose= Child’s weight in lbs/150 X Adult Dose

• Ex) An MD orders Cefzil for a child that weighs 38 lbs. The adult dose is 500 mg q24h.

• Using Clark’s Rule to solve: 38 lb/150 lb X500 mg= 126.67 mg

• The medication is available as 125 mg/5 ml. Using this information what is the correct dose to give to the above child?

• 125 mg/5 ml X 126.67 mg/X

• 125x=633.35

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MORE “RULES”…

• Fried’s Rule: Pediatric dosage used for infants up to 12 months old; based on age and also considering normal adult dose

• Infant’s Dose= Age in months X Adult dose/150

• Ex) An MD orders Amoxicillin for a 6 month old. The usual adult dose is 500 mg. What is the dose to be given to this child?

• Using Fried’s Rule: 6 mo X 500 mg/150= 20 mg

• Young’s Rule: Pediatric dosage for children 1 yr to 12 yrs old

• Child’s Dose= Age in years X Adult Dose/ Age in Years+12

• Ex) An MD orders Amoxicillin for a 5 yr old. The adult dose is 500 mg. What is the dose to be given to this child?

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PRACTICE PROBLEMS

• 1) An MD orders Milk of Magnesia for a 3 yo child. The adult dose is 30 mL. What volume of medication should be given to this child?

• Young’s Rule: 3 x 30 mL/ 3+12= 6 mL

• 2) An 8 mo old has a staph infection. The MD order Cephalaxin. The adult dose is 500 mg q12h. The available medication is 125 mg/5 mL. What dose should be given to this child? What is the volume of medication to be given q12h? • Fried’s Rule: 8 X 500mg/150= 26.67 mg

• Volume: 125mg/5 mL X 26.67 mg/X • 125x=133.33

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CALCUL ATION OF MEDS MEASURED IN UNITS,

MILLIEQUIVALENTS AND PERCENT OF

CONCENTRATION

• Units and milliequivalents are used to measure medications such as penicillin, insulin and heparin

• Label will read “units/mL”

• Milliequivalents are the measurement of strength of ion

concentration in a medication which indicates the number of grams per solute

• Percentages are the concentration of the weight of a substance or medication dissolved in a solute (volume in the case of

liquids, weight with solids)

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SAMPLE PROBLEMS

• The math is essentially the same except the medication is measured in units and mEq

• Ex) An MD orders penicillin G potassium 750,000 units to be added to IV fluids. The available medication is 1,000,000 units

• Using the formula method to solve: dose desired/dose on hand X quantity

• 750,000 u/1,000,000u X 1 dose= 0.75 mL

• Ex) An MD orders Pfizerpen 800,000 units to be added to IV fluids. The label states adding 4.0 mL of diluent for a medication strength of 250,000 units/mL. • What volume of medication should be prepared for the patient?

• 800,000 u/250,000 u X 1= 3.2 mL

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INSULIN AND HEPARIN

• Most insulin preparations are only available in U-100 strength, which means that each mL of insulin contains 100 units of

medication

• Insulin preparations are labeled by the type of insulin in the vial; “R” is rapid, short-acting regular insulin; “N” is

intermediate acting with a longer onset of action

• Heparin= injectable anticoagulant which is measured in units • Usually available in strengths from 10 units/mL for pediatrics,

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CALCULATIONS

• Can use ratio/proportion, dimensional analysis, or formula method to solve

• Ex) An MD orders heparin sodium 5000 units subcutaneously stat. The

medication is available as heparin sodium 5000 units/mL. What is the dose to be given?

• R & P: 5000 u:1 mL=5000 u: X

• 5000x=5000

• X= 1 mL

• DA: 5000 u/1mL X 5000 u/x

• X= 1 mL

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MEDICATIONS MEASURED IN

MILLIEQUIVALENTS (MEQ)

• Used as a term of measurement for most electrolytes such as potassium, sodium, and chlorides; a direct conversion between mEq and other

measurement systems cannot be made

• Ex) A physician orders potassium chloride 20 mEq IV. The medication label reads 2 mEq/mL. What volume of medication should be given?

