1. A Brief Overview of Pharmacy Calculations for Pharmacy Technicians
Maine Pharmacy Association and
Maine Society of Health-System Pharmacists Sesquicentennial Anniversary Meeting
October 14, 2017
Tianzhi Yang, BS, PhD Associate Professor of Pharmaceutical Sciences Husson University School of Pharmacy

2. LEARNING OBJECTIVES
Discuss and explain ratio and proportion and dimensional analysis methods to solve pharmacy calculation problems
Recognize and express units among the pharmacy math systems, especially the metric systems
Define and express percentage strength of a medication expressed as weight/weight, weight/volume or volume/volume
Discuss and explain parenteral IV flow rate calculations

3. PHARMACEUTICAL CALCULATIONS
   
TWO BASIC METHODS USED:
1. Ratio and Proportion
2. Dimensional Analysis

4. RATIO AND PROPORTION Ratio: A relation of two numbers
expressed as a fraction. e.g. 1/3 or 1:3
Proportion: The equality of two ratios
If a = c
b d
Then a = bc or b = ad
d c
or c = ad or d = bc
b a
Ratio and Proportion method is based on the concept that one component is in proportion to another.
   

5. RATIO AND PROPORTION
Example 1: If one capsule contains 50 mg of Benadryl, how many milligrams of Benadryl would be contained in 10 capsules?
1 capsule = 10 capsules
50 mg x mg
x = 10 x = 500 mg 1

6. RATIO AND PROPORTION
   
Example 2: If a solution of NaCl contains 2 g in 50 ml of solution, how many grams of NaCl in 100 ml of solution?
2 g = x g
50 ml ml
x = 2 X = = 4 g

7. RATIO AND PROPORTION – REAL LIFE
One person's error killed Elisha Crews Bryant, hospital officials said:
“a miscalculation overdosed the pregnant 18-year- old with a magnesium sulfate meant to slow her labor. She got 16 grams when she should have gotten 4 grams. The young mother began having trouble breathing, went into cardiac arrest and could not be revived.”
Pregnant 18 Year Old Dies After Hospital Error (2006)

8. RATIO AND PROPORTION – REAL LIFE
Patient received 16 grams Magnesium Sulfate, fatal dose.
Patient should have received 4 grams.
The hospital had 25 g/50 ml Magnesium Sulfate.
How many milliliters of this Magnesium Sulfate (25 g/50 ml) should the patient have received to obtain 4 grams?

9. RATIO AND PROPORTION – REAL LIFE
How many milliliters of this Magnesium Sulfate (25 g/50 ml) should the patient have received to obtain 4 grams?
25 g = 4 g
50 ml x ml x = 8 ml

10. TRY ONE – RATIO AND PROPORTION
How many milliliters must be drawn from an ampule of Prochlorperazine labeled "10 mg/2 ml" in order to obtain a dose of 7.5 mg?
10 mg = mg
2 ml x ml
x = 1.5 ml

11. DIMENSIONAL ANALYSIS Based on cancelling out the units of measures
   
Based on cancelling out the units of measures
Steps:
Step 1: find the ratio that is in the problem
Step 2: set up the problem around the ratio so that the units cancel out. The unit that is left (i.e. the unit that does not cancel out with the other units) should correspond to the unit needed for the answer to the problem
Especially useful in calculating IV flow rates

12. DIMENSIONAL ANALYSIS SCHEME
Conversion
Factor
as Needed
Given Quantity/ratio
Conversion
Factor
as Needed
? (Wanted Unit) =
= Conversion Computation
= Wanted Quantity and Unit

13. DIMENSIONAL ANALYSIS
Example 1: If an antibiotic preparation contains 5 g of penicillin V potassium in 200 mL of solution, how many milligrams of the antibiotic would be contained in each teaspoonful dose? 1) 5 g = 5000 mg 1 tsp = 5 ml 2) ? mg = 5000 mg x 5 ml 200 ml = 125 mg

14. DIMENSIONAL ANALYSIS Example 2:
An order is written for 375 mg of ampicillin to be given intravenously every 6 hours to a child weighing 15 kg. Ampicillin is available in a 1g/50 ml concentration. Calculate the volume in milliliters needed for a single 375 mg dose.
? ml = ml x g x 375 mg
1 g mg
= ml
   

