Pharmaceutical Calculation

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PHARMACEUTICAL CALCULATION

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PHARMACEUTICAL CALCULATION Introduction

Basic arithmetic is always involved in some manner in the solving of pharmaceutical calculations. Perfecting basic mathematical functions will help to attain the goal of 100 percent accuracy desired in pharmacy. Because of the need for 100 %, no partial credit will be given for setting a problem up correctly if the answer is incorrect. In pharmacy, the correct answer is more important than the method.

COMMON FRACTIONS, DECIMAL FRACTIONS AND PERCENT

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A Half can be written...

As a fraction: 1/2

As a decimal: 0.5

As a percentage: 50%

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Percent Decimal Fraction

1% 0.01 1/100

5% 0.05 1/20

10% 0.1 1/10

12½% 0.125 1/8

20% 0.2 1/5

25% 0.25 1/4

331/3% 0.333... 1/3

50% 0.5 1/2

75% 0.75 3/4

80% 0.8 4/5

90% 0.9 9/10

99% 0.99 99/100

100% 1  

125% 1.25 5/4

150% 1.5 3/2 5

CONVERSION

From Percent To Decimal  move the decimal point 2 places to the left, and remove the "%" sign.

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From Percent to Decimal

To convert from percent to decimal: divide by 100, and remove the "%" sign.The easiest way to divide by 100 is to move the decimal point 2 places to the left. So:                                                                                  

From Decimal to PercentTo convert from decimal to percent: multiply by 100, and add a "%" sign.The easiest way to multiply by 100 is to move the decimal point 2 places to the right.

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From Decimal To Percent  move the decimal point 2 places to the right, and add the "%" sign.

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From Fraction to Decimal

The easiest way to convert a fraction to a decimal is to divide the top number by the bottom number (divide the numerator by the denominator in mathematical language)

Example: Convert 2/5 to a decimalDivide 2 by 5: 2 ÷ 5 = 0.4

Answer: 2/5 = 0.4

Steps ExampleFirst, write down the decimal "over" the number 1

0.75 / 1

   Then multiply top and bottom by 10 for every number after the decimal point (10 for 1 number, 100 for 2 numbers, etc)

0.75 × 100 / 1 × 100

(This makes it a correctly formed fraction) = 75 / 100

Then Simplify the fraction 3 / 4

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From Decimal to FractionTo convert a decimal to a fraction needs a little more work.

Example: To convert 0.75 to a fraction 

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From Fraction to Percentage

The easiest way to convert a fraction to a percentage is to divide the top number by the bottom number. then multiply the result by 100, and add the "%" sign.

Example: Convert 3/8 to a percentageFirst divide 3 by 8: 3 ÷ 8 = 0.375,Then multiply by 100: 0.375 x 100 = 37.5Add the "%" sign: 37.5%

Answer: 3/8 = 37.5%

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Steps ExampleConvert 80% to a decimal (=80/100): 0.8

Write down the decimal "over" the number 1 0.8 / 1

   Then multiply top and bottom by 10 for every number after the decimal point (10 for 1 number, 100 for 2 numbers, etc)

0.8 × 10 / 1 × 10

(This makes it a correctly formed fraction) = 8 / 10

Then Simplify the fraction 4 / 5

From Percentage to FractionTo convert a percentage to a fraction, first convert to a decimal (divide by 100), then use the steps for converting decimal to fractions (like above).Example: To convert 80% to a fraction

EXPONENTIAL NOTATION

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There are 602,000,000,000,000,000,000,000 molecules in 18 grams of water. A shorter way of writing the same number is by using exponential notation to show all those zeros as a number to the power of ten:  

6.02 x 1023 is the shorter way of representing all those molecules. Such a number can be read "Six point zero two times ten to the twenty third."

A small number such as 0.0000000057 can be written as 5.7 x 10-9.  Such a number can be read "Five point seven times ten to the minus nine."

•To type 104, one way of doing it is to type ten: •Then use the xy button  and type 4 •The result should be ten thousand: 

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To remember which way to move (left or right), remember that negative exponents (like -4 in the example) represent very small numbers - numbers that are less than one.  As a result, they will always start with zero. 

