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BEPAABoard for the Education of People of African Ancestry
286 Convent Avenue New York, NY 10031Integrated Health Sciences ProgramFor H.S. students interested in pursuing careers in the health professions
MODULE II
Fundamentals of MathematicsWith clinical medicine, nursing and pharmacy application
Designed and Facilitated by
Marc Imhotep Cray, M.D.
mailto:[email protected]:[email protected]7/21/2019 BEPAA-Integrated Health Sciences (IHS) Curriculum, Fundamentals of Mathematics | Student Version
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2
There is an old adage that questions, Is it better to givea poor man a fish, or teach him how to fish?Of course, simply giving him the fish will satisfy his immediatehunger; but he will soon again be hungry. If you teach him how to
fish, on the other hand, he will never be hungry again.
Similarly, it is better to understand how to preform mathematicalfunctions in a general sense, and thereby feed yourself, than it is
to be dependent on someone else should an unfamiliar problem/equation come along.If you learn the basics of solving math problems and
equations, they will all become familiar.
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Fundamentals of Mathematics
3
Math is used extensively to solve problems and equations in the varioushealth professions
A thorough understanding of mathematics will be required as you pursueyour chosen health field
This mathematics review will help prepare you for the upcoming ChemistryModule lessons where math will be applied to understand Chemistry
concepts and principles and solving problems
Finally, this mathematic review module will contribute to your success oncurrent and upcoming high stakes exams, such as ACT, SAT, AP and Statestandardized test
Rational
MODULE II
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Unit Approach and Content
4
These Units/Lessons will review basic math concepts andapply math to the science concepts you have alreadylearned, including: learning dimensional analysis (conversion factor-method) drugs and dosage calculations and solving for physiologic variables like cardiac output,
stroke volume, and mean arterial blood pressure etc.
Companion Workbook/Clinical Calculations Ch. 1 and 2Source: Clinical Calculations Made Easy: Solving Problems Using DimensionalAnalysis, Fifth Edition; LLW 2012- Pg. 2-46
https://drive.google.com/file/d/0B-tlCbPSHvfZUUhjbDA0b3pfODA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZUUhjbDA0b3pfODA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZUUhjbDA0b3pfODA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZUUhjbDA0b3pfODA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZUUhjbDA0b3pfODA/edit?usp=sharing7/21/2019 BEPAA-Integrated Health Sciences (IHS) Curriculum, Fundamentals of Mathematics | Student Version
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Mathematics Units/Lessons
5
I. Basic Mathematical Definitions and FractionsII. Decimals, Percents, and Ratios
III. Exponents, Scientific Notation, and the Metric SystemIV.An Introduction to AlgebraV. An Introduction to EquationsVI.More Equation Forms
VII.Statistics and Graphs
Special Reading (cloud folded):Mathematicians of the African Diaspora-49 Special Articles and Web Pages
https://drive.google.com/file/d/0B-tlCbPSHvfZcDBLRlVPS21vYTA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZcDBLRlVPS21vYTA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZcDBLRlVPS21vYTA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZcDBLRlVPS21vYTA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZcDBLRlVPS21vYTA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZcDBLRlVPS21vYTA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZcDBLRlVPS21vYTA/edit?usp=sharing7/21/2019 BEPAA-Integrated Health Sciences (IHS) Curriculum, Fundamentals of Mathematics | Student Version
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Unit I
Basic Mathematical Definitions and Fractions
6
Objectives
Upon completion of this unit, you should be able to
Relate mathematics to everyday activities in the healthsciences and state the importance of a thoroughunderstanding of mathematics to a successful career in thehealth professions
Define basic numerical terms and types of numbersDefine and perform various operations with fractionalnumbers
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Key Terms
7
common denominator
common fraction
denominator
gram (g)improper fraction
least, or lowest, common
denominator (LCD)
liter (L)
meter (m)
metric system
mixed numbers
numerator
ophthalmometer (off-thal-MOM-et-er)proper fractions
reciprocal
respirometer (res-per-OM-et-er)
thermometer
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Mathematics in the health field
8
The ability to correctly calculate the dosage of a
drug for a patient may mean the difference betweenlife and death
Relate to mathematics: The vital signs of blood pressure, pulse, temperature, and respirationAll relate in some way to mathematics
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Mathematics in the health field (2)
9
Many measurements taken in
the health professions require
mathematical calculations.For example, by measuring theamount of air a patient is breathingover a period of time, you candetermine whether the patient needsto be placed on a breathing machine
(called a ventilator)
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Mathematics in the health field(3)
