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3
Chapter
Basic Math Review
Objectives
After reading this chapter, you will be able to:
• Identify whole numbers.
• Perform basic calculations using addition and subtraction.
• Perform basic calculations using multiplication and division.
• Set up and use fractions.
• Compare fractions, express them as decimals, and find common denominators.
• Perform basic mathematical calculations using fractions and decimals.
• Understand and use percentages.
• Perform basic mathematical calculations using percentages.
Key Terms
Addition
Common denominator
Cross-multiplication
Decimal number
Denominator
Difference
Division
Factor
Fractions
Improper fraction
Mixed number
Multiplication
Number
Numerator
Percentages
Product
Proper fraction
Quotient
Subtraction
Sum
Whole numbers
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UNIT One Foundation
77Chapter Overview
This chapter will review basic mathematical skills that are necessary to perform more
complex calculations. Calculations in the pharmacy setting require a basic understanding of addition, subtraction, division, multiplication, fractions, percentages, and
decimals.
77Numbers
A number is a quantity or amount that is made up of one or more numerals. There are
several kinds of numbers: whole numbers, fractions, and decimal numbers. Examples
of whole numbers include 2, 10, 100, 2,000, and 5,000,000. Examples of fractions
include 1⁄4, 3/10, and 7/8. Examples of decimals include 0.2, 0.75, and 2.6.
A number is represented by one or more numerals.
77Whole Numbers
Our number system is a base-10 number system. That is, it is based on the number
10. We use 10 digits, also called whole numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). Whole
numbers are counting numbers.
We use whole numbers to count.
77Addition and Subtraction
Many dosage calculations involve addition and subtraction. With addition, you add
something to a given number Figure 3.1 . The sum is the amount obtained when adding numbers (see Table 3.1 ). With subtraction, you take something away from a given
number Figure 3.2 . The difference is the amount obtained when subtracting numbers
(see Table 3.2 ). There is an inverse relationship between addition and subtraction.
Addition means that you are adding or
increasing the quantity by a given
number.
Adding a zero to a number does not change the value of
the original number. The value of zero is nothing.
Examples: 0 + 5 = 5; 10 + 0 = 10; 210 + 0 = 210.
Subtraction means that you are taking
away or reducing the quantity by a
given number.
Subtracting a zero from a number does not change the
value of the original number. The value of zero is
nothing. Examples: 7 – 0 = 7; 128 – 0 = 128; 21 – 0 =
21.
+5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Figure 3.1 Addition example using a number line.
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CHAPTER 3 Basic Math Review
Table 3.1 Addition Strategy
Fact Strategy
Example
Answer (Sum)
Plus 0
8+0
8
Plus 1
8+1
9
Plus 2
8+2
10
Plus 3
8+3
11
Plus 4
8+4
12
Plus 5
8+5
13
Plus 6
8+6
14
Plus 7
8+7
15
Plus 8
8+8
16
Plus 9
8+9
17
Plus 10
8 + 10
18
–5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Figure 3.2 Subtraction example using a number line.
Table 3.2 Subtraction Strategy
Fact Strategy
Example
Answer (Difference)
Minus 0
10 – 0
10
Minus 1
10 – 1
9
Minus 2
10 – 2
8
Minus 3
10 – 3
7
Minus 4
10 – 4
6
Minus 5
10 – 5
5
Minus 6
10 – 6
4
Minus 7
10 – 7
3
Minus 8
10 – 8
2
Minus 9
10 – 9
1
Minus 10
10 – 10
0
77Multiplication and Division
Multiplication and division are used constantly in performing pharmacy calculations.
Sometimes, you must enlarge or reduce the quantity or determine a part or portion of
a quantity needed. Multiplication and division are the mathematical processes used to
do this. The product is the amount obtained by multiplying numbers. The quotient
is the amount obtained by dividing one number by another number. Table 3.3 shows
a multiplication chart.
