UNZA Pharmacy Training Lecture notes

2. Outline of the Topics
 General Introduction to the concepts of pharmaceutical numeracy
 Relationships of Weights and Measures
 Mathematical Concepts for Pharmaceutical Calculations
 Pharmaceutical Dosage Units for Calculations
 Commonly Used quantification Systems
 Pharmaceutical Calculations requirements

3. General Introduction to the concepts of
pharmaceutical numeracy
weights and measures
 Weight is a measure or the gravitational force acting on a body;
weight is directly proportional to the body's mass - The latter, being a
constant based on inertia, never varies, whereas weight varies slightly
with latitude, altitude, temperature, and pressure.
 Measure is the determination of the volume or extent of a body -
Temperature and pressure have a pronounced effect, especially on
gases or liquids.

4. • Fractional and Multiple prefixes - In many experimental procedures,
including some in the pharmaceutical sciences, very small (and occasionally
very large) quantities of weight, length, volume, time, or radioactivity are
measured.
• To avoid the use of numbers with many zeros in such cases, the USA-NIST
recognizes prefixes to be used to express fractions or multiples of the
International System of Units (SI), which was established in 1960 by the
General Conference on Weights and Measures (see the foregoing
discussion).
Weights and measures (Continued)

16. Weights and measures (Continued)

17. Weights and measures (Continued)

18. Any Questions or Additions

19. The Relationships of Weights and Measures
(Practical Equivalents)
• Tables of weights and measures and a table of practical equivalents should
be kept in a conspicuous and convenient place in the prescription
department, and the following equivalents, which are given with practical
accuracy, should be committed to memory.
• Other equivalents may be calculated from these:

20. Approximate Measures
• In apportioning doses for a patient, the practitioner usually is compelled to
order the liquid medicine to be administered in certain quantities that have
been established by custom, and can be estimated as:
• Quantities have been practically approximated using the following calibrated
devices:

21. Density and Specific Gravity
(Specific Gravity)
• Several terms are used to express the mass (weight) of equal volumes of different
substances.
• Absolute density is the ratio of the mass of an object, determined in or referred to a
vacuum, at a specified temperature, to the volume of the object at the same temperature -
this relationship is expressed mathematically as:
• Apparent density differs from absolute density only in that the mass of the object is
determined in air; the mass is influenced by the difference in the buoyant effect of air on the
object being weighed, and on the standard masses (weights) used for comparison.
• Relative density is an expression sometimes employed to indicate the mass of 1 mL (not cc,
which is very slightly different) of a standard substance, such as water, at a specified
temperature, relative to water at 4°C taken as unity. Thus, at 4°C the relative density of water
is 1.0000, whereas its absolute density at the same temperature is 0.999973
• Specific gravity may be defined as the ratio of the mass of a substance to the mass of an
equal volume of another substance taken as the standard. For gases, the standard may be
hydrogen or air; for liquids and solids, it is water..

22. • Density is defined as the mass of a substance per unit volume - It has
the units of mass over volume.
• Specific gravity is the ratio of the weight of a substance in air to that of
an equal volume of water. In the metric system both density and specific
gravity may be numerically equal, although the density figure has units.
• The equations for calculating density, weight, and volume are:
Density and Specific Gravity
(Density)

23. Pharmaceutical Calculations
(Mathematical Principles)
• A few mathematical principles (e.g., common decimal fractions, exponents,
powers and roots, significant figures, and logarithms) will be reviewed, as
these are areas where students often become careless or have forgotten skills.
• Significant figures are digits that have practical meaning – In some
instances zeros are significant; in other instances they merely indicate the order
of magnitude of the other digits by locating the decimal point.
• For example, in the measurement 473 mL all the digits are significant, but in the
measurement 4730 mL the zero may not be significant.
• In the weight 0.0316 g the zeros are not significant but only locate the decimal
point.
• In any result the last significant figure is only approximate, but all preceding
figures are accurate.
• The Rules to these principle figures - When adding or subtracting, retain the sum or
remainder no more decimal places than the least number entering into the calculations -
For e:<ample.

