The document provides an overview of basic pharmaceutical measurement and calculation topics including: - Numbers, fractions, decimals, ratios, proportions, and percent calculations which are essential skills for pharmacy technicians. - Metric and household conversion factors and methods for calculating drug doses using ratios, proportions, and conversions between fractions, decimals, and percentages. - Key terms and examples related to measurement, calculations involving numbers in different forms, and applying these concepts to pharmacy practice.

1. The Pharmacy
Technician 4E
Chapter 6
Basic Pharmaceutical
Measurement Calculation

2. Topic Outline







Numbers
Fractions
Decimal Numbers
Significant Figures
Measurement
Equations & Variables
Ratio & Proportion





Percents & Solutions
Alligation
Powder Volume
Children’s Doses
Calculations for
Business

3. Comparison of Roman and Arabic
Numerals

4. Example:



xxx = 30 or 10 plus 10 plus 10
DC = 600 or 500 plus 100
LXVI = 66 or 50 plus 10 plus 5 plus 1
When the second of two letters has a value greater
than that of the first, the smaller is to be subtracted
from the larger.

5. Fractions
• When something is divided into parts, each part is
considered a fraction of the whole.
• If a pie is cut into 8 slices, one slice can be
expressed as 1/8, or one piece (1) of the whole
(8).
If we have a 1000 mg tablet,
• ½ tablet = 500 mg
• ¼ tablet = 250 mg

6. Fractions
Fractions have two parts:
• Numerator (the top part)
• Denominator (the bottom part)
1
8
1
8

7. Fractions
A fraction with the same numerator and the same
denominator has a value equivalent to 1.
In other words, if you have 8 pieces of a pie that has
been cut into 8 pieces, you have 1 pie.
8
=1
8

8. Terminology
Proper fraction
• A fraction with a value of less than 1.
• A fraction with a numerator value smaller than the
denominator’s value.
1
<1
4
Improper fraction
• A fraction with a value larger than 1.
• A fraction with a numerator value larger than the
denominator’s value.
6
>1
5

9. Adding or Subtracting Fractions
• When adding or subtracting fractions with unlike
denominators, it is necessary to create a common
denominator.
•This is like making both fractions into the same kind
of “pie.”
• Common denominator is a number that each of the
unlike denominators of two or more fractions can be
divided evenly.

10. Remember
Multiplying a number by 1 does not change
the value of the number.
5 ×1 = 5
Therefore, if you multiply a fraction by a
fraction that equals 1 (such as 5/5), you do
not change the value of a fraction.
5× 5 = 5
5

11. Guidelines for Finding a Common
Denominator
1. Examine each denominator in the given
fractions for its divisors, or factors.
2. See what factors any of the denominators
have in common.
3. Form a common denominator by multiplying
all the factors that occur in all of the
denominators. If a factor occurs more than
once, use it the largest number of times it
occurs in any denominator.

12. Example 1
Find the least common denominator of the following fractions.
Step 1.) Find the prime factors (numbers divisible only by 1 and
themselves) of each denominator. Make a list of all the different
prime factors that you find. Include in the list each different factor
as many times as the factor occurs for any one of the
denominators of the given fractions.
• The prime factors of 28 are 2, 2, and 7 (because 2 3 2 3 7 5
28).
•The prime factors of 6 are 2 and 3 (because 2 3 3 5 6).
The number 2 occurs twice in one of the denominators, so it must
occur twice in the list. The list will also include the unique factors 3
and 7; so the final list is 2, 2, 3, and 7.

13. Example 1
Find the least common denominator of the
following fractions.
Step 2. Multiply all the prime factors on your list.
The result of this multiplication is the least common
denominator.

14. Example 1
Find the least common denominator of the following fractions.
Step 3. To convert a fraction to an equivalent fraction with the
common denominator, first divide the least common
denominator by the denominator of the fraction, then multiply
both the numerator and denominator by the result (the
quotient).
•The least common denominator of 9⁄28 and 1⁄6 is 84. In the
first fraction, 84 divided by 28 is 3, so multiply both the
numerator and the denominator by 3.

15. Example 1
Find the least common denominator of the following
fractions.
In the second fraction, 84 divided by 6 is 14, so
multiply both the numerator and the denominator by
14.

16. Example 1
Find the least common denominator of the following
fractions.
The following are two equivalent fractions:

17. Example 1
Find the least common denominator of the following
fractions.
Step 4. Once the fractions are converted to contain
equal denominators, adding or subtracting them is
straightforward. Simply add or subtract the numerators.

18. Multiplying Fractions
• When multiplying fractions, multiply the
numerators by numerators and denominators
by
denominators.
• In other words, multiply all numbers above the
line; then multiply all numbers below the line.
• Cancel if possible and reduce to lowest terms.

