Instructor: Daniella Krantz, CPhT B. S. Biology Pharmacy Math Instructor: Daniella Krantz, CPhT B. S. Biology

Class Expectations Homework is due at the beginning of each class There will be a quiz at the beginning of each class Students may use a calculator, but only a basic four- function calculator Academic conduct is outlined in the syllabus If you are confused about a concept or unsure of how to solve the problems, do not wait until the day before the exam to ask for help.

Class expectations (cont’d) Writing and answers are to be legible. If I cannot read/understand the answer, it will be marked wrong If a problem requires several steps to obtain the answer, you will need to show your work. Just giving the answer is not enough; students need to demonstrate the ability to step through a problem

Topics Fractions Proportions Ratios Percents Metric Conversions/Unit conversions Liquids

Lecture 1 Introduction to pharmacy Math Roman Numerals Basic Algebra/math Fractions Decimals

Fractions A proper fraction is a fraction that has the numerator is smaller than the denominator (Ex. 1/3) An improper fractions is a fraction that has a numerator larger than the denominator (Ex. 8/3) A mixed fraction is a combination of whole numbers and a fraction (Ex. 1 1/3) Fractions can be made into decimals as well as a percent) (Ex. 25/100 = 0.25 also, 25/100 = 25%)

Fractions (cont’d) Fractions should be reduced to it’s lowest possible state Right: 1/4 Wrong: 25/100 If your answer on a quiz/exam is not completely reduced, there will be a 1-point deduction

Fractions (cont’d) Reduce these fractions 6/36 3/9 10/50 7/49 25/200 9/189 4/20 2/28 5/65 8/72

Fractions (cont’d) Converting mixed fractions back into improper fractions Why? In order to calculate with fractions you need to have a mixed fraction switched back to an improper fraction 12 5/8 = ? Multiply 12 by the denominator (8): 12 x 8 = 96 Add 96 to the numerator (5): 96 + 5 = 101 Place the 101 over the ORIGINAL denominator (8) Answer: 101/8

Fractions (cont’d) Convert these Mixed fractions to Improper fractions 1 1/2 4 1/3 5 7/8 6 3/4 3 1/3 2 11/24 9 1/4 5 1/12 11 3/4 9 1/2

Fractions (cont’d) Calculating with Fractions: Multiplying (probably the easiest) Just multiply straight across: Numerator with numerator Denominator with denominator 3/5 x 1/5 = 3/25 3/5 x 4/5 = 12/25 2/3 x 2/7 = 4/21 2/7 x 3/6 = 6/42  1/7 4/9 x 2/3 = 8/27

Fractions (cont’d) Multiply these fractions 1/3 x 2/4 2/5 x 3/2

FRACTIONS (cont’d) Calculating with Fractions: Addition: the denominators MUST be the same 3/5 + 1/5 = 4/5 3/5 + 4/5 = 7/5  1 2/5 2/3 + 2/7 = ? Find the lowest common denominator: 21 Change the numerators & denominators to reflect the new common denominator 2/3 x 7/7 =14/21 2/7 x 3/3 = 6/21 14/21 + 6/21 = 30/21  1 9/21  1 3/7

FRACTIONS (cont’d) Calculating with Fractions: Subtraction: the denominators MUST be the same 3/5 - 1/5 = 2/5 7/5 - 4/5 = 3/5 2/3 - 2/7 = ? Find the lowest common denominator: 21 Change the numerators & denominators to reflect the new common denominator 2/3 x 7/7 =14/21 2/7 x 3/3 = 6/21 14/21 - 6/21 = 8/21

decimals Fractions with a denominator being a power of 10 and the numerator expressed after the decimal point Ex: 0.3 is the same as 3/10 10 is a “power of ten” 0.7 = 7/10 0.25 = 25/100 also 1/4 when reduced to lowest terms 0.675 = 675/1000 also 27/40 0.3579 = 3579/10000

tens ones 67.123 Decimals (cont’d) 1/10 (tenths) 1/100 (hundredths) 1/1000 (thousandths)

Decimals (cont’d) Converting decimals back to fractions: IF 0.23 = 23 hundredths, THEN 23/100 is the fraction form of the decimal Ex: 0.234 = 234/1000 = 117/500 0.25 = 25/100 0.500 = 500/1000 = 1/2 0.333 = 333/1000 however, we would probably leave this as a decimal or memorize it as the fraction 1/3 0.680 = 68/100 the zero trailing the 8 is unnecessary in the conversion back to fraction