24. Fractions (Common Fractions)
• An example of a common fraction is 3/8 – this is read as "three eighths and
indicates three parts divided by eight parts of the same thing, meaning the units
with both numbers must be the same measure.
• The following principles should be applied when using common fractions:
1. The value of a fraction is not altered by multiplying or dividing both numerator
and denominator by the same number.
2. Multiplying the numerator or dividing the denominator by a number, multiplies
the fraction by that number,
3. Dividing the numerator or multiplying the denominator by a number divided the
fraction by that number.
4. To add or subtract fractions, form fractions with the lowest common
denominator perform the arithmetical operation, and reduce to the lowest
common denominator.
5. To multiply fractions. multiply all numbers above the line to form the new
numerator and multiply all numbers below the line to form the new
denominator - Cancel if possible to simplify and reduce to the lowest common
denominator.
6. To divide by a fraction, multiply by the reciprocal of the fraction.

25. Fractions (Decimal Fractions)
• Fractions with the power of 10 as the denominator are known as decimal fractions and
are written by omitting the denominator and inserting a decimal point in the numerator
as many places from the last number on the right as there are ciphers of 10 in the
denominator.
• The following principles should be applied when using decimal fractions:
1. When adding or subtracting decimals, align the decimal points under each other.
2. When multiplying decimals, proceed as with whole numbers, then place the
decimal point in the product as many places from the first number on the right as
the sum of the decimal places in the multiplier and the multiplicand.
3. When dividing by a decimal fraction. move the decimal point to the right, in both
divisor and dividend, as many places as it is to the left in the divisor to form a
whole number in the divisor; proceed as with whole numbers, The decimal point in
the quotient should be placed immediately above the decimal point in the dividend.
4. When converting a common fraction into a decimal fraction, divide the numerator
by the denominator and place the decimal point in the correct place,
5. When converting decimal fraction into common fraction, place the entire number.
as the numerator, over the power of 10 containing the same number of ciphers of
10 as there are decimal places. Cancel, if possible, to simplify.

26. Exponents, Powers, and Roots
• In the expression 24 = 16, the following names are given to the terms: 16 is
called the power of the base 2 and 4 is the exponent of the power –
however, if the exponent is 1, it usually is omitted.
• The following laws should be recalled:
1. The product of two or more powers of the same base is equal to that base with an exponent
equal to the sum of the exponents of the powers; e.g. 25 x 23 = 28
2. The quotient of two powers of the same base is equal to that base with an exponent equal to the
exponent of the dividend minus the exponent of the divisor; e.g. 28 ÷ 23 = 25
3. The power of a power is found by multiplying the exponents; e.g. (28)3 = 224
4. The power of a product equals the product of the powers of the factors;
e.g. (2 x 3 x 4)2 = 22 x 32 x 42
5. The power of a fraction equals the power of the numerator divided by the power of the
denominator; e.g.

27. Exponents, Powers, and Roots
(continued)
• The root of a power is found by dividing the exponent of the power by the index of
the root; e.g.
• Any number other than 0 with an exponent 0 equals 1; e.g., 20= 1. A number with a
negative exponent equals one divided by the number with a positive exponent
equal in numerical value to the negative exponent; for example,

28. Logarithms
• Logarithms (logs) were invented to facilitate the solution or involved and
lengthy problems.
• Many calculations that are difficult by ordinary arithmetical processes are
performed rapidly and easily with the aid or logs; the advent of modern
calculators and computer spreadsheet programs has made this use or logs
obsolete.
• Logs still appear, however, in many chemical and pharmacokinetic
equations.
Y = ax
log
a
Y = x