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UNIT One Foundation
Table 3.3 Multiplication Chart
1
2
3
4
5
6
7
8
9
10
11
12
2
4
3
6
6
8
10
12
14
16
18
20
22
24
9
12
15
18
21
24
27
30
33
36
4
8
12
16
20
24
28
32
36
40
44
48
5
10
15
20
25
30
35
40
45
50
55
60
6
12
18
24
30
36
42
48
54
60
66
72
7
14
21
28
35
42
49
56
63
70
77
84
8
16
24
32
40
48
56
64
72
80
88
96
9
18
27
36
45
54
63
72
81
90
99
108
10
20
30
40
50
60
70
80
90
100
110
120
11
22
33
44
55
66
77
88
99
110
121
132
12
24
36
48
60
72
84
96
108
120
132
144
Division is the inverse relationship of multiplication.
Example:
9 ¥ 5 = 45
45 ÷ 9 = 5
45 ÷ 5 = 9
Example:
7 ¥ 6 = 42
42 ÷ 7 = 6
42 ÷ 6 = 7
When multiplying numbers, any number multiplied by zero always equals zero.
Example:
Example:
Example:
126 ¥ 0 = 0
546 ¥ 0 = 0
4¥0=0
When dividing numbers, any number divided by zero always equals zero.
Example:
Example:
Example:
978 ÷ 0 = 0
69 ÷ 0 = 0
78 ÷ 0 = 0
77Fractions
Many dosage calculations use amounts other than whole numbers—for example,
fractions. Pharmacy technicians work with these amounts to fill orders and complete
pharmaceutical calculations. Pharmacy technicians must be able to recognize fractions
and understand what amounts they represent.
Fractions express a quantity or portion that is a part of a whole number Figure 3.3 .
It is a ratio of a part to the whole. The number above the fraction line is called the
numerator. The number below the fraction line is called the denominator. The denominator tells how many equal parts are in the whole or set. The numerator tells
you the number of those parts you are expressing.
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CHAPTER 3 Basic Math Review
Numerator
Fractions express a quantity that is a part of a whole number.
1
The numerator is the number above the fraction line.
§4
Denominator
Figure 3.3 Defining a fraction.
The denominator is the number below the fraction line.
Another way to think about fractions is by dividing a pie into parts.
For example, imagine a pie that has been divided into eight pieces or
parts. Each piece is a fraction, or 1⁄8 of the whole pie Figure 3.4 .
A number line can be a useful tool when thinking about fractions Figure 3.5 .
A proper fraction has a numerator that is smaller than the denominator. The value of this type of fraction is always less than 1. Examples
of a proper fraction include 1⁄5, 2⁄3, and 3⁄8.
0
1
§4
1
§2
§4
3
1
Figure 3.5 Fraction example using a
number line.
The value of a proper fraction is always less than 1.
An improper fraction has a numerator that is equal to or larger
than the denominator. The value of this type of fraction is always
equal to or greater than 1. When the numerator and denominator are
the same, the value of the fraction is always 1, because any number
divided by itself always equals 1. Examples of an improper fraction
include 2⁄2, 4/3, and 9/7.
The value of an improper fraction is always equal to or greater than 1.
A mixed number contains both whole numbers and fractions. Examples of mixed numbers include 12⁄3 and 24/7.
Figure 3.4 Fraction example using a
pie.
A mixed number has whole numbers and fractions.
You can convert a mixed number to an improper fraction by multiplying the denominator by the whole number and adding the numerator Figure 3.6 .
You can rewrite a fraction greater than 1 as a mixed number or as a
whole number Figure 3.7 . For example, to write 7/4 as a mixed number,
follow these steps:
1.Divide the numerator by the denominator.
2.Use the remainder to write the fraction part of the quotient.
A fraction is in its simplest form when its numerator and denominator have no common factor other than 1 Figure 3.8 . A factor is a number
that you multiply by another number, which is also a factor, to make
yet another number. For example, the simplest form of 6/18 is 1⁄3.
123 =
((3 × 1) + 2) = 5
2
3
Figure 3.6 Converting a mixed fraction
to an improper fraction.
