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CALCULATIONSChapter 6

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Roman Numerals

• Quantities for tablets or capsules are often written in Roman numerals.

• All Roman numerals are written using a combination of 8 letters:

S = 1/2I = 1V = 5X = 10

L = 50C = 100D = 500M = 1000

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Roman Numerals

• Here’s a trick for remembering the order of Roman numeral values greater than 10:

• Little Cows Drink Milk

L = 50C = 100D = 500M = 1000

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Roman Numeral Rules• When a smaller numeral is placed

after a larger numeral (or equal), add all of the numerals together.– Example: XVI = 10 + 5 + 1 = 16

• When a smaller numeral is placed before a larger numeral, subtract the lower value from the higher one.– Example: IV = 5 – 1 = 4

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Roman Numeral Rules (cont.)• If a smaller numeral is placed

between two larger numerals, add the two greater numerals together, then subtract the smaller numeral from the total.– Example: XIV = (10 + 5) – 1 = 14

• Add or subtract any other numeral according to its placement. Example: XCIV = (100 – 10) + (5 – 1) = 94

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Roman Numeral Practice

IV

VIII XXV

XL

LXX

LXXV

LXXX

XC

4

8

25

40

70

75

80

90

==

=

=

=

=

=

=

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FRACTIONS

• Common– Proper, improper, mixed

MixedImproper 4 ½ 9/2

Multiply whole number by denominator of mixed fraction, add to numerator, put that number over denominator

(4 x 2) + 1 = 9

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Fractions

• Adding/Subtracting– Find common denominator– Convert fractions: multiply numerator and denominator

by same number– Add/Subtract numerators

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Fractions

• Addition/Subtraction– 1. Find LCD

• Use multiple of largest denominator that all others will divide into evenly

5 2 18 4 6

+ +

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Fractions

• Addition/Subtraction– 2. Set up ratio: multiply numerators

with same number as denominators

5 2 18 4 6

+ +

5 = num.8 24

8 x ? = 24 ? = 35 x 3 = numerator

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Fractions

• Add/Subt– 3. Add numerators and put over

denominator

1524 24 24 24

+ + = =

5 2 18 4 6

+ +

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Fractions

• Multiplication– Multiply across top – Multiply across bottom– Reduce if possible

• Division– Multiply by reciprocal (invert)– Same rules as multiplication

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MEASUREMENTS

• Metric System: system of weights and measures based on decimal system

Largest to smallest:UNIT SYMBOL

• kilogram kg• gram g• milligram mg• microgram mcg

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Metric System

.

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Kg G Mg Mcg

• When converting…• Move your decimal point the direction you are going in.

• If converting Mcg to Mg: • Move decimal 3 places to the left.

• If you are converting Kg to G:• Move your decimal 3 places to the right.

. . . . . . . . .

Metric System

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Kg G Mg Mcg. . . . . . . . .

Move 3 decimals to the right, replace decimals With 0s to get the answer

1

Amoxil 1 gm = ___ mg Answer is 1000 mg

Metric System

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Kg G Mg Mcg. . . . . . . . .

Amoxil 75 mcg = ___ mg

Move 3 decimals to the left, replace decimals With 0’s to get the answer

Answer is 0.075 mg

75

Metric System

As units get larger, numbers get smaller

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Metric System

• Example:

1. 0 0 0gram to milligram

Larger unit to smaller unit--

decimal moves right 3 places

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Metric System

• Example:

1 0 0 0 .gram to milligram

1 gm = 1000 mg

As units get smaller, number gets larger!

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Metrics

Changing from smaller unit to larger, decimal goes to ?

