2. Review of Mathematical Concepts
for Pharmaceutics
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: 1/6 = (6/2) / (10/2) = 3/5
3. 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.
4. 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.
5. 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
6. 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
7. 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
106. 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?
107. 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?
10 Given a 20% w/v solution of chlorhexidine gluconate, what volume is required to
make 400 mL of a 2% w/v solution?
11. 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?
12. 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?
13 A patient taking 10.0 mL Erythroped suspension (250 mg/5 mL) qid will receive how
much erythromycin each day?
14. 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.
15. 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)
16 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.
108. 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.
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?
109. 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.
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.
110. 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?
111. 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.