7 ÷ 4 = 1 37
Figure 3.7 Division example using
fractions.
6 ÷3= 1
8
3
Figure 3.8 Simplifying a fraction.
When a fraction is in its simplest form, the numerator
and denominator have no common factor other than 1.
You can divide the numerator and denominator
by common factors until the only common factor
is 1.
To convert a mixed number to an
improper fraction, multiply the denominator by the whole number. Then add
the numerator to this calculated amount
and place it over the same denominato
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UNIT One Foundation
Comparing Fractions
Reduce every mixed number and
improper fraction.
To simplify a fraction, divide the
numerator and denominator by a
common factor until the only common
factor is 1.
Comparing fractions helps to ensure that correct
calculations are used in the practice of pharmacy and
in the normal work tasks for pharmacy technicians.
Fractions can be compared using denominators that
are the same or different, as well as numerators that
are the same or different. Using fractions in pharmacy
work is a simple and helpful mathematical concept
to be mastered by the pharmacy technician.
Fractions with Like Denominators
Comparing fractions with the same denominators
is similar to comparing whole numbers Figure 3.9 .
When comparing fractions that have the same denominator, the fraction with the larger numerator will have the larger value. For
example, 3⁄5 is greater than 2⁄5.
When two fractions have the same denominator, the fraction with the larger numerator has a
larger value.
Fractions with Like Numerators
When two fractions have the same numerator, the fraction with the smaller denominator has the larger value Figure 3.10 .
When two fractions have the same numerator, the fraction with the smaller denominator has a
larger the value.
If the numerator is the same, the smaller the denominator, the larger the
value Figure 3.11 .
If the denominator is the same, the smaller the numerator, the smaller the
value Figure 3.12 .
Figure 3.9 Same denominator fraction graph.
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CHAPTER 3 Basic Math Review
Figure 3.10 Different denominator fraction pie graph.
Figure 3.11 Which tablet is bigger?
Figure 3.12 Which tablet is smaller?
Adding and Subtracting Fractions
When you add and subtract fractions, it often helps
If both fractions have the same denomito find the lowest common denominator. A common
nators, add the numerators and keep
denominator is a number into which both denominathe denominator the same.
tors can divide evenly.
When adding fractions that have the same denominators, add the numerators and keep the denominator the same.
Example:
⁄7 + 2⁄7 = 3⁄7
1
Rules for Adding Fractions
When adding fractions, first convert the fractions to the lowest common denominator (if
necessary). Then, add the numerators, placing the sum over denominator. Finally, reduce the
resulting fraction.
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UNIT One Foundation
In order to add fractions with different denominators, you must first find the lowest common denominator. For example, suppose you want to add 1⁄3 + 2⁄4. The lowest
number that can be divided evenly by both 3 and 4 is 12. Therefore, the common
denominator is 12.
Example:
Answer:
⁄3 + 2⁄4
1
⁄3 ¥ 4/4 = 4/12
2
⁄4 ¥ 3⁄3 = 6/12
4
/12 + 6/12 = 10/12
1
When adding fractions with different
denominators, first find the lowest common denominator.
If both fractions have the same denominator, subtract the numerators and
keep the denominator the same.
You can further simplify this to 5/6 by dividing both
the numerator and the denominator by their common factor of 2:
10
/12 ÷ 2⁄2 = 5/6
When subtracting fractions that have the same
denominators, subtract the numerators and keep the
denominator the same.
5
Example:
/7 – 2⁄7 = 3⁄7
In order to subtract fractions with different denominators, you must first find the lowest common
denominator. For example, suppose you want to
subtract 8/10 – 2⁄5. The lowest number that can be
divided evenly by both 10 and 5 is 10. Therefore,
the common denominator is 10.
Example:
Answer:
When subtracting fractions with
different denominators, first find the
lowest common denominator.