10
Many medical instruments are used tomeasure patient parameters and, thus,require the user to understandmathematicsFor example,
thermometer(thermos meaningtemperature, meter meaning to
measure) is used to measure
temperature; ophthalmometer(ophthalmus meaning
eyes) is used to measure eyes; and
spirometeris used to measure
breathing
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The Nurse and MathematicsEvery nurse must know and practice the six rights ofmedication administration including the
1. Right drug2. Right dose3. Right route4. Right time5. Right patient
6. Right documentation
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The Nurse and Mathematics (2)
Although the right drug, route, time, patient, and documentation maybe readily identified, the right dose requires arithmetic skills thatmay be difficult for you
This lesson reviews the basic arithmetic skills (multiplication anddivision) necessary for calculating medication dosage problemsusing the problem-solving method of dimensional analysis
Calculating the right dose of medication to be administered to apatient is one of the first steps toward preventing medication errors
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Basic ArithmeticReview
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ObjectivesAfter completing this lesson, you will successfully be able to:
1. Express Arabic numbers as Roman numerals.
2. Express Roman numerals as Arabic numbers.3. Identify the numerator and denominator in a fraction.4. Multiply and divide fractions.5. Multiply and divide decimals.6.
Convert fractions to decimals.
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Most medication dosages are ordered by the physician in metricand household systems for weights and measures usingtheArabic number system with symbols called digits (i.e., 1, 2, 3, 4,5)
Occasionally, orders are received in the apothecaries system ofweights and measures using the Roman numeral system withnumbers represented by symbols (i.e., I, V, X)
The Roman numeral system uses seven basic symbols, andvarious combinations of these symbols represent all numbers in theArabic number system
Arabic Numbers and Roman Numerals
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Arabic Numbers and Roman Numerals (2)
Correctly identifying Roman numerals will assist in preventingmedication errors
Some medication orders may include a Roman numeral.Example: Administer X gr of aspirin, which is correctly
interpreted as administer 10 gr of aspirin
However, according to the Institute for Safe Medication Practices(ISMP), abbreviations increase the risk of medication errors
While some health care providers may still use roman numerals andthe apothecaries system, the ISMP recommends using the metricsystem
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Arabic Numbers and Roman Numerals (3)Table includes the seven basic Roman
numerals and the corresponding
Arabic numbers
The combination of Roman numeralsymbols is based on three specific
principles:1. Symbols are used to construct a
number, but no symbol may be usedmore than three times
2. The exceptionis the symbol for five(V),which is used only once becausethere is a symbol for 10 (X) and acombination of symbols for 15 (XV)
Also see Cloud Folder Notes:More RN Rules The Romannumerals controversy
https://drive.google.com/file/d/0B-tlCbPSHvfZOVZYcnEzRUxPaUE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZOVZYcnEzRUxPaUE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZOVZYcnEzRUxPaUE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZOVZYcnEzRUxPaUE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZOVZYcnEzRUxPaUE/edit?usp=sharing7/21/2019 BEPAA-Integrated Health Sciences (IHS) Curriculum, Fundamentals of Mathematics | Student Version
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Roman Numerals Subtractive Principle
18
Any numeral is counted positively unless there's a larger numeralanywhere to its right, in which case it is counted negatively
However, proper Roman numbers are subject to the followingrestrictions about the applicability of the subtractive principle
The use of the subtractive principle has always been optional. Its
systematic use is fairly modern
For example, it's acceptable to use IIII instead of IV, as is usuallydone on clockfaces (to "balance" their left and right halves, so
we're told).Source: Final Answers 2000-2014, Grard P. Michon, Ph.D.http://www.numericana.com/answer/roman.htm#convert
http://www.numericana.com/answer/roman.htmhttp://www.numericana.com/answer/roman.htmhttp://www.numericana.com/answer/roman.htm7/21/2019 BEPAA-Integrated Health Sciences (IHS) Curriculum, Fundamentals of Mathematics | Student Version
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Roman Numerals Subtractive Principle
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The subtractive principle (asubtrahend preceding a minuend) may apply:
Only to a numeral (the subtrahend) that is a power of ten (I, X or C).For example, "VL" is not a valid representation of 45 (XLV is correct)
Only when the subtrahend precedes a minuend no more than ten times larger
For example, "IL" is not a valid representation of 49 (XLIX is correct)
Only if any numeral preceding the subtrahend is at least ten times largerFor example, "VIX" is not a valid representation of 14 (XIV is correct), and
"IIX" is not correct for 8 (VIII is correct).