1.0

1 mg = ? g

As units get larger, numbers get smaller

0.10.010.001

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CONVERSIONS

• Definition: The change of one unit of measure

into another so that both amounts are equal.Unit is the key word

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AVOIRDUPOIS/APOTHECARY SYSTEMS

pound (lb) 16 oz454 g

grain (gr) ~60 or 65 mg

gram (g) ~15 gr

gallon (gal) 4 qt4000 ml

quart (qt) 2 pints

pint (pt) 16 oz473 (~480) ml

ounce (oz) ~30 ml28 (~30) g

teaspoon (tsp) 5 ml

tablespoon (tbsp) 3 tsp15 ml

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Ratios

Ratio: relationship between 2 quantities(same as fraction and percent)

1:1000 = 1/1000 or 0.001 or 0.1%

Proportion: expression of equality of 2 ratios or fractions to each other

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Ratios, Proportions and Equations

• A ratio states a relationship between two quantities

• Ex. 250mg/5ml

• Two equal ratios form a proportion• Ex. 250 mg = 500 mg

5 ml 10 ml

• In a proportion equation, all four terms are related to each other and the relationship of each term can be stated in different ways:

• Ex. 250mg/5ml = 500mg/10ml or 5ml/250mg = 10ml/500mg

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Ratios & Proportions• Therefore, if three of the four terms in a

proportion problem are known, an unknown fourth term (x) can also be calculated.

• Ex. 250 mg = X mg 5 ml 10 ml

• Conditions for using ratios and proportions:

1. Three of the four values must be known2. Numerators must have the same units3. Denominators must have the same units

• Ex. mg/ml = mg/ml

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• Steps for solving proportion equation1. Define the variable and correct ratios2. Set-up the proportion equation3. Cross multiply numerators and denominators4. Solve for X (variable)5. Express solution in correct units

Ratios, Proportions and Equations

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Let’s set up a basic algebra problem. 2 100---------------- = ----------------- 3 x

2x = 300---- ------- 2 2

Ratio and Proportion

100 * 3 = 300 2 * x = 2x

x=150

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mg (we have) mg (needed)----------- = ------------ml (we have) x

250mg 200mg---------- -----------5ml x ml

Divide both sides by 250 to get x by itself.X = 4 ml

Ratio and Proportion

= Cross multiply 250x = 1000

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Equations and Variables• A prescription calls for 200 mg of a

drug that you have in a 10 mg/15 ml concentration. How many ml of the liquid do you need?

Match units and solve for x 10 mg = 200 mg 15 ml x Cross Multiply 15 ml x 200 mg = 3000 ml 3000 ml ÷ 10 mg = 300 mlX = 300 ml

“short-cut”

Do Ratios & Proportions Handout # 1-4

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Augmentin 400mg/5ml mg (we have) mg (needed)Sig: 500mg po tid x 10 days ----------- = ------------ ml (we have) x400mg 500mg---------- ----------- 5ml x

Cross multiply 400x = 2500 ------ ------ Divide both sides by 400 to get 400 400 x by itself. X = 6.25 ml

6.25ml * 3 (daily dose ) * 10 days = 187.5 or 188 ml needed

Augmentin 400mg/5ml comes in bottles of 75ml, 100mland 150ml. Which size would you give?

2 bottles of 100

Ratio and Proportion

Reminder:

Keep mg with mg, ml with ml.

=

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Ratios in Reconstituted Drugs

• Amoxil 250mg/5ml Rx: Give 300 mg po tid

for 10 days

– How many ml per dose?– How many ml total needed?

For more help in solving ratio/proportion problems, read pages 110-117 in green textbook, 120-127 in purple book.

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0011 0010 1010 1101 0001 0100 1011When the directions are written in teaspoons, convert the teaspoons to milliliters before performing Days Supply calculations.

Days Supply: Teaspoons

½ teaspoon = 2.5ML

1 teaspoon = 5ML

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Days Supply: Teaspoons

_______________ Days Supply=Quantity Dispensed

Daily Dose

_______________75ML dispensed

7.5ML per day

10 Days Supply=

Example: Give ½ tsp po tid

75ML

Convert teaspoons to milliliters: ½ teaspoon = 2.5ML per dose

Calculate daily dose: 2.5ML per dose x 3 doses per day = 7.5ML per day

Calculate day supply:

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_______________ Days Supply=Quantity Dispensed

Daily Dose

_______________100ML dispensed

12.5ML per day

8 Days Supply=

Example: Amoxicillin Suspension 100MG/5ML, 100ML

Give 1 & ¼ tsp po bid

Convert teaspoons to milliliters: 1 ¼ teaspoon = 6.25 ML per dose

Calculate daily dose: 6.25 ml per dose x 2 doses per day = 12.5 ML per day

Now do Ratios & Proportions 2nd page # 1-4

Days Supply: Teaspoons

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Back to Conversions

• Problem: Dr. orders ¼ gr Codeine. How many milligrams is this equivalent to?