• R & P: 2 mEq:1 mL= 20 mEq:x

• 2x=20

• x=10 mL

• DA: 2 mEq/1 mL x 20 mEq/x

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PERCENTAGE AND RATIO STRENGTH

• Usually used for liquid or semisolid medications

• When interpreting labels for percentage medications, the amount of medication us either in weight/weight (solid in solid) like 1%

hydrocortisone cream; weight/volume (solid in liquid) like 0.9% normal saline; or in volume/volume (liquid in liquid) as in 70% isopropyl alcohol • **Medications that are labeled weight/weight, the number of grams of

solid solute is always 100 g of solvent

• Ex) 1% hydrocortisone cream is equal to 1g of hydrocortisone in 100 g of solvent; 0.9% normal saline equals 0.9g sodium chloride in 100 mL

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INTERPRETING LABELS

• Ex: epinephrine solution 1:10,000 • 1g in 10,000 mL OR 0.01g/100 mL • Ex) 0.5% glycerol solution

• 0.5g in 100 mL OR 0.5g/100mL • Ex)1:50 Chloraseptic solution

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CALCULATION OF IV MEDS

• _{Mainly found in hospital environment to administer medications}

continuously, for fluid replacement, or to allow administration over a longer period of time vs. repeated injections of parenteral medications

• _{Common prepared IV fluids are dextrose and normal saline; medications}

can then be added to these fluids to administer

• _{Fluids are labeled with a percentage of solute in the container or the} amount of solute found in solvent

• Common IV Solutions: NS (normal saline, 0.9% sodium chloride); ½ NS

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SAMPLE PROBLEMS

• Ex) An IV bag reads 10% Dextrose and 0.9% Sodium Chloride injection USP in a 500 mL bag. What is the weight of Sodium Chloride in these fluids?

• 0.9g/100mL= x/500mL

• 100x=450

• X=4.5g NaCl

• What is the weight of Dextrose?

• 10g/100mL=x/500mL

• 100x=5000

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ANOTHER EXAMPLE

• One liter of D-5-NS contains Potassium Chloride. The label for the vial of KCl reads 40 mEq (or 2 mEq/mL) in a 20 mL single vial dose. The entire vial is used and the vein infiltrates after 900 mL have been infused. How many mEq of KCl are added to the fluids?

• 2mEq/1mL=x/20/mL

• X=40 mEq (or simply read the label, if the entire vial is used then 40mEq is given!)

• How many g of Dextrose did the patient receive after filtration?

• 5g/100mL=x/900

• 100x=4500

• X=45 g Dextrose

• How many g of NaCl were received after filtration?

• 0.9g/100mL=x/900mL

• 100x=810

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IV FLOW RATES IN DROPS/MINUTE

• Flow Rate= Amount of fluid x Calibration on administration set/ Infusion Time

• OR Flow Rate (gtts/min)= mL ordered x (gtts/mL)/minutes for infusion

• Ex) A physician orders 1.5L of Lactated Ringer’s solution to be infused over 8 hours. The infusion administration set reads 20 gtts/mL. What is the flow rate? • 1.5L=1500 mL

• 8 hours= 8x60min=480 minutes • FR= 1500mL x 20 gtts/ml / 480 min • FR=30,000gtts/480 min

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MORE EXAMPLES

• A physician orders 2L of Lactated Ringer’s solution over 12 hours. The drop factor is 20 gtts/mL. How many drops/minute would be infused?

• 2L=2000 mL

• 12 hours=12x60 min=720 minutes • FR=2000mL x 20 gtts/mL / 720 min • FR=40,000/720

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COMBINATION PROBLEM

• A physician orders 1000mL D-10-1/2NS to be administered over 12 hours. The drop factor is 10 gtts/mL. How many drops per minute would be infused?

• 12 hours= 12x60=720 min

• FR= 1000 mLx10gtts/mL /720 min

• FR=13.8 or 14 gtts/min

• How many g of Dextrose are in these fluids?

• 10g/100mL=x/1000 mL

• X=100g Dextrose

• How many g of NaCl are in these fluids?

• 0.45g/100mL=x/1000mL

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CALCULATING IV INFUSION TIMES

• Use the Flow Rate formula and plug in what you know to find out the infusion time

• Ex) A physician orders 1000mL D-5-NS to be administered at 20gtts/min. The drop factor is 15 gtts/mL. What is the infusion time for these fluids?

• 20gtts/min= 1000mL x 15gtts/mL / X • 20gtts/minX=15000gtts

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ANOTHER EXAMPLE FOR INFUSION

TIME

• A physician orders 2500 mL D-5-1/2NS to be infused at 30gtts/min. The drop factor is 10gtts/mL. What is the infusion time?

