Pharmaceutical Calculations Lecture 1 introduces key concepts in pharmaceutical calculations including: - Symbols and their meanings commonly used in pharmaceutical calculations. - Converting between common fractions, decimal fractions, percentages, and exponential notation. - Calculating ratios, proportions, and using dimensional analysis to solve pharmaceutical calculation problems. - Estimating answers to check the reasonableness of calculation results.

1. Pharmaceutical Calculations
Lecture 1
Laith Al-Asadi

2. Introduction
• Pharmaceutical calculations
Is the area of study that applies the basic
principles of mathematics to the preparation
and safe and effective use of pharmaceuticals.
• It is the application of the mathematics that
requires study.

3. Symbol and meaning
symbol Meaning Symbol meaning
% percent; parts per hundred ᾳ varies as; is proportional to
‰ parts per thousand gm Gram
± add or subtract; plus or minus;
positive or negative
mg milligram
≠ is not equal to µg microgram
≈ is approximately equal to ng nanogram
≡ is equivalent to w/v Weight by volume
: ratio symbol (e.g., a:b) °C Degree centigrade
® Trade mark BP British pharmacopeia

4. Symbol and meaning
Symbol meaning Symbol meaning
℞ Prescription mark T.P.N Total parenteral nutrition
wf With food q.s A sufficient quantity
.Y.O Years old rept repeats
p.o Taken orally tsp Teaspoonful
p.r Rectally om Every morning
Vag. vaginally on Every night
top topically NS Normal saline
sid Once time daily GW 5% Glucose water
bid Twice daily G/S Glucose saline
tid Three time daily mEq Milliwquivillent
tiw Three time a week IV Intravenous
qid Four time daily IM Intramuscular
qod Every other day SC Subcutaneous
IU International unit IP Intraperitonial

5. Common fractions
• Common fractions are portions of a whole,
expressed at 1 ⁄ 3, 7 ⁄ 8, and so forth.
• They are used only rarely in pharmacy
calculations nowadays. It is recalled, that
when adding or subtracting fractions, the use
of a common denominator is required. The
process of multiplying and dividing with
fractions is recalled by the following examples.

6. Common fractions
• Examples: If the adult dose of a medication is
2 teaspoonful (tsp.), calculate the dose for a
child if it is 1 ⁄ 4 of the adult dose.
¼ × 2 tsp = ½ tsp
• If a child’s dose of a cough syrup is 3 ⁄ 4
teaspoonful and represents 1 ⁄ 4 of the adult
dose, calculate the corresponding adult dose?

7. Decimal Fraction
• A decimal fraction is a fraction with a
denominator of 10 or any power of 10 and is
expressed decimally rather than as a common
fraction. Thus, 1/10 is expressed as 0.10 and 45
/100 as 0.45.
• Decimal fractions often are used in
pharmaceutical calculations. To convert a
common fraction to a decimal fraction, divide the
denominator into the numerator.
• Thus, 1 /8 = 1 ÷ 8 = 0.125.

8. Decimal Fraction
• To convert a decimal fraction to a common fraction,
express the decimal fraction as a ratio and reduce.
Thus, 0.25 = 25 /100 = 1 /4

9. Percent
• The term percent and its corresponding sign, %, mean
‘‘in a hundred.’’ So, 50 percent (50%) means 50 parts in
each one hundred of the same item.
• Common fractions may be converted to percent by
dividing the numerator by the denominator and
multiplying by 100. Example: Convert 3 ⁄ 8 to percent.
3 ⁄ 8 × 100 = 37.5%, answer.
• Decimal fractions may be converted to percent by
multiplying by 100.
• Example: Convert 0.125 to percent.
0.125 ×100 =12.5%, answer.