19. Multiplying Fractions
Dividing the denominator by a number is the
same as multiplying the numerator by that
number.
3 × 5 15 3
=
=
20
20 4

20. Multiplying Fractions
Dividing the numerator by a number is the same
as multiplying the denominator by that number.
6
6 1
=
=
4 × 3 12 2

21. Dividing Fractions
To divide by a fraction, multiply by its reciprocal,
and then reduce it if necessary.
1 1× 3 3
=
= =3
1/ 3
1
1

22. Reciprocals




Reciprocals are two different fractions that equal 1 when
multiplied together.
Every fraction has a reciprocal (except those fractions
with zero in the numerator). The easiest way to find the
reciprocal of a fraction is to switch the numerator and
denominator, or just turn the fraction over.
To find the reciprocal of a whole number, just put 1 over
the whole number.
EXAMPLE:
 The reciprocal of 2 is 1/2.

23. Example 2
Multiply the two given fractions

24. Decimal Places
1000
mg
Whole
500
mg
0.5
tenths
(1 place to
the right)
50
mg
0.05
hundredths
(2 places to
the right)
5
mg
0.005
thousands
(3 places to
the right)
0.5
mg
0.0005
ten thousands
(4 places to
the right)

25. Decimals
Adding or Subtracting Decimals
• Place the numbers in columns so that the decimal points are
aligned directly under each other.
• Add or subtract from the right column to the left column.
Multiplying Decimals
• Multiply two decimals as whole numbers.
• Add the total number of decimal places that are in the two
numbers being multiplied.
• Count that number of places from right to left in the answer,
and insert a decimal point.

26. Decimals
Dividing Decimals
1. Change both the divisor and dividend to whole numbers
by moving their decimal points the same number of
places to the right.
• divisor: number doing the dividing, the denominator
• dividend: number being divided, the numerator
2. If the divisor and the dividend have a different number of
digits after the decimal point, choose the one that has
more digits and move its decimal point a sufficient
number of places to make it a whole number.

27. Decimals
Dividing Decimals
3. Move the decimal point in the other number the same number
of places, adding zeros at the end if necessary.
4. Move the decimal point in the dividend the same number of
places, adding a zero at the end.
1.45 ÷ 3.625 = 0.4
1.45 1450
=
= 0. 4
3.625 3625

28. Decimals
Rounding to the Nearest Tenth
1.
2.
Carry the division out to the hundredth place
If the hundredth place number ≥ 5, + 1 to the
tenth place
3. If the hundredth place number ≤ 5, round the
number down by omitting the digit in the
hundredth place:
5.65 becomes 5.7 4.24 becomes 4.2

29. Decimals
Rounding to the Nearest Hundredth or
Thousandth Place
3.8421 = 3.84
41.2674 = 41.27
0.3928 = 0.393
4.1111 = 4.111

30. System International Prefixes






Micro - One millionth (basic unit × 10–6 or unit ×
0.000,001)
Milli - One thousandth (basic unit × 10–3or unit × 0.001)
Centi - One hundredth (basic unit × 10–2 or unit × 0.01)
Deci - One tenth (basic unit × 10–1 or unit × 0.1)
Hecto - One hundred times (basic unit × 102 or unit × 100)
Kilo - One thousand times (basic unit × 103 or unit × 1000)

31. Common Metric Units: Weight
Basic Unit
Equivalent
1 gram (g)
1000 milligrams (mg)
1 milligram (mg)
1000 micrograms
(mcg)
1000 grams (g)
1 kilogram (kg)

32. Common Metric Conversions



kilograms (kg) to grams (g)
 Multiply by 1000 (move decimal point three places to the
right).
 Example: 6.25 kg = 6250 g
grams (g) to milligrams (mg)
 Multiply by 1000 (move decimal point three places to the
right).
 Example: 3.56 g = 3560 mg
milligrams (mg) to grams (g)
 Multiply by 0.001 (move decimal point three places to the
left).
 Example: 120 mg = 0.120 g

33. Common Metric Conversions


Liters (L) to milliliters (mL)
 Multiply by 1000 (move decimal point three
places to the right).
 Exmaple: 2.5 L = 2500 mL
Milliliters (mL) to liters (L)
 Multiply by 0.001 (move decimal point three
places to the left).
 Example: 238 mL = 0.238 L

34. Avoirdupois System



1 gr (grain)
1 oz (ounce)
1 lb (pound)
- 65 mg
- 437.5 gr or 30 g (28.35 g)
- 16 oz or 7000 gr or 1.3 g

35. Household Measure: Volume







1 tsp (teaspoonful)
1 tbsp (tablespoonful)
1 fl oz (fluid ounce)
1 cup
1 pt (pint)
1 qt (quart)
1 gal (gallon)
- 5 mL
- 3 tsp (15 mL)
- 2 tbsp (30 mL (29.57 mL)
- 8 fl oz (240 mL)
- 2 cups (480 mL)
- 2 pt (960 mL)
- 4 qt (3840 mL)

36. Household Measure: Weight



1 oz (ounce)
1 lb (pound)
2.2 lb
- 30 g
- 16 oz (454 g)
- 1 kg

37. Numerical Ratios
Ratios represent the relationship between:
• two parts of the whole
• one part to the whole
Written as follows:
1:2 “1 part to 2 parts” ½
3:4 “3 parts to 4 parts” ¾
Can use “per,” “in,” or “of,” instead of “to”
• Proportions are frequently used to calculate drug
doses in the pharmacy.
• Use the ratio-proportion method any time one ratio is
complete and the other is missing a component.