/10 – 2⁄5
8
/10 ¥ 1⁄1 = 8/10
2
⁄5 ¥ 2⁄2 = 4/10
8
/10 – 4/10 = 4/10
8
You can further simplify this answer to 2⁄5 by dividing both the numerator and the denominator by the
least common factor, which is 2:
/10 ÷ 2⁄2 = 2⁄5
4
Rules for Subtracting Fractions
When subtracting fractions, first convert the fractions to the lowest common denominator (if
necessary). Next, subtract the numerators, placing the difference over the denominator.
Finally, reduce the resulting fraction.
Multiplying and Dividing Fractions
It is important to understand the mathematical concepts used in multiplying and
dividing fractions. Many dosage calculations will require the pharmacy technician to
calculate the amount of drug required using multiplication and division with fractions
prior to a final check by the pharmacist.
Multiplying Fractions
To multiply fractions, multiply the numerators of each fraction and the denominators
of each fraction.
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CHAPTER 3 Basic Math Review
Example:
Answer:
⁄3 ¥ 4/5
2¥4=8
3 ¥ 5 = 15
8
/15
2
Rules for Multiplying Fractions
When multiplying fractions, first convert any mixed numbers into improper fractions (if
necessary). Next, multiply the numerators to obtain a new numerator. Then multiply the
denominators to obtain a new denominator. Finally, reduce the resulting fraction.
Dividing Fractions
To divide fractions, invert or reverse the numbers of the second fraction in the equation.
Then multiply the numerators of each fraction and the denominators of each fraction,
as you do when multiplying fractions. This is known as “invert and multiply.”
Example:
Answer:
⁄3 ÷ 1⁄8
⁄3 ¥ 8/1
1
⁄3 ¥ 8
8
/3, which simplifies to 22⁄3
1
1
Rules for Dividing Fractions
When dividing fractions, first convert any mixed numbers into improper fractions (if necessary). Next, find the reciprocal of the divisor (the second fraction). Then multiply the first
number by the reciprocal. Finally, reduce the resulting fraction.
77Decimals
Not every number is a whole number. Decimal numbers are numbers that are written using place value. The decimal system includes 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8,
9). Using these digits, you can express numbers of all sizes using a decimal point.
The decimal point, which looks like a period, is the central character in the decimal
system. Any value can be expressed using a combination of these numerals. Digits to
the left of the decimal point indicate whole numbers, and digits to the right indicate
fractions of a whole. The value of a digit increases by a multiple of 10 each time it
moves one space to the left. Equally, the value of a digit decreases by a multiple of 10
each time it moves one space to the right.
The scheme of the decimal system is depicted in Table 3.4 . In this case, the number
63,207.5184 is used to illustrate decimal-system notation.
The total value of a number expressed by the decimal system is the sum of all its
digits according to their place to the right or left of the decimal point. Therefore, in
the sample number, 63,207.5184 is equal to the following:
60,000.0000 (6 ten-thousands)
+ 03000.0000 (3 thousands)
77 + 00200.0000 (2 hundreds)
77 + 00000.0000 (0 tens)
77 + 00007.0000 (7 ones)
77
77
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UNIT One Foundation
Thousands
Hundreds
Tens
Ones
Decimal point
Tenths
Hundredths
Thousandths
Ten-thousandths
Sample
Number
Ten-thousands
Value
Table 3.4 Decimal System Notation
6
3
2
0
7
.
5
1
8
4
+ 00000.5000 (5 tenths)
77 + 00000.0100 (1 hundredth)
77 + 00000.0080 (8 thousandths)
77 + 00000.0004 (4 ten-thousandths)
77
Any fraction can be written in decimal form. The position of a number in relation to the decimal point indicates the value of that number. Digits to the left of the
decimal point indicate whole numbers, and digits to the right of the decimal point
indicate fractions of a whole. Each place value is multiplied by 10 as you move left
from the decimal point. Each place value is divided by 10 as you move right from
the decimal point.
In pharmacy practice, when using the
decimal
system to express a number less than
Think of money to help understand decimals and
1,
a
leading
zero (0) is placed to the left of
their place values. One dollar can be thought of as
the decimal point. Trailing zeros, or zeros
having 100 parts, with each part being equal to one
that are placed to the right of the final digit,
penny. A dollar can also be thought of as having 10
are not used in pharmacy. These techniques
parts, with each part then being equal to one dime.
help eliminate errors when reading decimals.