10. Exponential Notation
• Many physical and chemical measurements deal with
either very large or very small numbers. Because it
often is difficult to handle numbers of such magnitude
in performing even the simplest arithmetic operations,
it is best to use exponential notation or powers of 10 to
express them.
• We may express
121 as 1.21 × 102
1210 As 1.21 × 103
1,210,000 as 1.21 × 106
0.0121 as 1.21 × 10-2
0.00121 as 1.21 × 10-3

11. Exponential Notation
• In the multiplication of exponentials, the exponents are added.
For example, 102 × 104 = 106
. In the multiplication of numbers that are expressed in
exponential form, the coefficients are multiplied in the usual
manner, and then this product is multiplied by the power of 10
found by algebraically adding the exponents.
Examples: (2.5 × 102) × (2.5 × 104) = 6.25 × 106
In the division of exponentials, the exponents are subtracted. For
example, 105 ÷ 102 = 103
• In the division of numbers that are expressed in exponential
form, the coefficients are divided in the usual way, and the
result is multiplied by the power of 10 found by algebraically
subtracting the exponents.
(7.5 ×105) ÷ (2.5 ×103 ) = 3.0 ×102

12. Ratio
• Ratio The relative magnitude of two quantities
is called their ratio. Since a ratio relates the
relative value of two numbers, it resembles a
common fraction except in the way in which it
is presented. Whereas a fraction is presented
as, for example 1 ⁄ 2, a ratio is presented as
1:2 and is not read as ‘‘one half,’’ but rather as
‘‘one is to two.’’

13. Ratio
• All the rules governing common fractions
equally apply to a ratio. Of particular
importance is the principle that if the two
terms of a ratio are multiplied or are divided
by the same number, the value is unchanged,
the value being the quotient of the first term
divided by the second. For example, the ratio
20:4 or 20⁄ 4 has a value of 5; if both terms
are divided by 2, the ratio becomes 10:2 or
10⁄ 2, again the value of 5.

14. Ratio
• A proportion is the expression of the equality
of two ratios. It may be written in any one of
three standard forms:
(1) a:b = c:d
(2) a /b = c/d
If a /b = c/d , then a = bc / d , b = ad / c , c =ad /
b , and d = bc / a .

15. Dimensional Analysis
• When performing pharmaceutical
calculations, some students prefer to use a
method termed dimensional analysis (also
known as factor analysis, factor-label method,
or unit-factor method). This method involves
the logical sequencing and placement of a
series of ratios (termed factors) into an
equation

16. Dimensional Analysis
= 169.09 fl. Oz.

17. Variation
• In the preceding examples, the relationships were
clearly proportional. Most pharmaceutical
calculations deal with simple, direct relationships:
twice the cause, double the effect, and so on.
Occasionally, they deal with inverse relationships:
twice the cause, half the effect, and so on, as
when you decrease the strength of a solution by
increasing the amount of diluent. Here is a
typical problem involving inverse proportion:
• If 10 pints of a 5% solution are diluted to 40 pints,
what is the percentage strength of the dilution?

18. Rounding
Rules for Rounding
• When rounding a measurement, it would be correct to
record a measurement of 11.3 centimeters but not 11.32
centimeters, because the 2 (tenths) is uncertain and no
figure should follow it.
• When adding or subtracting approximate numbers,
include only as many decimal places as are in the number
with the fewest decimal places. For example, when adding
162.4 grams + 0.489 grams + 0.1875 grams + 120.78
grams, the sum is 283.8565 grams, but the rounded sum
is 283.9 grams

19. Estimation
• Estimation One of the best checks of the
reasonableness of a numeric computation is
an estimation of the answer. If we arrive at a
wrong answer by using a wrong method, a
mechanical repetition of the calculation may
not reveal the error. But an absurd result, such
as occurs when the decimal point is put in the
wrong place, will not likely slip past if we
check it against a preliminary estimation of
what the result should be.

20. Estimation
• Example: Add the following numbers: 7428, 3652,
1327, 4605, 2791, and 4490.
• Estimation: The figures in the thousands column
add up to 21,000, and with each number on the
average contributing 500 more, or every pair 1000
more, we get 21,000 + 3000 = 24,000, estimated
answer (actual answer, 24,293).
• Example: Multiply 612 by 413.
• Estimation:4 × 6 = 24, and because we discarded
four figures, we must supply four zeros, giving
240,000, estimated answer (actual answer,
252,756).