29. Any Questions or Additions

30. Review of Mathematical Concepts
for Pharmaceutical Calculations
1. Proper, Improper Fractions and Mixed Numbers
Proper fraction: numerator is less than denominator; value is less than 1.
Example: 1/2
Improper fraction: numerator is greater than denominator; value is greater than 1.
Example: 4/3.
Or numerator denominator; value 1.
Example: 5/5
Mixed number: whole number a fraction; value is greater than 1.
Example: 1(1/2)
Complex fraction: numerator and/or denominator are composed of a fraction, decimal,
or mixed number; value is less than, greater than, or = 1.
Example: (1/2) / (1/50)
Any nonzero number divided by itself 1.
Example: (3/3) = 1
To reduce a fraction to lowest terms, divide both terms by the largest nonzero whole
number that will divide both the numerator and denominator evenly. Value remains the
same.
Example: 3/5 = (6/2) / (10/2) = 3/5

31. To enlarge a fraction, multiply both terms by the same nonzero number. Value remains
the same.
Example: 1/12 = (1x2) / (12x2) = 2/24
To convert a mixed number to an improper fraction, multiply the whole number by the
denominator and add the numerator; use original denominator in the fractional part.
Example: 1(1/3) = 4/3
To convert an improper fraction to a mixed number, divide the numerator by the
denominator.
Express any remainder as a proper fraction reduced to lowest terms.
Example: 21/9 = 2(3/9) = 2(1/3)
When numerators are equal, the fraction with the smaller denominator is greater.
Example: 1/2 is greater than 1/3
When denominators are equal, the fraction with the larger numerator is greater.
Example: 2/3 is greater than 1/3
2. To add or subtract fractions:
 Convert to equivalent fractions with least common denominators.
 Add or subtract the numerators; place that value in the numerator.
Use the least common denominator as the denominator.
 Convert the answer to a mixed number and/or reduce to lowest terms.

32. 3. Multiplication of Fractions
When multiplying a fraction by a nonzero whole number, the same rule applies as for
multiplying fractions.
First convert the whole number to a fraction with a denominator of 1; the value of the
number remains the same.
Example (2/3) x (4/1)
To multiply mixed numbers, first convert them to improper fractions, and then multiply.
Example 31/2 x 41/3 = 7/2 x 13/3
To divide mixed numbers, first convert them to improper fractions.
Example[1(1/2)] / (3/4) = (3/2) / (3/4) = (3/2) x (4/3) = (1/1) x (2/1) = 2
To multiply fractions, cancel terms, multiply numerators, and multiply denominators.
 To divide fractions, invert the divisor, cancel terms, and multiply.
 Convert results to a mixed number and/or reduce to lowest terms.

33. 4. DECIMALS
In a decimal number, whole number values are to the left of the decimal point, and
fractional values are to the right.
Zeros added to a decimal fraction before the decimal point of a decimal number less
than 1 or at the end of the decimal fraction do not change the value.
Example: .5 = 0.5 = 0.50. However, using the leading zero is the only
acceptable notation (such as, 0.5).
In a decimal number, zeros added before or after the decimal point may change the
value.
Example: 1.5 ≠ 1.05 and 1.5 ≠ 10.5.
To avoid overlooking the decimal point in a decimal fraction, always place a zero to the
left of the decimal point.
Example: .5 ← Avoid writing a decimal fraction this way; it could be mistaken
for the whole number 5.
Example: 0.5 ←. This is the required method of writing a decimal fraction with a
value less than 1
The number of places in a decimal fraction indicates the power of 10.
Examples:
0.5 = five tenths
0.05 = five hundredths
0.005 = five thousandths

34. Compare decimals by aligning decimal points and adding zeros.
Example:
Compare 0.5, 0.05, and 0.005.
0.500 = five hundred thousandths (greatest)
0.050 = fifty thousandths
0.005 = five thousandths (least)
To convert a fraction to a decimal, divide the numerator by the denominator.
To convert a decimal to a fraction, express the decimal number as a whole number in
the numerator and the denominator as the correct power of 10. Reduce the fraction to
lowest terms.
Example:
0.04 = 4 (numerator is a whole number)
100 (denominator is 1 followed by two zeros)
= 4/100
= 1/25