Only if any numeral following the minuend is smaller than the subtrahendFor example, "XCL" is not a valid representation of 140 (CXL is correct)
Modified from Final Answers 2000-2014, Grard P. Michon, Ph.D.http://www.numericana.com/answer/roman.htm#convert
http://www.numericana.com/answer/roman.htmhttp://www.numericana.com/answer/roman.htmhttp://www.numericana.com/answer/roman.htm7/21/2019 BEPAA-Integrated Health Sciences (IHS) Curriculum, Fundamentals of Mathematics | Student Version
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Arabic Numbers and Roman Numerals (4)
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PracticeExercise 1.1 ArabicNumbers and Roman
Numerals
Companion Workbook/Clinical Calculations Ch. 1 and 2Source: Clinical Calculations Made Easy: Solving Problems UsingDimensional Analysis, Fifth Edition; LLW 2012- Pg. 2-46
https://drive.google.com/file/d/0B-tlCbPSHvfZUUhjbDA0b3pfODA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZUUhjbDA0b3pfODA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZUUhjbDA0b3pfODA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZUUhjbDA0b3pfODA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZUUhjbDA0b3pfODA/edit?usp=sharing7/21/2019 BEPAA-Integrated Health Sciences (IHS) Curriculum, Fundamentals of Mathematics | Student Version
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Fractions
A common fraction is a comparison of two numbers.Fractionsusually are written , where ais called thenumeratorand bthe denominator.The denominator, or bottom number, tells how many total
partsit takes to make the wholeThe numerator, or top number, is the actual numberofparts of a whole being considered
QUOTES & NOTESFractionshave been used for more than 3,600 years.
They appear in an Egyptian handbook of mathematics
called the Rhind papyrus, written around the year 1650 B.C.
http://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus7/21/2019 BEPAA-Integrated Health Sciences (IHS) Curriculum, Fundamentals of Mathematics | Student Version
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Fractions
23
A fraction is a comparison of parts
(numerator) to a whole (denominator)
Source: Delmar/Cengage Learning
S f F i i H l h
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Summary of Fractions in HealthcareMedication dosages with fractions are occasionally ordered by the physician or used
by the pharmaceutical manufacturer on the drug label.
A fraction is a number that represents part of a whole number andcontains three parts:
1. Numeratorthe number on the top portion of the fractionthat represents the number of parts of the whole fraction
2. Dividing linethe line separating the top portion of thefraction from the bottom portion of the fraction
3. Denominatorthe number on the bottom portion of thefraction that represents the number of parts into which thewhole is divided
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Multiplying Fractions
The three steps for multiplying fractions are:1. Multiply the numerators2. Multiply the denominators
3. Reduce the product to the lowest possible fraction
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Multiplying Fractions(2) Examples
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PracticeExercise 1.2 Multiplying Fractions
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Dividing Fractions
The four steps for dividing fractions are:1. Invert (turn upside down) the divisor portion of theproblem (the second fraction in the problem)2. Multiply the two numerators.