Fraction  Decimal conversion In order to convert a fraction to a decimal you need to take the numerator and divide by the denominator 5/8 = 5 divided by 8 = 0.625 1/2 = 1 divided by 2 = 0.5 3/4 = 3 divided by 4 = 0.75 7/8 = 7 divided by 8 = 0.875 6/7 = 6 divided by 7 = 0.857

HOMEWORK Prepare for Quiz 1 Review basic concepts of lecture 2 Decimals Percents Metric Apothecary Milliequivalents

Lecture 2 Quiz 1 Metric Ratios Proportions

metric Three primary units We will focus on volume and weight Meter—length Liter—volume Gram—weight We will focus on volume and weight

Metric (cont’d)

Metric (cont’d)

Metric (cont’d)

Ratios A ratio is a comparison of two amounts This can be expressed several different ways Part to part (1:3)—for every 1 apple there are 3 oranges Part to whole (1:4)—for every 1 apple there are 4 pieces of fruit As a decimal (0.25)— 0.25 are apples As a percent (25%)—25% of the fruit is apples

Ratios (cont’d) There are 10 apples, 13 bananas, 16 oranges, and 7 peaches Write ratios 4 different ways comparing apples to each piece of fruit

Ratios (cont’d)

proportions A proportion is the expression of equality of two ratios or fractions to each other. A/B = C/D This is the concept you will most likely use the most in pharmacy—especially hospital pharmacy

Proportions (cont’d) If you have a label that reads as follows: 4gm magnesium in 100 mls saline How many mL will you need to add to the bag?

Proportions (cont’d) Your labels reads as follows: Dexmedetomidine 400 mcg/ 100 mL saline How many vials will you use?

Proportions (cont’d) You are given an order for two bags: Bag 1 (bolus): 150 mg Amiodorone in 100 mls dextrose Bag 2: 900 mg Amiodorone in 500 mls dextrose How many and which of the vials to the right would you use?

Proportions (cont’d) Your label reads as follows: Fentanyl PCA 2000mcg/100mls saline Your choices of ampules are: 100mcg/2ml 250mcg/5ml 1000mcg/20ml How many milliliters do you need, which vial/ampule, and how many would you choose?

Terms (pg 51) q. d. PRN b. i. d q. i. d Q. 12 (or any number) STAT t. i. d IVPB Inj o. d. a. d. SC or SQ a. s. IM IV IVP P.O.

Terms (cont’d) A C Q DC, d/c, or disc N/V R/O NPO Dx Hx HA

HOMEWORK Prepare for Quiz 2 Review basic concepts of lecture 3 Percents Milliequivalents IV flow rates

Lecture 3 Quiz 2 Review: Proportions vs. Unit conversions Drug Orders Percents IV Flow Rates Milliequivalents

Proportion review 3 of the 4 amounts must be known in order for a proportion to work Start with what you are given Write down what you need to find The numerators must be the same units The denominators must be the same units

Unit conversion review Different than a proportion Convert your units first before setting up your proportion Start with what you are given Make sure your units will cancel out Make sure your final answer has the units in the numerator position

Parts of a drug order Name of the patient to receive the medication Name of the drug to be dispensed or administered Dose of the drug Route by which the drug is to be taken or administered Dosage regimen by which the drug is to be taken or administered Signature of the person writing the order or prescription Memorize for final exam

percents Percents are a “part of 100” Represented by the symbol % 25% 30% 80% 75% 10% 15%

Percents (cont’d) Percent strengths are used to describe IV solution & topical drugs The higher the concentration of the dissolved drug, the higher the strength of the drug will be Pg 97

Percents (cont’d) If you took a 135-question history exam & got 94 questions correct, what would be your score? Solve Pg 97

Percents (cont’d) Page 99, #2a #2b Do # 2 c & d on your own What percent of milk is a 12-fluid ounce glassful? 12 fl oz/128 fl oz = x/100 fl oz Why do you want all of your units to cancel out? #2b If patricia drank a pint of milk, what percent did she drink? 1 pint/8 pints = x/100 pints Do # 2 c & d on your own

Percents (cont’d) Percent weight-in-weight (w/w) or (wt/wt) grams/100 grams Percent volume-in-volume (v/v) or (vol/vol) mL/ 100 mL Percent weight-in-volume (w/v) or (wt/vol) grams/100 mL

Percents (cont’d) You can change a % to a ratio by dividing by 100. Then reduce the fraction using the 1st number in the ratio as the numerator and the 2nd number in the ratio as the denominator Ex: 75%  75/100  3/4  3: 4 35%  35/100  7/20  7:20 40%  40/100  2/5  2:5

Percents (cont’d) What is the concentration of drug per milliliter?