35. To multiply decimals, place the decimal point in the product to the left as many decimal
places as there are in the two decimals multiplied.
Example:
0.25 x 0.2 = 0.050 = 0.05 (Zeros at the end of the decimal are unnecessary).
To divide decimals, move the decimal point in the divisor and dividend the number of
decimal places that will make the divisor a whole number and align it in the quotient.
Example: 24 / 1.2
To multiply or divide decimals by a power of 10, move the decimal point to the right (to
multiply) or to the left (to divide) the number of decimal places as there are zeros in the
power of 10.
Examples:
5.06 x10 = 5.0.6 = 50.6
2.1 /100 = .02.1 = 0.021
When rounding decimals, add 1 to the place value considered if the next decimal place
is 5 or greater.
Examples:
Rounded to hundredths: 3.054 = 3.05; 0.566 = 0.57.
Rounded to tenths: 3.05 = 3.1; 0.54 = 0.5

36. Any Questions or Additions

45. Any Questions or Additions

72. Any Questions or Additions

73. Pharmaceutical Calculations
Reducing & Enlarging Formulas
Percentage Preparations
Ratio Strength
Dilution and Concentration

123. THANK YOU

124. STUDY QUESTIONS
FOR
PHARMACEUTICAL NUMERACY

125. Study Questions
• Define the following terms:
• [weight, measure, fraction, apothecary, density, gravity, decimal, microgram,
milligram, centigram, decigram, decagram, hectogram, exponent, logarithms,
semisolid, oral, intravenous, subcutaneous, buccal, sublingual, rectal, topical,
transdermal, pestle, mortar, beaker, numerator, denominator, dosage,
ingredients, excipients, Alligations, etc]
• Respond to the following questions:
 State and explain the quantitative variations both the weight and measuring entities
 Explain with examples the relationship between weights and measures of
pharmaceutical substances
 Give a detailed account of the existing practical types of densities for pharmaceutical
materials
 What is the difference between common and decimal fractions in pharmaceutical
numeracy
 Give a detailed account of proper, fractions, improper fractions and mixed numbers
in pharmaceutical numeracy
 Give a descriptive account of all practical tools that provide quantitative
measurements of pharmaceutical substances

126. • Group work discussional questions:
First set of Questions are based on:
 Design and work out for pharmaceutical answers from formulated pharmaceutical
problems based on the following categories using appropriate pharmaceutical formulas:
1. 3 problems & answers on DENSITY
2. 3 problems & answers on numerical ADDITIONS
3. 3 problems & answers on numerical SUBTRACTIONS
4. 3 problems & answers on numerical MULTIPLICATIONS
5. 3 problems & answers on numerical DIVISIONS
6. 3 problems & answers on numerical FIGURES CONVERSIONS
7. 3 problems & answers on numerical DOSAGE CALCULATIONS
8. 3 problems & answers on numerical REDUCTION & ENLARGEMENTS
9. 3 problems & answers on numerical percentages based on WEIGHT PER VOLUME systems
10. 3 problems & answers on numerical percentages based on WEIGHT PER WEIGHT systems
11. 3 problems & answers on numerical percentages based on PARTS PER MILLION systems
12. 3 problems & answers on numerical percentages based on DILUTION & CONCENTRATION of systems
13. 3 problems & answers on numerical percentages based on MIXING PRODUCTS of STRENGTH of systems
14. 3 problems & answers on numerical figures based on DILUTION & CONCENTRATION of systems
15. 3 problems & answers on numerical figures based on ALLIGATION ALTERNATE systems
16. 3 problems & answers on numerical figures based on SATURATED PHARMACEUTICAL SOLUTIONS systems
17. 3 problems & answers on numerical figures based on MILLIEQUIVALENTS of systems
18. 3 problems & answers on numerical figures based on TEMPERATURE of systems