Let’s set up our conversion equation!

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Other Conversions

Temperature: Converting between Celsius and Fahrenheit

9C=5F-160[9 x C = (5 x F) - 160]

0°C freezing pt water100°C boiling pt water

vs.~30-220°F scale

Temperature Conversions

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Temperature Conversions

So how many degrees C is 40 degrees F?

9C = 5F - 160

Some people prefer other 2 formulas on page 106 in green, 116 in purple textbook.

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Dosing

• BSA (body surface area)- black bk.p.54

– Usual dosage in m²• 1 Place a dot on patient’s weight on

vertical line to the right• 2 Place a dot on patient’s height on

vertical line to the left• 3 Connect dots by drawing a line with a

straight edge• 4 Read the BSA located on the center

vertical line at the intersections

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BSA

• So, say you figured the child’s body surface area to be 0.63 m², and the doctor gives you a dosage 50mg/m² ?

• Just set up a ratio and cross-multiply:50mg = x1m² 0.63m² x = 31.5mg

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0011 0010 1010 1101 0001 0100 1011MilliEquivalents mEqMilliEquivalents mEq• Measurement for electrolytes

(substances that conduct electrical currents, found in blood, tissue & cells) in a soln.

• Examples:– Electrolyte: salt (NaCl), KCl– Electrolyte solution: saline soln

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Each measurement is specific, based on electrolyte’s atomic wt & valence (# ion charges)

mEq or mEq ml L

In order to calculate the mEq for an electrolyte, the atomic weight and valence of the electrolyte must be known. The weight is then divided by the valence.

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Problem:A solution calls for 5 mEq of sodium

that you have in a 1.04 mEq/ml solution of NaCl. How many ml of it do you need?

1.04 mEq = 5mEq 1 ml x ml

X = 4.8 ml

MilliEquivalents mEqMilliEquivalents mEq

Do mEq handout from folder. For extra help, see pages 121-125 in green, 131-135 in purple textbook.

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Kg to Pound Conversion

Weight (lb)/2.2 lb = kg

How many kg would a 44 lb child weigh?

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Calculating mg/kg/day

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Weight (kg) x (mg/kg) = dose (mg)16.8 kg x 2 mg/kg = 33.6 mg This means that the patient has to take 33.6 mg daily. *Note* - In 3 doses (TID), not at once, so 33.6 mg divided by 3 = 11.2 mg

How many ml the patient should take?

How to set it up with ratios (crossed multiplication) :

Orapred’s strength is 15mg, 5ml

we need 11.2 mg

15mg = 11.2mg 5ml X

Solve for X 3.73 ml = X

The sig reads: 2 mg/kg/day TID for 4 days. The patient’s weight is 16.8 kg

Remember! Sometimes the patient’s weight could be reported in pounds (lbs). 1 kg=2.2 lbs. Do not forget the measurement conversion.Weight (lb)/2.2 lb = kg

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Children/Infant Dosing

Dose/kg

See example-Infant dose, p 127 green, 143 purple book

15mg/kg bid, baby weighs 18 lbs.How many mg per dose?

P. 137 green, 151 purple # 6

Weight (kg) x (mg/kg) = dose (mg)

Weight (lb)/2.2 lb = kg

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CALCULATIONS: mg/kg/day

Problem: p.139 green,152 purple #18

RX for cefadroxil, 30mg/kg once a day for 14 days. How many mg does child need for full course of medication? The child weighs 44 lbs.

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CALCULATIONS

Problem:Given that you need 8400 mg total of cefadroxil for 14 days, what is the smallest bottle that will provide enough medication?

A. 50 ml bottle of 125 mg/5 mlB. 50 ml bottle of 250 mg/5 ml

C. 75 ml bottle of 500 mg/5 mlD. 100 ml bottle of 500 mg/5 ml

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Solution

Now use cross multiplication to figure bottle size and dosing required.