3. Multiply the two denominators4. Reduce answer to lowest term (fraction or whole number)
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DividingFractions (2)
P ti
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PracticeExercise 1.3Dividing Fractions
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Simplifying Fractions to Lowest Terms Being able to simplify a fraction to the lowest equivalent fraction
(i.e., to the lowest terms) is extremely important
A most fundamental rule of fractions states that the numerator(the top number) and the denominator (the bottom number) canbe divided or multiplied by the same nonzero number without
changing the value of the fraction
Source: Modified from An Integrated Approach to Health Sciences 2e, Delmar Cengage Learning 2012: pg. 198
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Stop and Review
a. If 13 pills remain in a bottle originally containing 40 pills, what
fraction would apply?
b. If you gave a patient 20 mL of a 100-ml bolus (a concentratedamount of a drug) of Lidocaine, what fraction of the drug have yougiven to the patient?What fraction of the drug is left?
c. A bottle contains 300 mL of liquid. If 200 mL were given to apatient, what fraction does this represent? What fraction remains?
d. A bottle contains 800 tablets. The pharmacist asks that 350 tabletsbe sent to the nurses station. What fraction of the bottle was sent?
R l Lif
I d A li ti
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Source: Modified from An Integrated Approach to Health Sciences 2e, Delmar Cengage Learning 2012: pg. 207
Real Life Issues and ApplicationsHard Decisions with Fractions
Suppose for the sake of argument that you were chosen from the total population of theUnited States to dole out $100 billion for health care. This $100 billion would be the totalhealth care budget for our nation for 1 year. What fraction of this amount would you spend on new equipment for research or on new
equipment for diagnostics and treatment of disease? What fraction would you spend to find a cure for AIDS or a cure for cancer? What about prenatal care or the care of premature infants? How would you justify giving to one group and not to another? Would giving an equal amount to each group be a good idea, or would the resulting
lower amount of money given to each group prevent any one group from accomplishinganything?
Using what you have learned about fractions, devise a simple budget showing the actual amountof money to be given to each area of health care that you think is important. Then, explain thereasons for your decisions. If you have difficulty deciding to whom to give the money or even
how to divide up the money, you are not alone. State and federal governmental agencies have to
make these tough decisions on a regular basis.
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DecimalsMedication orders are often written using decimals, and
pharmaceutical manufacturers may use decimals when
labeling medications.
Therefore, you must understand the learning principles involving
decimals and be able to multiply and divide decimals.
N.B. A decimal point is preceded by a zero if notpreceded by a number to decrease the chance of an error
if the decimal point is missed
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Decimals(2)
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Rounding Decimals
Decimals may be rounded off If the number to the right of the decimal is greater than or
equal to 5, round up to the next number
If the number to the right of the decimal is less than 5, deletethe remaining numbers
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DECIMALS & PREVENTING MEDICATION ERRORS
Understanding the importance of a decimal point will assist in preventingmedication errors.
An improper placement of a decimal point can result in a serious
medication error
According to the Institute for Safe Medication Practices (ISMP):
Trailing zeros should not be used with whole numbersExample: Administer 1 mg of Xanax
WHY? If a decimal point and a zero are placed after the number (1.0 mg),the order could be misread as Administer 10 mg of Xanax
Leading zeros shouldalways
precede a decimal point when thedosage is not a whole numberExample: Administer 0.125 mg of Lanoxin
WHY? If a zero is not placed in front of the decimal point the order couldbe misread as Administer 125 mg of Lanoxin
Reference: Companion Workbook/Clinical Calculations Ch. 1 and 2
http://ismp.org/https://drive.google.com/file/d/0B-tlCbPSHvfZUUhjbDA0b3pfODA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZUUhjbDA0b3pfODA/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZUUhjbDA0b3pfODA/edit?usp=sharinghttp://ismp.org/http://ismp.org/http://ismp.org/7/21/2019 BEPAA-Integrated Health Sciences (IHS) Curriculum, Fundamentals of Mathematics | Student Version
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PracticeExercise 1.4 Rounding Decimals
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Multiplying Decimals When multiplying with decimals, the principles of
multiplication still apply
The numbers are multiplied in columns, but the numberof decimal points are counted and placed in the
answer, counting places from right to left
The answer to the problem before adding decimalpoints is 345 but when decimal points are correctlyadded (two decimal points are added to theanswer, counting two places from the right to theleft) then 3.45 becomes the correct answer
Practice
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PracticeExercise 1.5 Multiplying Decimals
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Dividing Decimals
When dividing with decimals, the principles of division still apply,
except that the dividing number is changed to a whole numberby moving the decimal point to the right
The number being divided also changes by accepting the samenumber of decimal point moves
N.B. Division problems can be written as
"Dividend Divisor = Quotient" or
"Dividend / Divisor = Quotient." Because multiplication is the inverse of
division, you can turn the equationaround to read
"Quotient * Divisor = Dividend."