38. Proportions
• An expression of equality between two ratios.
• Noted by :: or =
3:4 = 15:20
or
3:4 :: 15:20

39. Rules for Ratio-Proportion Method
• Three of the four amounts must be known
• The numerators must have the same unit of
measure
• The denominators must have the same unit of
measure

40. Steps for Solving for X
1. Calculate the proportion by placing the ratios in
fraction form so that the x is in the upper-left corner.
2. Check that the unit of measurement in the
numerators is the same and the unit of measurement
in the denominators is the same.
3. Solve for x by multiplying both sides of the proportion
by the denominator of the ratio containing the
unknown, and cancel.
4. Check your answer by seeing if the product of the
means equals the product of the extremes.

41. Remember
When setting up a proportion to solve a
conversion, the units in the numerators must
match, and the units in the denominators must
match.

42. Example 3 Solve for X

43. Percents
• The number of parts per 100 can be written as a
fraction, a decimal, or a ratio.
• Percent means “per 100” or hundredths.
• Represented by symbol %.
30% = 30 parts in total of 100 parts
30:100, 0.30, or
30
100

44. Percents in the Pharmacy
• Percent strengths are used to describe IV
solutions and topically applied drugs.
• The higher the % of dissolved substances, the
greater the strength.
• A 1% solution contains
• 1 g of drug per 100 mL of fluid
• Expressed as 1:100, 1/100, or 0.01

45. Equivalent Values
45
100
0.5
100

46. Converting a Ratio to a Percent
1. Designate the first number of the ratio as the
numerator and the second number as the
denominator.
2. Multiply the fraction by 100%, and simplify as
needed.
3. Multiplying a number or a fraction by 100%
does not change the value.

47. Converting a Ratio to a Percent
5:1 = 5/1 × 100% = 5 × 100% = 500%
1:5 = 1/5 × 100% = 100%/5 = 20%
1:2 = 1/2 × 100% = 100%/2 = 50%

48. Converting a Percent to a Ratio
1. Change the percent to a fraction by dividing it
by 100.
2. Reduce the fraction to its lowest terms.
3. Express this as a ratio by making the
numerator the first number of the ratio and
the denominator the second number.

49. Converting a Percent to a Ratio
2% = 2 ÷ 100 = 2/100 = 1/50 = 1:50
10% = 10 ÷ 100 = 10/100 = 1/10 = 1:10
75% = 75 ÷ 100 = 75/100 = 3/4 = 3:4

50. Converting a Percent to a Decimal
1. Divide by 100% or insert a decimal point two
places to the left of the last number, inserting
zeros if necessary.
2. Drop the % symbol.

51. Converting a Decimal to a Percent
1. Multiply by 100% or insert a decimal point
two places to the right of the last number,
inserting zeros if necessary.
2. Add the the % symbol.
symbol

52. Percent to Decimal
4% = 0.04
15% = 0.15
200% = 2
4 ÷ 100% = 0.04
15 ÷ 100% = 0.15
200 ÷ 100% = 2
Decimal to Percent
0.25 = 25%
1.35 = 135%
0.015 = 1.5%
0.25 × 100% = 25%
1.35 × 100% = 135%
0.015 × 100% = 1.5%

53. Example 4
How many milliliters are there in 1 gal, 12 fl oz?
According to the values in Table 5.7, 3840 mL are found in
1 gal. Because 1 fl oz contains 30 mL, you can use the ratioproportion method to calculate the amount of milliliters in
12 fl oz as follows:

54. Example 4
How many milliliters are there in 1 gal, 12 fl oz?

55. Example
A solution is to be used to fill hypodermic syringes,
each containing 60 mL, and 3 L of the solution is
available. How many hypodermic syringes can be filled
with the 3 L of solution?
1 L is 1000 mL. The available supply of solution is therefore
Determine the number of syringes by using the ratio-proportion
method:

56. Example
How many hypodermic syringes can be filled with the 3
L of solution?

57. Example
You are to dispense 300 mL of a liquid preparation. If
the dose is 2 tsp, how many doses will there be in the
final preparation?
Begin solving this problem by converting to a
common unit of measure using conversion values.