A. 50 ml, 125 mg/5 ml

125 mg = x mg 125 x 50 = 5 x 5 ml 50 mlx = 1250 mg, not enough

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Solution continued

B. 50 ml, 250 mg/5 ml

250 mg = x mg 5 ml 50 ml

x = 2500 mg, not enough

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Solution continued

C. 75 ml, 500 mg/5 ml

500 mg = x mg 5 ml 75 ml

x = 7500 mg, not enough

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Solution continued

D. 100 ml, 500 mg/5 ml

500 mg = x mg 5 ml 100 ml

x = 10,000 mg, enough to cover 8400 mg needed

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Children’s Dosing

• Clark’s RuleChild’s weight x Adult dose = Child’s dose 150 lb

• Young’s RuleAge of child Average adult Child’s

Age of child + 12 dose dosex =

See Children’s Dosages Worksheet #2

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Flow Chart

Multiply following fractions: 7/3 x 2/7 x 3/4You can make a flow chart:

-Each row is multiplied across. The numerator is divided by the denominator. So all numerators are divided by all denominators (any order). -Any number divided by itself is 1. So any number in a numerator box and be divided by the same number in a denominator box and “cancel out”.

4284=

7 2 3

3 7 4 = 0.5

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Flow Chart

• You can do the same thing with letters (or units).

– Problem gives you:

– Asks for:

C D A D B C

AB

AB=

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Flow Chart• You can manipulate the letters or units!

– Put what answer you need at end and move everything else around to be left with only what you need. You want to be left with A in the numerator and B in the denominator.

– FYI: you can invert a fraction!– Hint: Start with a fraction that has part of the answer already!

– Problem gives you:– Asks for:

D D A C B CA

B

AB=

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Flow Chart• Make flow chart with desired units at

end. Enter information so units (other than desired) cancel out.

– Problem gives you:

– Asks for:

gram ml drop min gram ml

dropsmin

dropsmin

=

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FLOW RATES-IV Rates

• IV is ordered: – ml/hr– gtts/min– volume/time period

• IV tubing varies from 10 to 60 gtts/ml. This size will be provided.

See handout “Flow Rates”

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IV Rate

• Over or Per means divided by:– Give D5W 1000 ml over 10 hours is:

1000 ml10 hrs

• Make flow chart with desired units at end. Enter information so units (other than desired) cancel out.

Remember, this can also be written: 10 hrs

1000 ml

Given Convert Convert Desired

Dosage Vol. units Time units Units

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Flow Rates

• A solution is to be administered by IV infusion at a rate of 140 ml/hr. How many gtts/min should be infused if 1 ml = 25 gtts?

25 gtts

140 ml

1 hour

1 ml 1 hour 60 min

• 25 x 140 x 1 = 3500 1 x 1 x 60 = 60

3500 ÷ 60 = 58.3 gtts/min

gtts

min

=

Now let’s do more from Flow Rate handouts!

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CALCULATIONS

Problem:RX for cefadroxil, 30mg/kg once a day for 14 days. How many mg does child need for full course of medication? The child weighs 44 lbs.

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create flow chart where units cancel out :

30 mg x kg x 44 lb = mg dailykg 2.2 lb

600 mg needed daily x 14 days = 8400 mg total needed

Solution:

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Reducing/Enlarging Formulas

• IV Concentration Conversions– Hyperalimentation/TPN (Total

Parenteral Nutrition)-all nutrients administered by IV

Dextrose 50% soln 357 ml

KCl 10 ml

Sterile Water q.s. ad. 500 ml

q.s. ad. means sufficient quantity to make add up to

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CompoundingBenzyl benzoate 250 ml

Triethanolamine 5 ml

Oleic acid 20 ml

Purified water, to make

1000 ml

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Percents & Solutions• Percent is concentration of soln

– Weight to volume: X g/100ml– Volume to volume: X ml/100ml– Weight to weight: X g/100g

X = %• So, when there is a percentage, it means

there are that many grams per 100ml.– Ex. 50% dextrose = 50g/100ml

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• Example: If there is 50% dextrose in a 1000ml IV bag, how many grams of dextrose are there in the bag?