F f ll t lk th h
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For full steps walk throughsee or your cloud
http://www.math.com/school/subject1/lessons/S1U1L6GL.html
Basic Math & Pre-Algebra for Dummies.pdf
http://www.math.com/http://www.math.com/https://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttp://www.math.com/school/subject1/lessons/S1U1L6GL.htmlhttp://www.math.com/school/subject1/lessons/S1U1L6GL.htmlhttps://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttp://www.math.com/school/subject1/lessons/S1U1L6GL.htmlhttp://www.math.com/https://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttp://www.math.com/school/subject1/lessons/S1U1L6GL.html7/21/2019 BEPAA-Integrated Health Sciences (IHS) Curriculum, Fundamentals of Mathematics | Student Version
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Source: Basic Math & Pre-Algebra for Dummies.pdf
https://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharing7/21/2019 BEPAA-Integrated Health Sciences (IHS) Curriculum, Fundamentals of Mathematics | Student Version
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Source: Basic Math & Pre-Algebra for Dummies.pdf
P i
https://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZSG53OWRXelgwckE/edit?usp=sharing7/21/2019 BEPAA-Integrated Health Sciences (IHS) Curriculum, Fundamentals of Mathematics | Student Version
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PracticeExercise 1.6 Dividing Decimals
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Converting Fractions to Decimals
When problem solving with dimensional analysis, medicationdosage calculation problems may frequently contain both fractionsand decimals Some of you may have fraction phobia and prefer to convert
fractions to decimals when solving problems
RULES:
To convert a fraction to a decimal, divide the numerator portionof the fraction by the denominator portion of the fraction
When dividing fractions, remember to add a decimal point and azero if the numerator cannot be divided by the denominator
Converting Fractions to Decimals (2)
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Converting Fractions to Decimals (2)Example
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Understanding the importance of converting fractions to decimals
will assist in preventing medication errors
Many medication errors occur because of a simple arithmetic errorwith dividing
Every nurse should have a calculator to recheck answers foraccuracy. N.B. If a recheck results in a different answer, the next recheck
should include consulting with another nurse or pharmacist
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PracticeExercise 1.7 Converting Fractions to Decimals
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Basic Units ofMeasurement
Exponents, Scientific Notation, and the Metric System
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Upon completion of this lesson, you should be able to
Evaluate numbers having positive and negative exponentsRepresent numbers in scientific notation
Change units within the metric systemConvert units between systems of measurement
Objectives
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Key Terms
52
basecenti
deca (da)
deci (d)
English system ofmeasurement
exponent
factor-label method
giga (G)
gram (g)
hecto (h)
kilo (k)
liter (L)
mega (M)
meter (m)
metric system of measurement
micro (mc)milli (m)
nano (n)
pico (p)
scientific notationunits
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Introduction
53
In health care we deal with very large numbers, such as thenumber of cells in your body, or very small numbers, such as themicroscopic size of the cell
Also, health care professionals regularly use the metric system ofmeasurement in their clinical practice
Therefore, this unit focuses on scientific notation and the metric
system of measurement,both of which are based on the powerof 10, and therefore can be related to the decimal system,discussed earlier and to follow
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Introduction(2)
54
Working with scientific notation and the metric system
requires a basic understanding of the powers of 10 and of
where to move the decimal point
This is especially true for conversions within the metric system
Also discussed in this unit are conversions between the Englishsystem of measurement, which is used predominately in UnitedStates, and the metric system of measurement, which is used
worldwide and in the health professions
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Exponents and Base
55