Percents (cont’d) What is the concentration of drug per milliliter? 1 gm /100 mL = x-mg/ 1mL 1000mg/100 mL = x-mg/ 1mL Cross multiply & divide 10 mg/ 1 mL

Percents (cont’d) What is the concentration of drug per milliliter?

Percents (cont’d) What is the concentration of drug per milliliter? 2 gm /100 mL = x-mg/ 1mL 2000mg/100 mL = x-mg/ 1mL Cross multiply & divide 20 mg/ 1 mL

Percents (cont’d) What is the concentration of drug per milliliter?

Percents (cont’d) What is the concentration of drug per milliliter? 0.5 gm /100 mL = x-mg/ 1mL 500 mg/100 mL = x-mg/ 1mL Cross multiply & divide 5 mg/ 1 mL

Percents (cont’d) What is the concentration of drug per milliliter?

Percents (cont’d) What is the concentration of drug per milliliter? 0.5 gm /100 mL = x-mg/ 1mL 500 mg/100 mL = x-mg/ 1mL Cross multiply & divide 5 mg/ 1 mL

Percents (cont’d) What is the concentration of drug per milliliter?

Percents (cont’d) What is the concentration of drug per milliliter? 0.25 gm /100 mL = x-mg/ 1mL 250 mg/100 mL = x-mg/ 1mL Cross multiply & divide 2.5 mg/ 1 mL

Percents (cont’d) What is the concentration of drug per milliliter?

Percents (cont’d) What is the concentration of drug per milliliter? 0.75 gm /100 mL = x-mg/ 1mL 750 mg/100 mL = x-mg/ 1mL Cross multiply & divide 7.5 mg/ 1 mL

milliequivalents This is the concentration of electrolytes in a solution Abbreviated (mEq) Common electrolytes Sodium chloride Potassium chloride Sodium phosphate Potassium phosphate Sodium bicarbonate

Milliequivalents (cont’d) Many times you will make IVs of electrolytes. Commonly it will read: 20 mEq potassium chloride/100 mL normal saline All you need to know is how many milliequivalents each milliliter contains Ex. Potassium Chloride: 2 mEq/mL How much would you put into an IV bag if the order is asking for 20 mEq of KCl? In general, you will use milliequivalents as just another unit of measure

Milliequivalents (cont’d) Your order reads: 2 gm CaCl in 100 mL NaCl How many mL will you use? How many vials will you use? Do you need to use a “percent” in this equation? Milliequivalents?

Milliequivalents (cont’d) Your order reads: 150 mEq Sodium Bicarbonate in 1000 mL NaCl How many mL will you use? How many vials will you use? Do you need to use a “percent” in this equation?

Milliequivalents (cont’d) Your order reads: 15 mEq Potassium Acetate in 50 mL NaCl How many mL will you use? How many vials will you need?

Milliequivalents (cont’d) Your order reads: 20 mEq NaCl in 250 mL Dextrose 5% How many mL will you use? How many vials will you need? Do you need to use a “percent” in this equation?

IV Flow rates Objective Be able to give an approximate length of time that an IV bag will last Rates will be calculated in mL/hour All answers for rates will be in mL/hour

IV flow rates You will use proportions to work these problems There is an order given to give a patient 1 liter of saline at the rate of 100ml/hour How long will this bag last? 1000 mL/x-hrs = 100 mL/1 hr Cross multiply & divide 1000mL/100mL = 10 hours There is an order given to give a patient 1 liter of saline over 8 hours What is the rate per hour? 1000 mL/8 hours = x / 1 hour 1000 mL/8 hrs = 125 mL/1 hour

IV flow rates You will use proportions to work these problems Give 750 mg Keppra in 100 mL saline over 1/2 hour Q8. How many mL of Keppra will you use? How many vials of Keppra will you use? What is the rate of this IV? If this IV is given at 9:00 am, when will the next bag be started?

IV flow rates You will use proportions to work these problems Give 500mg Venofer in 100 mL saline over 1 hour. How many mL of Venofer will you use using the vial shown? How many vials of Venofer will you use? What is the rate of this IV?

IV flow rates You will use proportions to work these problems Give 12.5 mg Phenergan in 50 mL saline over 15 mins. How many mL of Phenergan will you use, using the vial shown? How many vials of Phenergan will you use? What is the rate of this IV?

IV flow rates (cont’d) Your order reads: Give 2gm CaCl in 100 mL saline over 1/2 hour PRN. How many mL CaCl will you use? How many vials CaCl will you use? What is the rate of this IV bag? If this IV is given at 9:00 am, when will the next bag be started?

IV flow rates (cont’d) Your order reads: Give 150 mEq Sodium Bicarbonate in 1000 mL NaCl 200 mL/hr How many mL will you use? How many vials will you use? Do you need to use a “percent” in this equation? How long will this bag last?

