1. Math for the Pharmacy Technician: Concepts and Calculations
Egler • Booth
Chapter 1: Numbering Systems and Mathematical Review
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2. Numbering Systems and Mathematical Review
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3. Learning Outcomes
When you have successfully completed Chapter 1, you will have mastered skills to be able to:
Identify and determine the values of Roman and Arabic numerals.
Understand and compare the values of fractions in various formats.
Accurately add, subtract, multiply, and divide fractions and decimals.
Convert fractions to mixed numbers and decimals.
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4. Learning Outcomes (con’t)
Recognize the format of decimals and measure their relative values.
Round decimals to the nearest tenth, hundredth, or thousandth.
Describe the relationship among percents, ratios, decimals, and fractions.
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5. Introduction
Basic math skills are building blocks for accurate dosage calculations.
You must be confident in your math skills.
A minor mistake can mean major errors in the patient’s medication.
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6. Arabic Numbers Arabic numbers include all numbers used today.
Numbers are written using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
You can write whole numbers, decimals, and fractions by simply combining digits.
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7. Arabic Numbers (con’t)
The Arabic digits 2 and 5 can be combined to write:
The whole number 25
The decimal 2.5
The fraction 2/5
The same two digits are used in each of the above Arabic numbers but each have different values.
Example
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8. Roman Numerals Commonly used Roman numerals
ss = ½
I = 1
V = 5
X = 10
They may be written in lower or uppercase
Are used sometimes in drug orders
You need to understand how to change Roman numeral into Arabic numbers in order to do dosage calculations
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9. Roman Numerals Symbol Value I 1 (unus) V 5 (quinque) X 10 (decem)
L 50 (quinquaginta)
C (centum)
D (quingenti)
M 1,000 (mille)

10. Combining Roman Numerals
When reading a Roman numeral containing more than 1 letter, follow these two steps:
If any letter with a smaller value appears before a letter with a larger value, subtract the smaller value from the larger value.
Add the value of all the letters not affected by Step 1 to those that were combined.
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11. Combining Roman Numerals (con’t)
Example
Example
Roman numerals from 1 to 30 are the ones you are most likely to see in doctors’ orders.
Be familiar with these to read orders correctly.
IX = 10 –1 = 9
XIV = 10 + (5-1) = 14
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12. Fractions and Mixed Numbers
Measure a portion or part of a whole amount
Written two ways:
Common fractions
Decimals
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13. Common Fractions Represent equal parts of a whole
Consist of two numbers and a fraction bar
Written in the form:
Numerator (top part of the fraction) = part of whole
Denominator (bottom part of the fraction) represents the whole
one part of the whole
the whole
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14. Common Fractions (con’t)
Scored (marked) tablet for 2 parts
You administer 1 part of that tablet each day
You would show this as 1 part of wholes or ½
Read it as “one half”
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15. Check these equations by treating each fraction as a division problem.
Fraction Rule
When the denominator is 1, the fraction equals the number in the numerator.
Example
Check these equations by treating each fraction as a division problem.
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16. Mixed Numbers Example 2 (two and two-thirds)
Mixed numbers combine a whole number with a fraction.
Example
2 (two and two-thirds)
Fractions with a value greater than 1 are written as mixed numbers.
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17. Mixed Numbers (con’t)
If the numerator of the fraction is less than the denominator, the fraction has a value of < 1.
If the numerator of the fraction is equal to the denominator, the fraction has a value =1.
If the numerator of the fraction is greater than the denominator, the fraction has a value > 1.
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18. Only applied when the numerator is greater
Mixed Numbers (con’t)
To convert a fraction to a mixed number:
Divide the numerator by the denominator. The result will be a whole number plus a remainder.
Write the remainder as the number over the original denominator.
Combine the whole number and the fraction remainder. This mixed number equals the original fraction.
Only applied when the numerator is greater
than the denominator
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19. Mixed Numbers (con’t) Example Convert to a mixed number.
Divide the numerator by the denominator
= 2 R3 (R3 means a remainder of 3)
3. The result is the whole number 2 with a remainder of 3
4. Write the remainder over the whole ¾
5. Combine the whole number and the fraction 2+ ¾
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20. Mixed Numbers (con’t) 5x3 = 15
To convert a mixed number ( ) to a fraction:
1. Multiply the whole number (5) by the denominator (3) of the fraction ( )
5x3 = 15
Add the product from Step 1 to the numerator of the fraction
15+1 = 16
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21. Mixed Numbers (con’t)
To convert a mixed number to a fraction:
Write the sum from Step 2 over the original denominator
The result is a fraction equal to original mixed number. Thus
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22. What is the denominator in ?