58. Example 6
If the dose is 2 tsp, how many doses will there be in the
final preparation?
Using these converted measurements, the solution can be
determined one of two ways:
Solution 1: Using the ratio proportion method and the
metric system.

59. Example 6
If the dose is 2 tsp, how many doses will there be in the
final preparation?

60. Example 7
How many grains of acetaminophen
should be used in a Rx for 400 mg acetaminophen?
Solve this problem by using the ratio-proportion method.
The unknown number of grains and the requested
number of milligrams go on the left side, and the ratio of
1 gr 65 mg goes on the right side, per Table 5.5.

61. Example 7
How many grains of acetaminophen
should be used in the prescription?

62. Example 8
A physician wants a patient to be given 0.8 mg of
nitroglycerin. On hand are tablets containing
nitroglycerin 1/150 gr. How many tablets should the
patient be given?
Begin solving this problem by determining the number of
grains in a dose by setting up a proportion and solving
for the unknown.

63. Example 8
How many tablets should the patient be given?

64. Common Calculations in the
Pharmacy
Calculations of Doses
Active ingredient (to be administered)/solution
(needed)
•
=
Active ingredient (available)/solution
(available)

65. Example 9
You have a stock solution that contains 10 mg of active
ingredient per 5 mL of solution. The physician orders a
dose of 4 mg. How many milliliters of the stock solution
will have to be administered?

66. Example 9
How many milliliters of the stock solution will have to
be administered?

67. Example 10
An order calls for Demerol 75 mg IM q4h prn pain. The
supply available is in Demerol 100 mg/mL syringes.
How many milliliters will the nurse give for one
injection?

68. Example 10
How many milliliters will the nurse give for one
injection?

69. Example 11
An average adult has a BSA of 1.72 m2 and requires
an adult dose of 12 mg of a given medication. A child
has a BSA of 0.60 m2.
If the proper dose for pediatric and adult patients is a
linear function of the BSA, what is the proper
pediatric dose? Round off the final answer.

70. Example 11
What is the proper pediatric dose?

71. Example 11
What is the proper pediatric dose?

72. Example 12
A dry powder antibiotic must be reconstituted for
use. The label states that the dry powder occupies
0.5 mL. Using the formula for solving for powder
volume, determine the diluent volume (the amount
of solvent added). You are given the final volume for
three different examples with the same powder
volume.

73. Example 12
Using the formula for solving for powder volume,
determine the diluent volume.

74. Example 12
Using the formula for solving for powder volume,
determine the diluent volume.

75. Example 13
You are to reconstitute 1 g of dry powder. The label
states that you are to add 9.3 mL of diluent to make
a final solution of 100 mg/mL. What is the powder
volume?

76. Example 13
What is the powder volume?
Step 1. Calculate the final
volume. The strength of the
final solution will be 100
mg/mL.
mg/mL

77. Example 13
What is the powder volume?

78. Example 14
Dexamethasone is available as a 4 mg/mL
preparation. An infant is to receive 0.35 mg. Prepare
a dilution so that the final concentration is 1 mg/mL.
How much diluent will you need if the original
product is in a 1 mL vial and you dilute the entire
vial?

79. Example 14
How much diluent will you need if the original product is in a 1 mL vial
and you dilute the entire vial?

80. Example 14
How much diluent will you need if the original product
is in a 1 mL vial and you dilute the entire vial?

81. Example 15
Prepare 250 mL of dextrose 7.5% weight in volume
(w/v) using dextrose 5% (D5W) w/v and dextrose
50% (D50W) w/v. How many milliliters of each will
be needed?

82. Example 15
How many milliliters of each will be needed?
Step 1. Set up a box arrangement and at the upper-left
corner, write the percent of the highest concentration
(50%) as a whole number.

83. Example 15
How many milliliters of each will be needed?
Step 2. Subtract the center number from the upper-left
number (i.e., the smaller from the larger) and put it at
the lower-right corner. Now subtract the lower-left
number from the center number (i.e., the smaller from
the larger), and put it at the upper-right corner.

84. Example 15
How many milliliters of each will be needed?
50
2.5 mL parts D50W
7.5
5
42.5 mL parts D5W
45 mL total parts D7.5W

85. Example 15
How many milliliters of each will be needed?

86. Example 15
How many milliliters of each will be needed?

87. Example 15
How many milliliters of each will be needed?

88. Example 15
How many milliliters of each will be needed?

89. Example 15
How many milliliters of each will be needed?

90. Terms to Remember
1. Body surface area
2. Concentration
3. Conversions
4. Denominator
5. Flow rate
6. Least common
denominator
7. Milliequivalent (meq)
8. Nomogram
9. Numerator
10. Positional notation
11. Qs ad
12. Total parenteral nutrition
13. Usual and customary (U&C)
14. Valence
15. Variable