IV flow rates (cont’d) Your order reads: Give 15 mEq Potassium Acetate in 50 mL NaCl over 30 mins How many mL will you use? How many vials will you need? What is the rate of this IV bag?

IV flow rates (cont’d) Your order reads: Give 20 mEq NaCl in 250 mL Dextrose 5% over 10 hours How many mL will you use? How many vials will you need? Do you need to use a “percent” in this equation? What is the rate of this bag?

IV flow rates (cont’d) If you have a label that reads as follows: Give 3gm magnesium in 100 mls saline over 30 mins PRN How many mL will you use? How many vials will you need? Do you need to use a “percent” in this equation? What is the rate of this bag?

HOMEWORK Prepare for Quiz 3 We will discuss alligations Lecture 4 will be review session for the exam Come prepared with paper, calculator, and your notes

Lecture 4 Quiz 3 Child doses Commercial calculations Questions Review

Alligations These can be used to work on dilutions and compounding The simpler version of the alligation equations is (Old Volume) x (Old %) = (New Volume) x (New %) (OV) X (O%) = (NV) X (N%) ***Any problems on the math exam for alligations will come out of your textbook or the slides***

Alligations (cont’d) OV stands for Old Volume (This is the volume you are starting with.) O% stands for Old% (This is the percent concentration of the Old Volume.) NV stands for New Volume (This is the “new” volume you are making.) N% stands for New% (This is the “new” percent of the NV after the dilution.)

Alligations (cont’d) The percent of a concentrated stock solution will always be larger than the percent of the dilution made from that stock solution? True or False?

Alligations (cont’d) How many milliliters of a 40% stock solution drug would you need to prepare 500 mL of a 6% solution using that drug?

Alligations (con’td) ( ? ) ( 40% ) = ( 500 mL ) ( 6% ) 40X = 3000 How many milliliters of a 40% stock solution drug would you need to prepare 500 mL of a 6% solution using that drug? ( ? ) ( 40% ) = ( 500 mL ) ( 6% ) 40X = 3000 Divide both sides by the number with X to figure out what the ( ? ) number is. 40X/40 = 3000/40 3000 ÷ 40 = 75 ( 75 mL)

Alligations (cont’d) (75mL ) ( 40% ) = ( 500 mL ) ( 6% ) By plugging this number into the equation where the ( ? ) is, you solve the problem. The 75 mL answer means you would measure 75 mL of the 40% stock solution, and add enough water to it to make 500 mL of a 6 % solution.

Alligations (cont’d) How much water will you add to the 75 mL of the 40 % stock solution to make the dilution of 500 mL of 6 % ? The final volume is 500 mL and you are going to use 75 mL of the 40 % stock solution, then 500 mL - 75 mL = 425 mL. Answer: 425 mL of water must be added

Alligations (cont’d) You have 4 fluid ounces of 10 % aluminum acetate solution as a stock solution. How many milliliters of a 2 % solution of aluminum acetate can be prepared from the volume you have on hand? ( 120 mL ) ( 10 % ) = ( ? ) ( 2 % ) 1200/2 = 2X/2 X = 600 ( NV ) = 600 mL

Alligation (cont’d) How much water would you add to the stock solution to prepare the dilution in the last problem? 600 mL – 120 mL = 480 mL

Alligation (cont’d) You have on hand 100 mL of concentrated dextrose injection 50 %. What is the resulting percent of dextrose if you mixed the dextrose injection with 400 mL of water for injection? ( 100 mL ) ( 50 % ) = ( 500 mL ) ( N % ) 5000/5 = 500X/5 X = 10 N% = 10%

Alligation (cont’d) Using 100 mL of concentrated dextrose injection 50%, how much 5% dextrose could be prepared from the dextrose injection? ( 100 mL ) ( 50 % ) = (NV) ( 5% ) 5000/5 = 5X/5 X = 1000 (NV) = 1000 mL

Child doses Child dosing still uses proportions—it just may take and extra step or two. Make sure to be vigilant when preparing infant and child doses, preparing the wrong dose can have devastating effects As technicians, you need to be careful when filling all prescriptions

Child doses (cont’d) Given: Question: Package size is 20mg/2mL Original concentration is 10mg/1mL Question: If you are told to begin with the original volume of 2mL and add 2mL of diluent, what will be your final concentration?

Child doses (cont’d) Given: Answer: Package size is 20mg/2mL Original concentration is 10mg/1mL Answer: Concentration is still 20mg; New volume is 4mL Therefore: 20mg/4mL = 5mg/1mL

Lecture 5 Exam: Pharmacy Math