Percents & Solutions

50g 100ml

= Xg 1000ml

This means that there is 50 grams of dextrose in 100ml liquid and 500 grams in 1000ml of same liquid. It is a 50% strength solution.

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Percents & Solutions• Percent is concentration of soln

– Weight to volume: Xg/100ml

• Green Book p. 119, Purple p. 129 #7-12• Now do problem #13

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Percents & Solutions• Percent is concentration of soln

– Weight to volume: Xg/100ml

• Package says Dextrose 50% (0.5g/ml). Does that make sense?

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Percent Solution Formula

Final volume or weight x % strength = gm or ml of substance to be dissolved

FV x % = gm or ml100ml x 0.5 = 50g

FYI--This is same equation in different format.

50% = 50g 100ml

Percent strength

Substance to be dissolved

Final Volume

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Percent Solution Formula

Final volume or weight x % strength = gm or ml of substance to be dissolved (FV x % = gm or ml)

See Solutions worksheet #1-4

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Percent Solution Formula

See Solutions worksheet #5-11

(OV) (O%) = (NV) (N%)Black book chapter 9, p.122

Same FV = IS As: IV FS

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How do we know which formula to use??

• Figure out total number of strengths (percents) involved in question. (Including answer—if they give you 2 strengths and ask for another, that’s 3 total).– One strength: use FV x % = gm or ml– Two strengths: use FV/IV = IS/FS– Three strengths: use alligations

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Alligation-Alternate

• Method for calculating the amount of 2 different strength drugs needed to make a third, final strength compounded mixture.

• FYI-water is 0%• See Handout.

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Alligation-Alternate1. Make a tic-tac-toe board.2. Enter stronger solution in upper left hand

corner.3. Enter weaker solution in lower left hand

corner.4. Enter desired strength in middle box.5. Let x equal the upper right hand corner-this

will be the difference between the lower left corner and the center square.

6. Let y equal the lower right corner-this is the difference between the upper left corner and center square.

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Alligation-Alternate

Let’s take a look at the example in the next slide before

we have a nervous breakdown!

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Highest60%

Middle 40%

Lowest25%

Subtract 40 -25 15

Subtract 60 – 40 20

How many ml of 25% dextrose are needed to prepare 500 ml of 40% dextrose if

you are to prepare 40% dextrose from 25% dextrose and 60% dextrose?

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Alligation Alternate• X represents parts of higher

strength drug• Y represents parts of lower

strength• Take x and y:

1.The portion of higher strength needed will be equal to x/(x + y).

2.The portion of the lower strength needed will be = y/(x + y).

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• How many ml of 25% dextrose are needed to prepare 500 ml of 40% dextrose if you are to prepare 40% dextrose from 25% dextrose and 60% dextrose?

Highest60%

15=x x x+y

Middle 40%

Lowest25%

20=y yx+y

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Alligation-Alternate3. The resulting fractions or ratios are

multiplied by final volume desired to get the actual amount of each drug to be used.

The top volume represents the amount of the higher strength drug to be used.

The bottom volume represents the amount of the lower strength drug to be used.

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• How many ml of 25% dextrose are needed to prepare 500 ml of 40% dextrose if you are to prepare 40% dextrose from 25% dextrose and 60% dextrose?

Highest60%

15=x15 x 500ml35

Middle 40%

Lowest25%

20=y20 X 500ml35

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Alligation Medial

Volume A x Percent A= Total AVolume B x Percent B= Total BTotal Volume Total A + B

Total A + B = New %Total Volume See

bottom page 133 in black

book

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DEA

• Prescriber’s DEA must be on controlled prescriptions– 2 letters + 7 numbers– Formula:Add sum of first, third and fifth digits to twice

the sum of the second, fourth and sixth digits; the total should be a number whose last digit is same as last digit of DEA number.

See example page 41 green, 45 purple book/Handout

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MATH REVIEW

• Pages 134-141 in green textbook.• Pages 150-154 in purple book.• Pages 202-212 in white book.• Pages 167-210 in black book.• Math Review Cumulative Test.

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The End

Thank You!