The health professions often require use of both extremely largenumbers and extremely small numbers
For example, there are about 25,000,000,000 blood cells
circulating in an average adults body, and many organismsfound in the body are microscopic in size
Thus, it convenient to write these numbers in an abbreviated
form known asscientific notation
Doing this requires an understanding of certain terminology
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Scientific notation terminology
Exponents and Base
56
Consider the expression bn, where bis called the baseand nthe exponent The nrepresents the number of times that bis
multiplied by itself: bn is read the nth power of b. Exponents of 2 and 3 are read squared and cubed
That is,32
is read 3 squared
, while43
is read 4 cubed
Examples:
34= 3 3 3 3 = 81
5
3
= 5 5 5 = 125
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Stop and Review
57
Evaluate each of the following
a. 32b. 43
c. 25d. 103e. 105
Note: When the exponent is positive, the resultingnumber is relatively large
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SN for small numbers less than 1
58
Small numbers less than 1 canalso be represented in scientificnotation In these cases, negative
exponents are used A negative exponent
can be thought of as afraction
To do this, you simply put binthe denominator and thepositive version of nin thenumerator
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Stop and Review
59
Write each of the following using a positive
exponent, and then change each to a fractional form
a. 4 3
b. 3 4c. 10 1d. 10 2
e. 104
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The Power of 10Base 10
60
Of special interest are the powers of 10, or base 10
When b is 10, the number of zeros in the answer
equals the exponent
Example:102= 10 10 = 100103= 10 10 10 = 1,000104= 10 10 10 10 = 10,000105= 10 10 10 10 10 = 100,000
Evaluate each of the followinga. 106b. 107
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Scientific Notation(SN)
61
To write a number in scientific notation, write it as a product of anumber greater than or equal to 1 and less than 10, with acorresponding power of 10 In other words, only one number should be to the left of the
decimal point
As an example, consider those 25,000,000,000 blood cellsin your body We can shorten this number to 2.5 1010
The exponent is 10 because in order to ensure that only one
number is to the left of the decimal point, we need to movethe decimal point 10 placesfrom the end of the number tobetween the 2 and 5
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The process of scientific notation illustrated
62
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Scientific Notation Summary
63
So, in scientific notation, the decimal point is moved so that onlyone nonzero digit is to the left of the decimal; the number ofplaces that the decimal is moved is the exponent
Further, as is illustrated in the preceding example and slide , if the decimal point is moved to the left, the exponent is
positive; if the decimal point is moved to the right, the exponent
is negative Notice also that
the exponent is positive if the number is large and negative if the number is small (less than 1)
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Systems of Measurement:Basic Definitions
64
Several systems are used for measuring.
The one we commonly use in the United States is theEnglish system
In the English system, distanceis measured in inches, feet, yards, and miles; weightis measured in ounces, pounds, and tons; and volumeis measured in pints, quarts, and gallons
English system is not the system of choice usedthroughout the world and within the health professions
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International System of Units
65
Most scientific and medical measurements utilize what is commonlyreferred to as the metric systemThe system accepted as the metric system is the InternationalSystem of Units
The metric system uses only three basic units of measure:1. the meter(m) for length,2. the liter(L) for volume, and3. the gram(g) for weight
The metric system has only one basic unit of measure for each typeof measurement.Compare this to the English system, which has many units of measurefor distance, volume, and weight
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Source: Modified from An Integrated Approach to Health Sciences 2e , Delmar
Cengage Learning 2012: FIGURE 1-14, pg. 197
In the metric system, the prefixes are used to indicate the differentlengths, volumes, and weights The conversion is the same for each type of measurement.