If a leading zero is not included or a trailing
zero is included, and the decimal point may not be read. That means a dose may be
misinterpreted by a factor of 10, 100, or even 1,000.
For example, consider the following: 2⁄5 = 2 ÷ 5 = 0.4. This would not be written
as .4, .40, or 0.40. In this case, if 0.4 represented a dose, and a leading zero were not
included, the value could be misinterpreted as 4 or 40, resulting in respective overdoses of 10 or 100 times the intended dose. Or, if a
dose of 10mg was written as 10.0, the value could
Always include a leading zero when
easily be interpreted as 100mg.
expressing doses that are less than 1.
A leading zero is placed to the left of the decimal point
when expressing a number less than 1.
Do not include a trailing zero in
pharmacy calculations, except when it
is critical to the accuracy of the
measurement. Never use a trailing zero
when a dose is expressed as a whole
number.
The exception to the rule of no trailing zeros
is when the zero is considered significant to the
dose or calculation, or when rounding results in a
trailing zero—for example, 0.799 rounded to the
nearest hundredth is 0.80 (written with a trailing
zero). Another example is the use of a trailing zero
in calculations involving currency Figure 3.13 . For
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CHAPTER 3 Basic Math Review
THIS IS NOT LEGAL TENDER
AND SHALL NOT BE USED FOR SUCH
F285937448B
6
6
6
6
F285937448B
+
+
=
W A SH ING TO N
1 dollar
+
0.1 dollar
+
0.01 dollar
=
1.11 dollars
Figure 3.13 Practical decimal demonstration.
Table 3.5 Different Ways to Express a Decimal
Form
Example
Standard
45.38
Word
forty-five and thirty-eight hundredths
Expanded
(4 ¥ 10) + (5 ¥ 1) + (3 ¥ 0.1) + (8 ¥ 0.01)
example, 10 dollars and 70 cents is written mathematically as $10.70, not as $10.7,
without the trailing zero.
There are many ways to write the same decimal, as shown in Table 3.5 .
Adding and Subtracting Decimals
When adding and subtracting decimals, the numbers
should be placed in columns so that the decimal
points are all aligned. For example:
Align the decimal points before adding
and subtracting decimal numbers.
4.3
2.16
+ 3.289
9.749
Multiplying Decimals
When multiplying decimals, you multiply them as
whole numbers, and then move the decimal the total
Multiply the decimals as whole numbers
number of places that were in the two numbers being
first and then move the decimal over
multiplied Figure 3.14 . There is an implied decimal
the total number of places.
point for all whole numbers, which is placed at the
end of the smallest or last digit in the number. For
example, the whole number 76 is also the same as 76.0, with the decimal point and
zero being the implied decimal. Thus, when multiplying decimals, the starting point
for moving the decimal begins with the implied decimal point at the end of the whole
number. For example, to multiply 23.6 ¥ 45.29, you first multiply 236 ¥ 4529 =
1,068,844. The implied decimal point is 1,068,844.0 and the movement of the decimal 1068.844.
places begins after the number four in the ones digit place. Next, count from right to
Figure 3.14 Moving
left three decimal places (because there are three total decimals in 23.6 and 45.29)
decimals.
and place the decimal point there: 1,068.844.
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UNIT One Foundation
Dividing Decimals
Change each decimal to a whole number
by multiplying each number by the
same factor of 10 before dividing
decimals.
When dividing decimals, first change each decimal
to a whole number by multiplying each number by
the same factor of 10. Then proceed with the division operation.
Answer:
Example:1.34 ÷ 2.1
1.34 ¥ 100 = 134
2.1 ¥ 100 = 210
134 ÷ 210 = 0.638
Example:
Answer:
2.5 ÷ 1.25
2.5 ¥ 100 = 250
1.25 ¥ 100 = 125
250 ÷ 125 = 2
77Percentages
Pharmacy calculations often require the use of percentages. A percentage is represented by the % symbol and is used to express the number of parts of one hundred.
It represents the same number as a fraction whose denominator is 100. Examples
include the following:
0.9% = 0.9/100.
77 2% = 2/100
77 5/100 = 5%
77
77Tech Math Practice
Question: What is the sum of 143 + 211?
Answer: 354
Question: What is the sum of 29 + 33?
Answer: 62
Question: What is the difference of 187 – 96?
Answer: 91
Question: What is the difference of 435 – 78?
Answer: 357
Question: What is 8 ¥ 6?
Answer: 48
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CHAPTER 3 Basic Math Review
Question: What is 12 ¥ 11?
Answer: 132
Question: What is 99 ÷ 9?
Answer: 11
Question: What is 108 ÷ 9?
Answer: 12
Question: What type of fraction is 4/3?
Answer: Improper fraction
Question: What type of fraction is 2⁄7?
Answer: Proper fraction
Question: What type of fraction is 17⁄8?
Answer: Mixed number
Question: What is the simplest form of the fraction 20/40?
Answer: 20/40 ÷ 2⁄2 = 10/20 ÷ 2⁄2 = 1⁄2
Question: What is the simplest form of the fraction 8/24?
Answer: 8/24 ÷ 4/4 = 2⁄6 ÷ 2⁄2 = 1⁄3
Question: What is the simplest form of the fraction 12/16?
Answer: 12/16 ÷ 4/4 = 3⁄4
Question: Which of the following doses of drug is the smallest dose? 3⁄10 mg tablet, 7⁄10
mg tablet, or 2⁄10 mg tablet?
Answer: The 2⁄10 mg tablet is the smallest dose because when fractions have the same
denominator, the fraction with the smaller numerator has the smallest value.
Question: Which of the following doses of drug is the largest dose? 1/15 mg tablet, 3/15
mg tablet, or 7/15 mg tablet?
Answer: The 7/15 mg tablet is the largest dose because when fractions have the same
denominator, the fraction with the larger numerator has the largest value.
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UNIT One Foundation
Question: Which of the following tablets is the smallest dose? 3/10 mg tablet, 3/12 mg
tablet, or 3/20 mg tablet?
Answer: The 3/20 mg tablet is the smallest dose because when fractions have the same
numerator, the fraction with the larger denominator has the smallest value.
Question: Which of the following tablets is the largest dose? 4/7 mg tablet, 4/12 mg
tablet, or 4/15 mg tablet?
Answer: The 4/7 mg tablet is the largest dose because when fractions have the same
numerator, the fraction with the smallest denominator has the largest value.
Question: What is the sum of 1⁄6 + 4/6?
Answer: 5/6
Question: What is the sum of 3⁄8 + 2⁄8?
Answer: 5/8
Question: What is the sum of 1⁄5 + 2/10?
Answer: To obtain the sum, first determine the lowest common denominator: 1⁄5 ¥ 2⁄2 = 2/10.
Then complete the addition operation: 2/10 + 2/10 = 4/10, which simplifies to 2⁄5.
Question: What is the sum of 2⁄3 + 3⁄7?
Answer: To obtain the sum, first determine the lowest common denominator: 2⁄3 ¥ 7/7 = 14/21
and 3⁄7 ¥ 3⁄3 = 9/21. Then complete the addition operation: 14/21 + 9/21 = 23/21. Note that 23/21 is
an improper fraction, and can be simplified to 112/21.
Question: What is the difference of 8/9 – 3⁄9?
Answer: 5/9
Question: What is the difference of 3⁄5 – 1⁄5?
Answer: 2⁄5
6
/9 – 3/9 = 3/9 = 1/3
Question: What is the sum of 3⁄8 + 3/12?
Answer: To obtain the sum, first determine the lowest common denominator: 3⁄8 ¥ 3⁄3 = 9/24
and 3/12 ¥ 2⁄2 = 6/24. Then complete the addition operation: 9/24 + 6/24 = 15/24.
Question: What is the difference of 4/12 – 2⁄6?
Answer: To obtain the difference, first determine the lowest common denominator: 4/12 ¥ 1⁄1
= 4/12 and 2⁄6 ¥ 2⁄2 = 4/12. Then complete the subtraction operation: 4/12 – 4/12 = 0.
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CHAPTER 3 Basic Math Review
Question: What is the product of 2⁄5 ¥ 3⁄4?
Answer: 6/20, which simplifies to 3/10.
Question: What is the product of 3⁄5 ¥ 4/7?
Answer: 12/35
Question: What is the product of 4/12 ¥ 3⁄8?
Answer: 12/96, which simplifies to 1⁄8
Question: What is the quotient of 4/7 ÷ 3⁄7?
Answer: 28/21, which simplifies to 11⁄3.
Question: What is the quotient of 2/11 ÷ 1⁄2?
Answer: 4/11
Question: What is the quotient of 4/9 ÷ 5/8?
Answer: 32/45
Question: What is the value of the digit 4 in 369.045?
Answer: The digit 4 is in the hundredths place. It has a value of 0.04, or 4 hundredths.
Question: What is the value of the digit 2 in 364.0782?
Answer: The digit 2 is in the ten-thousandths place. It has a value of 0.0002, or 2 tenthousandths.
Question: What is the sum of 3.42 + 2.76 + 3.2?
Answer: 3.42
2.76
+ 3.2
9.38
Question: What is the sum of 2.3 + 1.4 + 1.22?
Answer: 2.3
1.4
+ 1.22
4.92
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UNIT One Foundation
Question: What is the difference of 12.9 – 3.5 – 1.1?
Answer: 12.9
3.5
– 1.1
8.3
Question: What is the quotient of 6.6 ÷ 2.2?
Answer: To determine the quotient, first multiply the dividend (the first number) by 10:
6.6 ¥ 10 = 66. Next, multiply the divisor (the second number) by 10: 2.2 ¥ 10 = 22. Finally,
solve the problem: 66 ÷ 22 = 3.
Question: What is the quotient of 4.8 ÷ 1.2?
Answer: To determine the quotient, first multiply the dividend by 10: 4.8 ¥ 10 = 48. Next,
multiply the divisor by 10: 1.2 ¥ 10 = 12. Finally, solve the problem: 48 ÷ 12 = 4.
Question: What is the quotient of 4.9 ÷ 0.7?
Answer: To determine the quotient, first multiply the dividend by 10: 4.9 ¥ 10 = 49. Next,
multiply the divisor by 10: 0.7 ¥ 10 = 7. Finally, solve the problem: 49 ÷ 7 = 7.
Question: How do you write ¼ as a percent?
Answer: To write ¼ as a percent, first find the equivalent fraction with a denominator of
100: 1⁄4 ¥ 25/25 = 25/100. 25/100 = 25%. Another method is to divide the top of the fraction (the
numerator) by the bottom of the fraction (the denominator) and multiply the result by
100: 1 ÷ 4 = 0.25 ¥ 100 = 25%.
Question: How do you write 2/20 as a percent?
Answer: To write 2/20 as a percent, first find the equivalent fraction with a denominator of
100: 2/20 ¥ 5/5 = 10/100. 10/100 = 10%.
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CHAPTER 3 Basic Math Review
WRAP UP
77Chapter Summary
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Numbers can be represented as whole numbers, fractions, and decimals.
Each type of number can be used in addition,
subtraction, multiplication, and division.
Addition and subtraction involve adding numbers to each other or taking numbers away
from each other, respectively.
Multiplication involves enlarging the quantity
of number by multiplying a number by another number.
Division involves reducing the quantity of a
number to determine a part or portion of a
quantity needed.
Cross-multiplication is a mathematical concept
that is very useful to the pharmacy technician
when performing calculations.
Fractions are used to express a quantity or
portion that is a part of a whole number. It is
a ratio of a part to the whole. The numerator
is the number above the fraction line, while
the denominator is the number below the
fraction line. The denominator tells you how
many equal parts are in the whole or set. The
numerator tells you the number of those parts
you are expressing.
Decimals are numbers that are used to represent a place value. Any fraction can be represented using a decimal point.
Percentages are used to express the number of
parts of 100. A percentage represents the same
number as a fraction whose denominator is
100, and is denoted by the % symbol.
Fractions, decimals, and percentages can be
converted to each other and used in addition,
subtraction, multiplication, and division equations.
When adding fractions, first convert the fractions to the lowest common denominator (if
necessary). Then, add the numerators, placing
the sum over denominator. Finally, reduce the
resulting fraction.
When subtracting fractions, first convert the
fractions to the lowest common denominator
(if necessary). Next, subtract the numerators,
placing the difference over the denominator.
Finally, reduce the resulting fraction.
77 When multiplying fractions, first convert any
mixed numbers into improper fractions (if
necessary). Next, multiply the numerators to
obtain a new numerator. Then multiply the
denominators to obtain a new denominator.
Finally, reduce the resulting fraction.
77 When dividing fractions, first convert any
mixed numbers into improper fractions (if
necessary). Next, find the reciprocal of the divisor (the second fraction). Then multiply the
first number by the reciprocal. Finally, reduce
the resulting fraction.
77 Pharmacy technicians need to be comfortable
performing basic mathematical equations with
all the concepts in this chapter.
77Learning Assessment Questions
1. Which of the following fractions has the highest value?
A. 11⁄3
B. 2⁄3
C. 4/9
D. 5/9
2. What is a fraction with a value of less than 1
called?
A. Mixed number
B. Proper fraction
C. Improper fraction
D. Whole number
3. What is a fraction with a value greater than 1
called?
A. Mixed number
B. Proper fraction
C. Improper fraction
D. Whole number
4. What is 12/16, reduced to its simplest form?
A. 12/16
B. 6/8
C. 3⁄4
D. 4/7
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UNIT One Foundation
5. Which dose is larger?
A. 1⁄4 of a 300mg tablet
B. 1⁄2 of a 300mg tablet
C. 1⁄3 of a 300mg tablet
D. 1/10 of a 300mg tablet
6. What is the value of the digit 3 in 2,476.9432?
A. 0.3
B. 0.003
C. 0.03
D. 0.0003
7. A percentage represents the same number as a
fraction whose denominator is?
A. 50
B. 25
C. 100
D. 0
8. What is the product of 9 ¥ 7?
A. 54
B. 63
C. 72
D. 45
9. What is the sum of 10.3 + 2.7 + 3.65?
A. 16.65
B. 16.62
C. 12.67
D. 16.66
10.What is the value of the digit 7 in 3,045.2783?
A. 0.7
B. 0.07
C. 0.007
D. 70
11.What is the value of the digit 4 in the number
567.431?
A. 4 tenths
B. 4 hundredths
C. 4 thousandths
D. 40
12.What is the product of 0.25 ¥ 0.2?
A. 0.005
B. 50
C. 0.5
D. 0.05
13. 12/20 is the same as what percent?
A. 50%
B. 60%
C. 30%
D. 20%
14. 3⁄4 is the same as what percent?
A. 25%
B. 55%
C. 75%
D. 33%
15.Convert 0.40 to a fraction.
A. 8/10
B. 1⁄5
C. 4/10
D. 40/10
16.Convert 4/8 to a decimal.
A. 0.5
B. 0.2
C. 0.4
D. 0.8
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CHAPTER 3 Basic Math Review
WRAP UP
17.What is the difference in 12.8 – 3.4?
A. 9.4
B. 9.3
C. 6.6
D. 4.9
18.What is the product of 3.62 ¥ 3.3?
A. 11.469
B. 11.649
C. 11.946
D. 19.4
19.What is the sum of 987 + 0?
A. 0
B. 978
C. 980
D. 987
20.What is the quotient of 978,657 ÷ 0?
A. 978,657
B. 987,756
C. 0
D. 978,665
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