127. • Group work discussional questions:
Second set of Questions are based on:
 Solving for pharmaceutical solutions based on the following pharmaceutical problems:
1. A liquid medicine is supplied in a concentration of 20 mg/5 mL. A patient requires 40 mg
orally three times daily for 5 days, then 20 mg three times daily for 5 days, then 20 mg twice
daily for 5 days and then 20 mg once daily for 5 days. Calculate the total volume of liquid
medicine that is precisely to be dispensed
2. You are required to make 350 g of a paste that contains 15% w/w zinc oxide. What is the
amount of zinc oxide required?
3. A 1 in 10 000 solution of potassium permanganate contains how much of the quantity from
the given ration
4. How much of the volumes of an adrenaline 1 in 100 solution would be given by
intramuscular injection to a 2-year-old child for treatment of anaphylaxis if the dose were
120 micrograms stat?
5. How much of copper sulphate is required to make 400 mL of an aqueous stock solution,
such that, when the stock solution is diluted 50 times with water, a final solution of 0.1%
w/v copper sulphate is produced?
6. A child requires a single oral daily dose of 7.0 mg/kg body weight of drug A. The child’s
weight is 8.0 kg. How much of the oral daily doses of drug A is received by this child?
7. A patient in one of the residential homes to which you supply medication is going on holiday
and needs her prescriptions made up for the 5 days that she will be away. If she usually
takes ranitidine 150 mg twice daily and atenolol 50 mg in the morning, calculate the
appropriate combinations of Zantac syrup (75 mg ranitidine/5 mL) and Tenormin syrup (25
mg atenolol/5 mL) that would be supplied?

128. • Group work discussional questions:
Second set of Questions are based on:
 Solving for pharmaceutical solutions based on the following pharmaceutical problems:
8. Potassium permanganate solution 1 in 8000 is prepared from a stock of 10 times this strength.
How much potassium permanganate will be needed to make sufficient stock solution if a
patient uses 200 mL of the diluted solution each day for 20 days?
9. What volume of phenytoin suspension 30 mg/5 mL is required to be added to a suitable
diluent to obtain 150 mL phenytoin suspension 20 mg/5 mL?
9. Given a 20% w/v solution of chlorhexidine gluconate, what volume is required to make 400 mL
of a 2% w/v solution?
10. You are presented with a prescription for allopurinol tablets 100 mg at a dose of 300 mg each
day for 14 days, reducing to 200 mg for a further 7 days. How many packs of 28 tablets should
you supply?
11. An injection solution contains 0.5% w/v of active ingredient. How much of the active
ingredient is needed to prepare 500 L of solution?
12. A patient taking 10.0 mL Erythroped suspension (250 mg/5 mL) qid will receive how much
erythromycin each day?
13. Calculate the number of days a 150 mL bottle of nitrazepam 2.5 mg/5 mL suspension will last
a patient prescribed nitrazepam 5 mg at bedtime for insomnia.
14. Calculate the number of tablets required to fulfil the following prescription:
 Prednisolone 5 mg e/c tablets
 Take 25 mg daily for 4 days, then reduce by 5 mg every 4 days until the course is finished
(total course: 20 days)

129. • Group work discussional questions:
Second set of Questions are based on:
 Solving for pharmaceutical solutions based on the following pharmaceutical problems:
15. The number of drops per minute required if 720 mL of 5% w/v glucose is to be given
intravenously to a patient over a 12-hour period. It is known that 20 drops = 1 mL.
16. You receive a prescription for phenindione tablets 50 mg with the following instructions:
‘200 mg on day 1, 100 mg on day 2 and then 50 mg daily thereafter’. Mitte: 56 days’ supply.
Which of the following is the correct quantity to supply?
17. An ointment contains 1% w/w calamine. Which of the following is the amount of calamine
powder that should be added to 200 g of the ointment to produce a 4% w/w calamine
ointment?
18. A patient weighing 30 kg requires a single oral daily dose of 9 mg/kg of drug B. This drug is
available only as a suspension of 15 mg/5 mL. How much suspension would you supply?
Which of the following is the volume of a 6% w/v solution that is required to give a single
dose of 12 mg?
19. Which of the following is the concentration of a solution prepared by dissolving 400 mg
potassium permanganate in water and making up to a final volume of 4.0 L.
20. The volume of amoxicillin syrup 125 mg/5 mL required by a child prescribed 250 mg
amoxicillin orally three times daily for 5 days.
21. The volume of a 5% w/v solution required to give a dose of 40 mg.
22. The volume required to give a 15 mg dose of haloperidol from a 2 mL ampoule containing 10
mg haloperidol/mL.