• 30gtts/min=2500mL x 10gtts/mL/ X • 30gtts/minx=25,000gtts

• X=25,000gtts/30gtts/min • X=833.3 min OR13.9 hours

• What is the metric weight of Dextrose? • 5g/100mL=x/2500mL

• X= 125 g Dextrose

• What is the metric weight of NaCl? • 0.45g/100mL=x/2500mL

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STOCK SOLUTIONS

• _{These medications are stored in concentrated amounts, for storage} purposes, and are diluted later on based on physician orders

• _{Calculation can be solved using ratio and proportion, dimensional} analysis or by using the dilution formula

• _{Weight in Weight Calculation:}

• Ex) An MD asks that you prepare 100g of 2.5% hydrocortisone cream

from available (stock) hydrocortisone cream 10%. What is the weight of hydrocortisone in the stock medication? 10%=10g in100g stock

• _{What weight of the 10% stock is needed to prepare 2.5%?}

• _{10g/100g=2.5g/x}

• _10x=250

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STOCK SOLUTIONS

• Volume in Volume Calculations:

• Ex) An MD orders 15 mL of a 10mcg/mL dilution of a drug for a child. The stock medication is 40 mcg/mL. How many mcg of the drug will be needed?

• You must first find the weight of medication in the MD order and then use that weight to solve for the stock solution

• 10mcg/1mL= x/15mL

• x=150 mcg

• How many mL of the stock medication is needed?

• 40 mcg/1 mL=150 mcg/x

• 40x=150

• x-=3.75 mL

• How many mL of diluent are necessary?

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EXAMPLE PROBLEMS

• Epinephrine is available in 5% solution. How many mL are

necessary to prepare 10 mL of a 2.5% solution of epinephrine? • 2.5g/100mL=x/10mL

• 100x=250

• X=0.25 g Now use this weight to solve the equation • 5g/100mL=0.25g/x

• 5x=25

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EXAMPLE PROBLEMS

• Available is a 15% solution of sodium hypochlorite. The MD wants a 1L of a 0.15% solution. How many mL of sodium hypochlorite would be

necessary?

• 0.15g/100mL=x/1000mL

• 100x=150

• X=1.5g

• 15g/100mL=1.5g/x

• 15x=150

• X=10mL of 15% solution for 1L of 0.15% solution

• How many mL of solvent should be added?

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DILUTION FORMULA FOR STOCK

SOLUTIONS

• Amount of stock solution needed=Amount sol’n prescribed x strength prescribed / strength of stock solution

• Ex) Prepare 1L of Lysol 3% from a stock solution of Lysol 10%. How many mL of stock solution are necessary for this preparation?

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ALLIGATION METHOD TO CALCULATE

DILUTIONS

• Alligation method is used to solve calculations involving the mixing of solutions or compounds possessing different percentage strengths.

• “Tic-Tac-Toe” board to solve: • 1) Prepare the tic-tac-toe graph

• 2) Put the strength to be calculated in the center box

• 3) Place the highest percentage concentration in the left upper corner • 4) Place the lowest percentage concentration in the left lower corner

• 5) Subtract the center square from the upper left square and place in the lower right square to reveal the parts of the lowest percent concentration to be used • 6) Subtract the lower left square from the center square and place in the upper

right corner to reveal the parts of the higher percent concentration to be used • 7) Add the calculated parts together to find the total parts of the two

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ALLIGATION EXAMPLE

• Prepare 1L solution of 70% alcohol from 50% alcohol and 95% alcohol.

95 20

70

50 25

45g/1000mL=20g/x 45x=20,000

X=444.4mL of 95% soln

45g/1000mL=25g/x 45x=25,000

X=555.6mL of 50% soln

444.4+555.6=1000mL 45 g

PHARMACY MATH

ROMAN NUMERALS, TEMPERATURE

RATIO AND PROPORTION

DIMENSIONAL ANALYSIS

DOSES

SOLID MEDICATIONS IN DIFFERENT

CALCULATION OF ORAL AND

CALCULATIONS WITH LIQUID

MEDICATIONS

RECONSTITUTION OF POWDERS

RECONSTITUTION CALCULATIONS

CALCUL ATIONS OF MEDICATIONS FOR SPECIAL

EXAMPLE PROBLEMS

PEDIATRIC “RULES” FOR

MORE “RULES”…

PRACTICE PROBLEMS

CALCUL ATION OF MEDS MEASURED IN UNITS,

SAMPLE PROBLEMS

INSULIN AND HEPARIN

CALCULATIONS

MEDICATIONS MEASURED IN

MORE SAMPLE PROBLEMS

PERCENTAGE AND RATIO STRENGTH

INTERPRETING LABELS

SAMPLE PROBLEMS

ANOTHER EXAMPLE

IV FLOW RATES IN DROPS/MINUTE

MORE EXAMPLES

CALCULATING IV INFUSION TIMES

STOCK SOLUTIONS

STOCK SOLUTIONS

EXAMPLE PROBLEMS

MORE EXAMPLE PROBLEMS

DILUTION FORMULA FOR STOCK

ALLIGATION EXAMPLE