Review and Practice
What is the numerator in ?
Answer = 17
What is the denominator in ?
Answer = 100
Twelve patients are in the hospital ward. Four have type A blood. What fraction do not have type A blood?
Answer =
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23. Equivalent Fractions
Two fractions written differently that have the same value = equivalent fractions.
Example
same as same as
Find equivalent fractions for
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24. Equivalent Fractions (con’t)
To find an equivalent fraction, multiply or divide both the numerator and denominator by the same number.
Exception:
The numerator and denominator cannot be multiplied or divided by zero.
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25. Equivalent Fractions (con’t)
To find missing numerator in an equivalent fraction:
Example
Divide the larger denominator by the smaller one: divided by 3 = 4
Multiply the original numerator by the quotient from Step a: 2x4=8
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26. Equivalent Fractions (con’t)
Find 2 equivalent fractions for
Answers
Find the missing numerator
Answer 128
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27. Reducing Fractions to Lowest Terms
To reduce a fraction to its lowest terms, find the largest whole number that divides evenly into both the numerator and denominator.
When no whole number except 1 divides evenly into them, the fraction is reduced to its lowest terms.
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28. Simplifying Fraction to Lowest Terms (con’t)
Example
Reduce
Both 10 and 15 are divisible by 5
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29. Equivalent Fractions (con’t)
Reduce the following fractions:
Answer
Answer
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30. Common Denominators
Any number that is a common multiple of all the denominators in a group of fractions
To find the least common denominator (LCD):
List the multiples of each denominator.
Compare the list for common denominators.
The smallest number on all lists is the LCD.
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31. Common Denominators (con’t)
To convert fractions with large denominators to equivalent fractions with a common denominator:
List the denominators of all the fractions.
Multiply the denominators. (The product is a common denominator.) Convert each fraction to an equivalent with the common denominator.
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32. Common Denominators (con’t)
Find the least common denominator:
Answer 21
Answer 144
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33. Adding Fractions To add fractions:
Rewrite any mixed numbers as fractions.
Write equivalent fractions with common denominators. The LCD will be the denominator of your answer.
Add the numerators. The sum will be the numerator of your answer.
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34. Subtracting Fractions
To subtract fractions:
Rewrite any mixed numbers as fractions.
Write equivalent fractions with common denominators. The LCD will be the denominator of your answer.
Subtract the numerators. The difference will be the numerator of your answer.
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35. Multiplying Fractions
To multiply fractions:
Convert any mixed numbers or whole numbers to fractions.
Multiply the numerators and then the denominators.
Reduce the product to its lowest terms.
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36. Multiplying Fractions (con’t)
To multiply
multiply the numerators and multiply the denominators
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37. Multiplying Fractions (con’t)
To cancel terms when multiplying fractions, divide both the numerator and denominator by the same number, if they can be divided evenly.
Cancel terms to solve 1 1 3 2
Answer will be
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38. CAUTION! Avoid canceling too many terms.
Each time you cancel a term, you must cancel it from one numerator AND one denominator.
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39. Multiplying Fractions (con’t)
Find the following products:
Answer
Answer
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40. Multiplying Fractions (con’t)
A bottle of liquid medication contains 24 doses. The hospital has 9 ¾ bottles of medication. How many doses are available?
Answer 234
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41. Dividing Fractions divided by by the reciprocal of Multiply
You have bottle of liquid medication available and you must give bottle to your patient. How many doses remain in the bottle?
divided by
by the reciprocal of
Multiply
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42. Dividing Fractions (con’t)
Convert any mixed or whole number to fractions.
Invert (flip) the divisor to find its reciprocal.
Multiply the dividend by the reciprocal of the divisor and reduce.
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43. CAUTION! Write division problems carefully to avoid mistakes.
Convert whole numbers to fractions, especially if you use complex fractions.
Be sure to use the reciprocal of
the divisor when converting
the problem from division to
multiplication.
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44. Dividing Fractions (con’t)
Find the following quotients:
divided by
Answer
divided by
Answer
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45. Dividing Fractions (con’t)
A case has a total of 84 ounces of medication. Each vial in the case holds 1¾ ounce. How many vials are in the case?
Answer 48
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46. Decimals
Decimal system provides another way to represent whole numbers and their fractional parts
Pharmacy technicians use decimals daily
Metric system is decimal based
Used in dosage calculations
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47. Working with Decimals
Location of a digit relative to the decimal point determines its value.
The decimal point separates the whole number from the decimal fraction.
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48. Working with Decimals (con’t)
Table 1-3 Decimal Place Values
The number 1, can be represented as follows:
Whole Number
Decimal Point
Decimal Fraction
Thousands
Hundreds
Tens
Ones .
Tenths
Hundredths
Thousandths 1, 5 4 2 6 7
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49. Decimal Place Values The number 1,542.567 is read: (1) - one thousand
(5) - five hundred
(42) - forty two and
(67) – sixty-seven thousandths
One thousand five hundred forty two and five hundred sixty-seven thousandths
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50. Writing A Decimal Number
Write the whole number part to the left of the decimal point.
Write the decimal fraction part to the right of the decimal point. Decimal fractions are equivalent to fractions that have denominators of 10, 100, 1000, and so forth.
Use zero as a placeholder to the right of the decimal point. For example, represents 0 ones, 2 tenths, 0 hundreds, and 3 thousandths.
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51. Decimals
Always write a zero to the left of the decimal point when the decimal number has no whole number part.
Using zero makes the decimal point more noticeable and helps to prevent errors caused by illegible handwriting.
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52. 0.003 is or three thousandths
Comparing Fractions
The more places a number is to the right of the decimal point the smaller the value.
For example:
0.3 is or three tenths
0.03 is or three hundredths
0.003 is or three thousandths
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53. Comparing Fractions (con’t)
The decimal with the greatest whole number is the greatest decimal number.
If the whole numbers of two decimals are equal, compare the digits in the tenths place.
If the tenths place are equal, compare the hundredths place digits.
Continue moving to the right comparing digits until one is greater than the other.
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54. Review and Practice Write the following in decimal form: = 0.2 Answers
= 0.17
= 0.023
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55. Rounding Decimals
You will usually round decimals to the nearest tenth or hundredth.
Underline the place value to which you want to round.
Look at the digit to the right of this target. If 4 or less do not change the digit, if 5 or more round up one unit.
Drop all digits to the right of the target place value.
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56. Review and Practice Answer 14.3 9.293 Answer 9.3
Round to the nearest tenth:
14.34
Answer 14.3
9.293
Answer 9.3
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57. Converting Fractions into Decimals
To convert a fraction to a decimal, divide the numerator by the denominator. For example:
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58. Converting Decimals into Fractions
Write the number to the left of the decimal point as the whole number.
Write the number to the right of the decimal point as the numerator of the fraction.
Use the place value of the digit farthest to the right of the decimal point as the denominator.
Reduce the fraction part to its lowest term.
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59. Review and Practice Answer 100.4 Answer
Convert decimals to fractions or mixed number:
1.2
Answer or
100.4
Answer or
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60. Adding and Subtracting Decimals
Write the problem vertically. Align the decimal points.
Add or subtract starting from the right. Include the decimal point in your answer.
2.47
+0.39
2.86
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61. Adding and Subtracting Decimals (con’t)
7.300
1.005
6.295
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62. Review and Practice Add or subtract the following pair of numbers:
Answer
– 1.625
Answer
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63. Multiplying Decimals
First, multiply without considering the decimal points, as if the numbers were whole numbers.
Count the total number of places to the right of the decimal point in both factors.
To place the decimal point in the product, start at its right end and move the decimal point to the left the same number of places.
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64. Multiplying Decimals (con’t)
Multiply 3.42 x 2.5
3.42
X 2.5
1710
684
8.550
There are three decimal places so place the decimal point between 8 and 5 (8.55).
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65. Review and Practice Answer 7.5 x 5 7.5 X 5 37.5
A patient is given 7.5 milliliters of liquid medication 5 times a day. How may milliliters does she receive per day?
Answer x 5
7.5
X 5
37.5
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66. Dividing Decimals Write the problem as a fraction.
Move the decimal point to the right the same number of places in both the numerator and denominator until the denominator is a whole number. Insert zeros.
Complete the division as you would with whole numbers. Align the decimal point of the quotient with the decimal point of the numerator, if needed.
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67. Review and Practice Answer: 32 divided by 0.4 Take 0.4 into 32
A bottle contains 32 ounces of medication. If the average dose is 0.4 ounces, how many doses does the bottle contain?
Answer: 32 divided by 0.4
Take 0.4 into 32
Add a zero behind the 32 for each decimal place
320 divided by 4 = 80 or 80 doses
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68. Review and Practice Answer Answer
Convert the following mixed numbers to fractions:
Answer
Answer
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69. Review and Practice Answer 7.1 7.091 Round to the nearest tenth:
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70. Review and Practice Add the following: 7.23 + 12.38 Answer 19.61
Multiply the following:
12.01 x 1.005
Answer
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71. Remember, you control the numbers!
Always ask for assistance if you are uncertain, the only bad question is the one not asked.
THE END
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