15. TRY ONE – DIMENSIONAL ANALYSIS
A patient receives a solution by IV infusion at a rate of 30 drops/min. How much solution (milliliters) is infused in 2 hours if the infusion set has a drop factor of 15 drops/mL?
? ml = 1 ml x 30 drops x 60 min x 2 hr = 240 ml
15 drops min hr

16. Conversions – Weights and Measures
Common prefixes in the metric system
Kilo (k) = 103 = as in kg
Deci (d) a 10th or as in dL
Centi (c) a hundredth or as in cm
Milli (m) a thousandth or as in mL or mg
Micro (μ) a millionth or as in mcg
Nano (n) a billionth or as in ng

17. EXAMPLES OF METRIC SYSTEM UNITS
 Weight
1 kg = 1000 g
1 g = 1 gram
1 mg = g (10-3 g)
1 mcg = 10-6 g
1 ng = 10-9 g
  Volume
1 L = 1 liter
1 dL = 0.1 L
1 mL = L (10-3 L)

18. CONVERSION EQUIVALENTS AMONG PHARMACY MATH SYSTEMS
Weight
1 kilogram = 2.2 lb
1 ounce = 28.4 (30) g
1 pound = 454 g
Volume
1 teaspoon = 5 mL
1 tablespoon = 15 mL
1 fl oz = 29.6 (30) mL
1 pint = 16 fl oz; 473 mL
1 quart = 2 pints; 946 ml
1 gallon = 4 quarts; 3785 mL
1 cup = 8 oz
Length
1 inch = 2.54 cm
1 foot = 12 inch

19. Conversions Example 1: How many fluid ounce are contained in 3 Liters?
1) Unit conversion factors : 1 L = 1000 ml
1 fluid oz = 30 ml
2) ? fl oz = fl oz x ml x 3 L
30 ml L
= 100 fl oz
   

20. CONVERSION
Example 2: A patient weighing 220 pounds is given a drug which is dosed at 5 mg per kg of body weight. Calculate the amount of drug needed for one dose for this patient.
  1) 220 lb = 100 kg  
2) 5 mg = x mg  x = 500 mg
1 kg kg
Using dimensional analysis:
? mg = 5 mg x 1 kg x 220 lb
1 kg lb
= 500 mg

21. TRY SOME CONVERSIONS 1 Liter = ? ml (2) 110 lb = ? kg (3) 1 tsp = ? ml
(4) 1 mg = ? mcg
1 liter = 1000 ml
110 lb = 50 kg
1 tsp = 5 ml
1 mg = 1000 mcg

22. Concentrations
Measure of how much of a given substance that is mixed with some other substances
Frequently expressed by:
Percentage strength
Ratio strength
Milliequivalents (per mL or L)
Molarity (mol/L or mmol/L)
Osmolarity (Osmol/L or mOsmol/L)

23. Concentrations – Percentage Strength
% – Parts per 100 parts
weight-in-volume (w/v) g / 100 mL
volume-in-volume (v/v) mL / 100 mL
weight-in-weight (w/w) g / 100 g
100 parts – Total product quantity
(weight or volume)

24. Percentage Strength
Example 1: How many grams of sodium chloride are provided by 500 ml of normal saline (hint: normal saline is 0.9% NaCl). 0.9 g = x g 100 ml 500 ml x = 4.5 g

25. Percentage Strength
Example 2: How many grams of hydrocortisone is available in a 60 g tube of 1% hydrocortisone cream? 1 g = x g 100 g 60 g x = 0.6 g

26. Percentage Strength
Example 3: A 30 ml of a liquid astringent was contained in 200 ml of lotion. Calculate the percent strength of the liquid astringent. 30 ml = x ml 200 ml 100 ml x = 15 ml --> 15%

27. TRY ONE - Percentage
Calculate the amount of alcohol in 1 liter of a 70% alcohol solution?
70 ml = x ml
100 ml ml
x = 700 ml