130. • Group work discussional questions:
Second set of Questions are based on:
 Solving for pharmaceutical solutions based on the following pharmaceutical problems:
23. You mix together 50 g of 0.5% w/w hydrocortisone cream and 25 g of 2% w/w sulphur cream
(the creams are compatible). What is the final concentration of each of the two drugs?
24. A patient weighing 50 kg requires a single oral daily dose of 9 mg/kg of drug Y. This drug is
available only as a suspension of 150 mg/5 mL. How much suspension would it be most
appropriate to supply to provide a single dose?
25. You have in your pharmacy a cream containing 0.5% w/w hydrocortisone. You have been
requested to use this cream as a base and to add in sufficient calamine such that the final
concentration of calamine in the new cream will be 10.0% w/w. What is the concentration of
hydrocortisone in the new cream?
26. A stock solution of drug G is available at 10%w/v. You need to dilute this with Syrup, BP in
order to supply a patient with a solution containing 5 mg/mL of drug G. Assuming no volume
displacement effects, what is your formula for the preparation of 100 mL of the final solution
27. A patient is on a continuous intravenous drip of drug B. He needs to be dosed at a rate of 25
mg/h. The drip is set to administer 10 drops of fluid/h, with 4 drops equaling 1 mL in volume.
Which of the following is the concentration of drug B in the intravenous fluid?
28. The amount of phytomenadione contained in a 0.2 mL ampoule of 10 mg/mL solution.
29. The weight of chlorhexidine contained in 2 mL of a 1 in 10 000 solution.

131. • Group work discussional questions:
Second set of Questions are based on:
 Solving for pharmaceutical solutions based on the following pharmaceutical problems:
30. The weight of ethambutol contained in 0.4 mL of 250 micrograms/mL solution.
31. What is the correct volume of a 5% w/v solution required to supply 150 mg of the active ingredient?
32. What is the amount of fluorescein sodium in 300 mL of a 2.8% w/v aqueous solution.
33. What is the amount of 5-aminolevulinic acid hydrochloride in 25 g of a 20% w/w cream.
34. How much active substance is required to manufacture a batch of granules for a compressed tablet
with a batch size of 420 kg, to produce tablets with a mean weight of 700 mg and an active
substance content of 600 mg?
35. Given that the relative molecular mass (RMM) of sodium chloride is 58.5 g/mol, what amounts of
sodium chloride powder would be required to prepare 300 mL of a solution containing 50 mmol/L?
36. A tablet labelled to contain 350 mg active ingredient has acceptable limits of 90–110% of that
amount. What are the corresponding figures in milligrams to the stated percentage range ?
37. How much of the amount of sodium ions does 50 mL sodium chloride solution 0.9% w/v intravenous
infusion contain if there are 150 mmol each of Na+ and Cl–/L of NaCl 0.9% w/v IV infusion.
38. The amount of sodium chloride required to make 500 mL of a 0.1 mol/L solution (relative atomic
mass [RAM]: sodium = 23; chlorine = 35.5.)
39. The amount of lymeycline contained in five Tetralysal 300 tablets. Each tablet contains 408 mg
lymecycline equivalent to 300 mg tetracycline.
40. What is the number of moles of 5-aminolevulinic acid hydrochloride in 50 mL of a 1 mol/L solution?

132. Reference:
1. Dosage Calculations: A Ratio-Proportion Approach, Gloria D. Pickar, EdD, RN 2007, 2nd Edition
2. Pharmaceutical and Clinical Calculations, Mansoor A.Khan. and Indra K. Reddy, 2000, 2nd Ed.