1. 1,000 meters in a kilometer,2. 1,000 grams in a kilogram, and
3. 1,000 liters in a kiloliter.For now, just be aware that a milliliter (mL) is a unit of volume, acentimeter (cm) a unit of length, and a kilogram (kg) a unit ofweight
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Systems of Measurement (3 systems)Metric System, Apothecaries System and Household System
67
1. The Metric System The metric system is a decimal
system of weights and measures
based on units of ten in whichgram,meter, and liter are the basic unitsof measurement
However, gram and liter are
the only measurements from themetric system that are used inmedication administration
The most frequently usedmetric units of weight and theirequivalents are summarized:
1 kilogram (kg)1 gram (g)1 milligram (mg)---------------------
1 microgram (mcg)1 kg= 1000 g1 g= 1000 mg1 mg= 1000 mcg
M t i t it f i ht d
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Metric system units of weight andequivalents
Source: Clinical Calculations Made Easy: SolvingProblems Using Dimensional Analysis, Fifth Ed:32
Metric System Units of Volume &
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Metric System Units of Volume &equivalents
1 liter (L)1 milliliter (mL)
1 L 1000 mLSource: Clinical Calculations MadeEasy: Solving Problems UsingDimensional Analysis, Fifth Ed:32
2 Th A th i S t
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2. The Apothecaries System
The apothecariessystem is a system ofmeasuring and
weighing drugs andsolutions in whichfractions are used to
identify parts of the
unit of measure
The basic units ofmeasurement in theapothecaries system
include weights andliquid volume
The Apothecaries System (2)
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e po eca es Sys e ( )
Apothecaries System Units of
Volume and Equivalents1 gallon (gal)1 quart (qt)1 pint (pt)1 fluid ounce (fl oz)
1 fluid dram (fl dr)1 minim (M)1 gal= 4 qt
1 qt =2 pt
1 pt =16 fl oz1 fl oz = 8 fl dr
1 fl dr = 60 M
1 fl oz =1 oz
1 fl dr = 1 dr
Apothecaries System Units of
Weight and Equivalents
1 pound (lb)1 ounce (oz)1 dram (dr)1 grain (gr)
1 lb =16 oz1 oz =8 dr
1 dr =60 gr
Apothecaries system of equivalents for weight and
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Apothecaries system of equivalents for weight andvolume Illustrated
Source: Clinical Calculations Made Easy: SolvingProblems Using Dimensional Analysis, Fifth Ed:35
Avoirdupois
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3. The Household System
The use of household measurements is consideredinaccurate because of the varying sizes of cups, glasses,and eating utensils, and this system generally has beenreplaced with the metric system.
However, as patient care moves away from hospitals,which use the metric system, and into the community, it isonce again necessary for the nurseto have an
understanding of the household measurement system tobe able to use and teach it to clients and families.
H h ld t t d
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Household measurement system andequivalents for volume Illustrated
Source: Clinical Calculations Made Easy: Solving ProblemsUsing Dimensional Analysis, Fifth Ed:32
Drugs and Dosage
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Drugs and DosageFormulas and Conversions
Volume60 minims = 1 dram = 5cc = 1tsp4 drams = 0.5 ounces = 1tbsp8 drams = 1 ounce16 ounces = 1pt.32 ounces = 1qt.
Drugs and Dosage
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Weight60 grains = 1dram1/60 grain=1mg8 drams = 1 ounce15 grains=1g12 ounces = 1 lb. (Troy weight/ Apothecaries )
2.2 lbs.=1kg
Drugs and DosageFormulas and Conversions (2)
NIST General Tables of Units of Measurement
Troy weight is a system of units of mass customarily used for precious metalsand gemstones. There are 12 troy ounces per troy pound rather than the 16ounces per pound found in the more common avoirdupois system.