Positive exponents (like 4 in the example), give numbers bigger than one. On scientific calculators, it is sometimes a bit of a trick to type these numbers in.                                               

•To type 10-4, type a one* and then hit the x 10x button:   Can't find this button on your calculator?  See the next box.•You should get this: •Now type a four •This number means 1 x 104 which is the same as 104.•If your calculator works like this, you can skip the next box.*Why did we type 1 and not 10?  •Let's see what happens when we try ten. •If you press =, you get •That's because the first screen (10.04) means 10 x 104. This number is the same a writing 105 which is 100000. 

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But there is a quicker way, using a special button which has been designed especially for exponential notation.                                                                                                 

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•Other calculators have "EXP" or "EE" buttons on them for scientific notation.  Certain calculators have a button which can be pushed directly, like the one on the right, others have this function available as a two-key operation. •On some calculators, you have to push the 2nd command (or Shift button) first.  Here is an example:                         •On these the expression "x 10x" is replaced by the letter "E".  •Let's take an example such as 6.23 x 1023 times 4.11.  •Type in 6.23 then the EE button, then 23.  After that, type times 4.11 and Enter.  Here is what it would look like on the calculator's screen: •Notice how the expression "x 10" has been replaced by the letter "E".  This is to save space on the small screen and to avoid confusion with multiplication operations.  

•The letter "E" is placed there to remind us that this number is written in "Exponential notation".

For a calculator, 6.23E23 equals  6.23 x 1023.•The answer above, 2.56E24 would be written on your paper 2.56 x 1024. 

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•Let's stay with calculators which have only one line on the screen which do not put the letter "E" at all but rather put the exponent up in the right hand corner of the screen in a smaller format.  A number such as 5.98 x 10-6 looks like this: •How do you type in numbers like this?  Let's try 0.0000371, which is written in scientific notation as 3.71 x 10-5. •First, type in three point seven one:                                                                                                                                                                                                                                    •Next, push the x 10x button to get into the scientific notation mode:                                                                                                                                                                                                                                   •To make the exponent negative, push the +/- button:  which on some calculators is a negative sign in parentheses: (-).•Important:  do NOT use the minus sign, this is a totally different operation!                                                                                                                              

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                                                                             •Type in five to get the final number:                                                                                                                                                                                                                                  •This is how your calculator says 3.71 x 10-5. If you get an answer like this from a calculation, be sure you write it the way humans do (3.71 x 10-5) and not the way your calculator does (3.71-05 or 3.71E-05).  Written on a piece of paper, the number 3.71-05 is equivalent to 0.00142 and not what we want, 0.0000371.  It's obvious that these are completely different numbers!

•A more simple example is that 5 x 102 is not the same as 52.  The first equals 500 and the second equals 25.

RATIO AND PROPORTION, VARIATIONS AND DIMENSIONAL ANALYSIS

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TEMPERATURE CONVERSIONS

FORMULAThe following formula works for converting both ways; that is, conversions can be made from Fahrenheit to Centigrade or from Centigrade to Fahrenheit using this formula:

5F = 9C + 160

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e.g :Convert 140ºF to ºC: 5F = 9C + 160º (formula) 5(140º) = 9C + 160º 700º = 9C + 160º

700º - 160º = 9C 540º = 9C

60º = C

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RATIO AND PROPORTIONDEFINITIONSa. Ratio: A ratio is the relationship of two

quantities. A ratio may be expressed as a ratio (1:8, 1:200, etc.) or as a fraction (1/8, 1/200, and so forth.).

b. Proportion: A proportion is the equality of two ratios.

For example: 1 = 32 6

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RATIO AND PROPORTIONFor example: If one aspirin tablet contains five

grams of aspirin, then how many grams would be required to make 40 tablets?

IF 1 tablet contains 5 grams of aspirin

Then 40 tablet contains X grams of aspirin

So X = 5 grams x 40 tablets = 200 grams

1 tablet

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RATIO AND PROPORTION

Another example:How many milliliters of tetracycline suspension, which contains 250 mg of tetracycline in each five milliliters, must be administered to give a patient 150 mg?

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The proportion should be as follows:

250 mg = 150 mg5 milliliters X milliliters 250 (X) = 5 (150)

250X = 750 X = 750

250 X = 3 ml

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Example problem: How many grams of potassium permanganate (KMnO4) are needed to make this prescription?

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Before calculation, the prescription must be analyzed.-the pharmacist should dispense a solution of potassium permanganate in distilled water. -the solution should have a strength equivalent to 1 gram per 5000 milliliters - and the pharmacist should dispense 120 milliliters of the solution.

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The expression of strength will be thefirst ratio of the proportion:

1 gram = _______________5000 ml

Assign the X value:1 gram = X gram5000 ml

Find the other known factor:1 gram = X gram5000 ml 120 ml

Then, cross-multiply:5000 (x) = 1 (120) 5000 x = 120

Solve for X: X = 120

5000 X = 0.024

Refer to the proportion for the units of X: X = 0.024 gram #

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Practical Exercises. Solve the following problems and check your answers.

(1) If 50 tablets contain 0.625 gram of an active ingredient, how many tablets can be prepared from 31.25 grams of the ingredient?

(2) If 50 tablets contain 1.5 gram of active ingredient, how much of the ingredient will 1375 tablets contain?

(3) If 3 doses of a liquid preparation contain 7.5 gram of a substance, how many doses will be needed to give a patient 80 grams of the substance?

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answers

(1) 2500 tablets(2) 41.25 gm(3) 32 doses

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REDUCTION AND ENLARGEMENT OF FORMULAS

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REDUCTION AND ENLARGEMENT OF FORMULAS

INTRODUCTIONMost of the preparations made in a pharmacy are from proven formulas that have been tested and are listed in the Pharmacopeia/National Formulary as official formulas. These formulas list the amount of each ingredient needed to make a certain amount of the preparation. It is necessary to reduce or enlarge a formula to satisfy the needs of your pharmacy.

1-42.

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RATIO AND PROPORTION METHODa. The formula:

IF Amount of each ingredient THEN *Amount of each in the official formula = Ingredient needed

Total quantity of the Total quantity official formula desired

*NOTE: Most of the time, the unknown factor will be the “Amount of each ingredient needed.”

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Sample Problem:

Calculate the amount of each ingredient needed to make 240 ml of Peppermint Spirit.

Peppermint Spirit

Peppermint Oil............ 100 mlPeppermint Powder.........10 gAlcohol.....qsad.......... 1000 ml

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Copying Formula

e.g. 1Peppermint Spirit For

1000 mlFor 240 ml

Rx

Peppermint OilPeppermint PowderAlcohol.....qsad

100 ml 10 g1000 ml

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(1) Solve first for the amount of peppermint oil needed:IF THEN

100 ml Peppermint oil = X ml peppermint oil 1000 ml spirit 240 ml spirit

(2) Cross multiply: (1000) X = 100 (240) 1000 X = 24000X = 24 ml of peppermint oil

(3) To solve for the amount of peppermint powder needed:

IF THEN10 g peppermint powder = X g peppermint powder 1000 ml of spirit 240 ml of spirit

(4) Cross multiply: (1000)X = 10 (240) 1000 X = 2400 X = 2.4 g of Peppermint powder

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Reduction of formula

e.g 2Calculate the amount of each ingredients needed to produce 100 ml of Codeine Linctus B.P.C.

Refer to formula in B.P.C, then copy; note that in BPC, the final volume is

1000 ml; - reduction of formula.

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Rx 1000 ml For 100 ml

Codeine phosphate 3 g

Tartrazine compound solution

10 ml

Benzoic acid 20 ml

Chloroform spirit 20 ml

Distilled water 20 ml

Lemon syrup 200 ml

Syrup B.P. to 1000 ml

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Rx 1000 ml For 100 ml

Codeine phosphate 3 g 0.3 g

Tartrazine compound solution

10 ml 1.0 ml

Benzoic acid 20 ml 2.0 ml

Chloroform spirit 20 ml 2.0 ml

Distilled water 20 ml 2.0 ml

Lemon syrup 200 ml 20.0 ml

Syrup B.P. to 1000 ml 100.0 ml

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Note that Codeine phosphate is in powder form; - First dissolve codeine phosphate in water Then add in other ingredients Finally add syrup B.P to 100 ml (final

volume)