28. Parenteral IV Fluid Flow Rate
IV fluid flow rate (infusion rate): rate that an IV medication leaves bag and enters patient’s blood stream; amount or volume of drug a patient will receive over a given period of time
Expressed as a volume or amount per unit time; e.g., ml/hour; ml/min; drops/min; mg/hr
Flow rate = volume / time
By manipulating the flow rate formula, there are three types of questions:
Flow rate
Volume
Time
Always be sure which time and volume units you are being asked to solve for
Is it ml/min ? Or l/hr? Something else?
Dimensional analysis method is recommended for solving the flow rate problems

29. Flow Rate Example
A patient receives 1 L of IV solution over a 3 hour period. Calculate the flow rate in ml/hr.
Flow rate = volume / time
= 1000 mL / 3 hours
= 333 mL/hour
Another way – dimensional analysis
? ml/hr = ml x L = 333 ml/hr
1 L hrs

30. Solve for Time By manipulating the rate formula, we can solve for time
The equation becomes:
Time = Volume / flow rate

31. Flow Rate - Time Example
If an IV is run at 50 ml/hr, how long will a 500-ml bag last?
Time = Volume / flow rate
= ml = 10 hours
50 ml/hr
Another way – dimensional analysis
? hr = hr x ml = 10 hrs
50 ml

32. Solve for Volume
By manipulating the rate formula, we can solve for volume
The equation becomes:
Volume = rate x time

33. Flow Rate - Volume Example
How many ml of IV solution would be required to run an IV for 12 hours at a rate of 60 ml/hr?
Volume = 60 ml x 12 hrs = 720 ml hr
Another way – dimensional analysis
? ml = ml × 12 hr = 720 ml
1 hr

34. ONE MORE EXAMPLE
A patient receives a solution by IV infusion at a rate of 36 drops/min. How much solution (milliliters) is infused in 3 hours if the infusion set has a drop factor of 15 drops/mL? Method 1 - use formula: Volume = rate x time rate = 36 drops x 1 ml = 2.4 ml/min min 15 drops Volume = 2.4 ml x 180 min = 432 ml min Method 2 - Another way “dimensional analysis” ? ml = 1 ml x 36 drops x 60 min x 3 hr = 432 ml 15 drops 1 min 1 hr

35. TRY ONE - FLOW RATE
You have a 1.5 liter bag that needs to be run over 12 hours. What is the flow rate in ml/min?
1) 1.5 liter = 1500 ml
5 hours = 5 x 60 min = 300 min
2) Flow rate = volume / time
= ml
300 min
= 5 ml/min

37. Self-Assessment Question #1
How much (in grams) potassium chloride is needed to prepare one liter of 3% potassium chloride solution?
A. 0.3 g
B. 1 g
C. 3 g
D. 30 g
Answer: D
3 g = x g
100 g ml x = 30 g

38. Self-Assessment Question #2
The dose of an antibiotic is 40 mg/kg once daily. How much (in mg) of the antibiotic per dose should be given to a patient who weighs 40 lb? A. 168 mg B. 727 mg C. 960 mg D mg
Answer: B
40 lb = kg
40 mg / kg x kg = 727 mg

39. Self-Assessment Question #3
Three liters of D5W is to be infused over 24 hours, using an IV set that delivers 15 drops/ml. Calculate the flow rate in drops/minute.
A. 31 drops/min
B. 55 drops/min
C. 69 drops/min
D. 82 drops/min
Answer: A
? drops = 15 drops x ml x 1 hour
min ml hours min
= 31 drops/min

40. Self-Assessment Question #4
Griseofulvin oral suspension contains 125 mg/5 mL. A physician prescribed 250 mg twice a day for 2 weeks for a patient. How many milliliters of griseofulvin should be dispensed in order to fill this prescription? A. 100 ml B. 150 ml C. 280 ml D. 473 ml
Answer: C
1) Total dose = 250 mg x 2 x 14 = 7000 mg
2) 125 mg = mg
5 ml x ml x = 280 ml

41. Thank you! Let me know if you have any questions.
Tianzhi Yang, BS, PhD Associate Professor of Pharmaceutical Sciences Husson University School of Pharmacy 1 College Circle, Bangor, ME Phone: Fax: