Download Chapter 01 - McGraw Hill Higher Education

Math for the Pharmacy Technician:
Concepts and Calculations
Egler • Booth
Chapter 1: Numbering
Systems and
Mathematical Review
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Numbering Systems and
Mathematical Review
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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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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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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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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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Arabic Numbers
(con’t)
Example
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.
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Roman Numerals
 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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Commonly used
Roman
numerals
 ss = ½
I = 1
V = 5
 X = 10
They may be
written in lower
or uppercase
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Combining Roman Numerals
When reading a Roman numeral
containing more than 1 letter, follow
these two steps:
1. If any letter with a smaller value appears
before a letter with a larger value, subtract
the smaller value from the larger value.
2. Add the value of all the letters not affected
by Step 1 to those that were combined.
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Combining Roman Numerals
(con’t)
Example
Example
IX = 10 –1 = 9
XIV = 10 + (5-1) = 14
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 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.
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Fractions and Mixed
Numbers
 Measure a portion or part of a whole
amount
 Written two ways:
 Common fractions
 Decimals
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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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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
2 wholes or ½
 Read it as “one half”
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Fraction Rule
When the denominator is 1, the fraction
equals the number in the numerator.
Example
4
1  4,
100
1  100
Check these equations by treating each
fraction as a division problem.
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Mixed Numbers
 Mixed numbers
combine a whole
number with a
fraction.
Example
2
2 3 (two and two-thirds)
 Fractions with a value greater than 1
are written as mixed numbers.
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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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Mixed Numbers (con’t)
To convert a fraction to a mixed number:
1. Divide the numerator by the denominator.
The result will be a whole number plus a
remainder.
2. Write the remainder as the number over
the original denominator.
3. 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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Mixed Numbers (con’t)
Example
Convert 11 to a mixed
4
number.
1. Divide the numerator by the denominator
11
2.
4 = 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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Mixed Numbers (con’t)
(5
1
3
To convert a mixed number
) to a
fraction:
1. Multiply the whole number (5) by the
denominator (3) of the fraction ( 13 )
5x3 = 15
2. Add the product from Step 1 to the
numerator of the fraction
15+1 = 16
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Mixed Numbers
(con’t)
To convert a mixed number to a fraction:
3.
Write the sum from Step 2 over the original
denominator 16
3
4.
The result is a fraction equal to original mixed
number. Thus 5 1  16
3
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Review and Practice
What is the numerator in
Answer = 17
What is the denominator
17
100
4
in 100
?
?
Answer = 100
Twelve patients are in the hospital ward. Four
have type A blood. What fraction do not have
type A blood?
8
Answer = 12
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Equivalent Fractions
Two fractions written differently that have
the same value = equivalent fractions.
Example
4
8
same as 3 same as
6
Find equivalent
fractions for
2
4
1
3
1 2 2
X 
3 2 6
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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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Equivalent Fractions (con’t)
To find missing numerator in an
equivalent fraction:
Example
2
?

3 12
a. Divide the larger denominator by the smaller one:
12 divided by 3 = 4
b. Multiply the original numerator by the quotient from
Step a: 2x4=8
2 8

3 12
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Equivalent Fractions
(con’t)
Find 2 equivalent fractions for
1
.
10
Answers
2 4
,
20 40
Find the missing numerator
?
8
16
Answer 128
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Reducing Fractions
to Lowest Terms
1. To reduce a fraction to its lowest
terms, find the largest whole number
that divides evenly into both the
numerator and denominator.
2. When no whole number except 1
divides evenly into them, the fraction
is reduced to its lowest terms.
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Simplifying Fraction
to Lowest Terms (con’t)
Example
Reduce
10
15
Both 10 and 15 are divisible by 5
10 2

15 3
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Equivalent Fractions
(con’t)
Reduce the following
fractions:
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8
10
4
Answer
5
27
81
Answer 1
3
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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):
1. List the multiples of each denominator.
2. Compare the list for common denominators.
3. The smallest number on all lists is the LCD.
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Common
Denominators
(con’t)
To convert fractions with large
denominators to equivalent fractions
with a common denominator:
1. List the denominators of all the fractions.
2. Multiply the denominators. (The product is
a common denominator.) Convert each
fraction to an equivalent with the common
denominator.
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Common Denominators (con’t)
Find the least common
denominator:
1
3
1
7
Answer 21
5
48
7
72
Answer 144
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Adding Fractions
To add fractions:
1. Rewrite any mixed numbers as fractions.
2. Write equivalent fractions with common
denominators. The LCD will be the
denominator of your answer.
3. Add the numerators. The sum will be the
numerator of your answer.
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Subtracting Fractions
To subtract fractions:
1. Rewrite any mixed numbers as fractions.
2. Write equivalent fractions with common
denominators. The LCD will be the
denominator of your answer.
3. Subtract the numerators. The difference
will be the numerator of your answer.
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Multiplying Fractions
To multiply fractions:
1. Convert any mixed numbers or whole
numbers to fractions.
2. Multiply the numerators and then the
denominators.
3. Reduce the product to its lowest terms.
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Multiplying Fractions
(con’t)
8
7
x
21 16
 To multiply
multiply the numerators and multiply
the denominators
8
7
8x7
56
x


21 16 21 x 16 336
56
1

336 6
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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
Answer will be
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1
6
81 7 1
x
213 16 2
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CAUTION!
Avoid canceling too many terms.
Each time you cancel a term, you must
cancel it from one numerator
one denominator.
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Multiplying Fractions (con’t)
Find the following products:
3 4
x
8 9
5
4
1 x7
6
5
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1
Answer
6
3
Answer 14
10
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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?
3
24 x 9
4
Answer 234
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Dividing Fractions
3
4
You have
bottle of liquid medication
1
available and you must give
16
bottle to your patient. How many
doses remain in the bottle?
Multiply
3
4
3
4
divided by
1
16
by the reciprocal of
1
16
3 16
x
 12 doses
4 1
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Dividing Fractions (con’t)
1. Convert any mixed or whole number to
fractions.
2. Invert (flip) the divisor to find its
reciprocal.
3. Multiply the dividend by the reciprocal
of the divisor and reduce.
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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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Dividing Fractions (con’t)
Find the following
quotients:
4
9
divided by
Answer
1
6
28
45
divided by
Answer
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5
7
3
4
2
9
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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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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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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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Working with Decimals (con’t)
Table 1-3 Decimal Place Values
The number 1,542.567 can be represented as follows:
Whole Number
Decimal
Point
Decimal Fraction
Thousands
Hundreds
Tens
Ones
.
Tenths
Hundredths
Thousandths
1,
5
4
2
.
5
6
7
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Decimal Place Values
The number 1,542.567 is read:
(1) - one thousand
(5) - five hundred
(42) - forty two and
(5) - five hundred
(67) – sixty-seven thousandths
One thousand five hundred forty two and
five hundred sixty-seven thousandths
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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, 0.203
represents 0 ones, 2 tenths, 0 hundreds,
and 3 thousandths.
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Decimals
1. Always write a zero to the left of the
decimal point when the decimal
number has no whole number part.
2. Using zero makes the decimal point
more noticeable and helps to prevent
errors caused by illegible handwriting.
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Comparing Fractions
The more places a number is to the right
of the decimal point the smaller the
value.
For example:
0.3
3
is 10
0.03
or three tenths
3
is 100
0.003
or three hundredths
3
is 1000
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or three thousandths
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Comparing Fractions
(con’t)
1. The decimal with the greatest whole
number is the greatest decimal
number.
2. If the whole numbers of two decimals
are equal, compare the digits in the
tenths place.
3. If the tenths place are equal, compare
the hundredths place digits.
4. Continue moving to the right
comparing digits until one is greater
than the other.
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Review and Practice
Write the following in decimal
form:
2
Answers
10
17
100
23
1000
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= 0.2
= 0.17
= 0.023
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Rounding Decimals
You will usually round decimals to
the nearest tenth or hundredth.
1. Underline the place value to which
you want to round.
2. 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.
3. Drop all digits to the right of the
target place value.
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Review and Practice
Round to the nearest tenth:
14.34
Answer 14.3
9.293
Answer 9.3
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Converting Fractions
into Decimals
To convert a fraction to a
decimal, divide the numerator
by the denominator. For
example:
3
4
 0.75
8
 1.6
5
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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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Review and Practice
Convert decimals to
fractions or mixed
number:
1.2
Answer
100.4
Answer
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2
1
or 1
1
10
5
4
2
or 100
100
10
5
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Adding and Subtracting
Decimals
1. Write the problem vertically.
Align the decimal points.
2. Add or subtract starting from the
right. Include the decimal point
in your answer.
2.47
+0.39
2.86
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Adding and Subtracting
Decimals (con’t)
Subtract 7.3 – 1.005
7.300
- 1.005
6.295
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1-61
Review and Practice
Add or subtract the
following pair of numbers:
13.561 + 0.099
Answer 13.66
16.250 – 1.625
Answer 14.625
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Multiplying Decimals
1. First, multiply without considering the
decimal points, as if the numbers were
whole numbers.
2. Count the total number of places to the
right of the decimal point in both factors.
3. 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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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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Review and Practice
A patient is given 7.5 milliliters of liquid
medication 5 times a day. How many
milliliters does she receive per day?
Answer 7.5 x 5
7.5
X5
37.5
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Dividing Decimals
1. Write the problem as a fraction.
2. 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.
3. 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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Review and Practice
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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Review and Practice
Convert the following
mixed numbers to
fractions:
3
39 13
2

Answer
18
18 6
9
9
10
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Answer
99
10
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Review and Practice
Round to the nearest tenth:
7.091
Answer 7.1
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Review and Practice
Add the following:
7.23 + 12.38
Answer 19.61
Multiply the following:
12.01 x 1.005
Answer 12.07005
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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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