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Enlargement of formulas

Rx For 250 ml

Light Kaolin 2 g

Light Magnesium Trisilicate 0.5 g

Sodium bicarbonate 0.5 g

Peppermint emulsion concentrate

0.25 ml

Chloroform double strength 5 ml

Distilled water to 10 ml

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Enlargement of formulas

Rx Untuk 250 ml

Light Kaolin 2 g 50 g

Light Magnesium Trisilicate 0.5 g 12.5 g

Sodium bicarbonate 05 g 12.5 g

Peppermint emulsion concentrate

0.25 ml 6.25 ml

Chloroform double strength 5 ml 125 ml

Distilled water to 10 ml to 250 ml

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Calculation of Concentration in Percentage

Term percent (%)refers to ‘by the hundreds’ or ‘in a hundreds’ Also can be expressed in ratio

50% means 50 parts in 100. In pharmaceutical; 3 types of

percentage Weight-in-volume (w/v) Volume-in-volume (v/v) Weight-in-weight (w/w)

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Definition of Percentage

According to USP

% w/v the number of grams of a constituent in 100 ml of solution/liquid preparation.

% v/v the number of millimeters of a constituent in 100 mL of solution/liquid preparation.

% w/w the number of grams of a constituent in 100 g of solution or preparation.

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Definition of Percentage

If a concentration of a preparation only be written as %

(without w/v or v/v or w/w)- the percentage should be considered as % w/v

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Example of percentage

1 % 1 part in 100 ml of solution- 1 grams of solute in sufficient amount

of solvent to produce 100 ml preparation.

* note that 1 % is NOT 1 gram of solute in 100 ml of solvent.

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Calculation of percentage

1:400 1 x 100 % = 0.25 % 400

1:1000 1 x 100 % = 0.1 % 1000

2:2500 ?4:10,000 ?

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Calculation of percentage w/v

Calculate the amount of Sodium Chloride needed to produce 1.5 L of Sodium Chloride 0.9% Solution.

Answer:0.9% - 0.9 g of NaCl in 100 ml of solutionIn 100 ml; contains 0.9 g NaClIn 1500 ml; contains ? g NaCl

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Therefore:In 1500 ml 0.9 g x 1500 ml

100 = 13.5 g

Amount Sodium Chloride needed is 13.5 g- 13.5 g NaCl to be completely dissolved in

a small part of water, then add the remaining water up to final volume (1500 ml)

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w/v practice problemsHow many grams of potassium permanganate should be used in compounding the following prescription?

Rx

Potassium permanganate 0.02 %

Purified water ad 250.0 ml

Sig. As directed

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Answer: 0.05 g

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w/v practice problems

How many grams of dextrose are required to prepare 4200 ml of 5% solution?

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Answer: 210 g

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Calculation of percentage v/v

How many milliliters of liquefied phenol should be used in compounding the following prescription?

Rx

Liquefied Phenol 2.5 %

Calamine lotion ad 240.0 ml

Sig. For external use.

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AnswerVolume (ml) x strength = milliliters of

active ingredients240 ml x 2.5 ml = 6 ml 100 ml

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e.g: 4 ml of an active drug are used to prepare 250

ml of a certain lotion. What was the % v/v of the active drug in the lotion?

Answer: 250 ml = 100 % 4 ml X %

X = 100 % x 4 ml = 1.6 % 250 ml

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Practice problem (% v/v)

Peppermint spirit contains 10 % v/v of peppermint oil. What volume of the spirit will contain 75 ml peppermint spirit?

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Answer: 750 ml

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Percentage Weight in Weight

%(w/w) - the number of grams of a constituent in 100 g of

solution or preparation.

RxSolute 1 gSovent to produce 100 g

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Calculation of Percentage w/w

e.g 1How many grams of a drug should be used to

prepare 240 g of a 5 % (w/w) solution in water?

Answer:Weight of solution (g) x % = g of solute 240 g x 5 g = 12 g 100 g

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Calculation of Percentage w/w

e.g: 2If 1.5 kg of a solution contain 75 g of a drug

substance, what is the percentage strength (w/w) of the solution?

Answer: 1.5 kg 1500 g 1500 g = 100 % ; x = 5 % (w/w) 75 g x %