D d D
http://localhost/var/www/apps/conversion/tmp/scratch_7/NIST%20General%20Tables%20of%20Units%20of%20Measurement.pdfhttp://localhost/var/www/apps/conversion/tmp/scratch_7/NIST%20General%20Tables%20of%20Units%20of%20Measurement.pdf7/21/2019 BEPAA-Integrated Health Sciences (IHS) Curriculum, Fundamentals of Mathematics | Student Version
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Household pothecary1tsp = 1 dram1tsp = 60 gtts (drops)
3tsp = 0.5 ounce1tbsp = 0.5 ounce
Drugs and DosageFormulas and Conversions (3)
D d D
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Household Apothecary Metric
1 tsp=5cc 1fl.dram=4cc 5cc=1tsp
3 tsp=1tbsp 4 drams=0.5oz 15cc=1tbsp1tbsp=0.5oz or15cc 8 drams=2tbsp (1oz) 30cc=2tbsp(1oz)2tbsp=1oz or 30cc 16 minims=1cc 1cc=16 minims
1pt.=16oz or 480cc 500cc=0.5L or 1pt.1qt=32oz or 960cc 1000cc=1L or 1qt.
Drugs and DosageFormulas and Conversions (4)
http://www.nurse-center.com/studentnurse/nur11.html
T C i M h d
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Temperature Conversion Method
To convert from Fahrenheit to Celsius:
C (F 32) 1.8To convert from Celsius to Fahrenheit:
F C 1.8 + 32
C temperature in degrees CelsiusF temperature in degrees Fahrenheit
ABBREVIATIONS USED IN CLINICAL CALCULATIONS
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Factor-Label Method of Units Conversion
81
The factor-label method works for all conversions of units, even
outside of the metric system
To use this method, simply write the original quantity as a fraction,and then treat the units as is done in the multiplication of fractions In other words, cancel the units
---------------------In Nursing Clinical Calculations this is called:Dimensional Analysis or the Conversion Factor Method
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Factor-Label Method of Units Conversion (2)
82
For example, to change from hours to seconds, first write hours as afraction over 1You may not know how many seconds are in one hour, but you do knowhow many minutes are in one hour
so you first convert to minutes by making a fraction that
represents this Since we are solving for seconds, we need to determinehow many
seconds are in a minute by making a representative fraction The next step is to cancel the like units that divide out
The final conversion should look like this:
Factor-Label Method
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Factor Label Methodof Units ConversionExamples
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Dimensional Analysis Terms and Steps
84
TERMS USED IN DIMENSIONAL ANALYSIS Given quantitythe beginning point of the problem Wanted quantitythe answer to the problem Unit paththe series of conversions necessary to achieve the answer to the problem. Conversion factorsequivalents necessary to convert between systems of measurement andallow unwanted units to be canceled from the problem
THE FIVE STEPS OF DIMENSIONAL ANALYSIS
1. Identify the given quantity in the problem2. Identify the wanted quantity in the problem3. Establish the unit path from the given quantity to the wanted quantity using equivalents asconversion factors
4. Set up the conversion factors to allow for cancellation of unwanted units5. Multiply the numerators, multiply the denominators, and divide the product of the numeratorsby the product of the denominators to provide the numerical value of the wanted quantity
Source: Clinical Calculations Made Easy: Solving Problems Using Dimensional Analysis,
Fifth Edition; LLW 2012- Pg. 48-59
DIMENSIONAL
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DIMENSIONALANALYSIS
Source: Clinical Calculations Made
Easy: Solving Problems Using
Dimensional Analysis, Fifth Edition;
LLW 2012- Pg. 48-59
DIMENSIONAL
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DIMENSIONALANALYSIS
Source: Clinical Calculations Made Easy:
Solving Problems Using DimensionalAnalysis, Fifth Edition; LLW 2012- Pg.48-59
Practice Dimensional Analysis
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Practice Dimensional AnalysisSource: Clinical Calculations Made Easy: Solving Problems Using Dimensional Analysis,
Fifth Edition; LLW 2012- Pg. 48-59
Learn and Practice moreSolving Problems Using DimensionalAnalysis-Ch. 3-Clinical Calculations.pdf
E l
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Example
88
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Decimals, Percents,and Ratios
89
Decimals, Percents, and Ratios
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Decimals, Percents, and Ratios
Objectives
Upon completion of this chapter, you should be able toWrite decimals in word formEstimate values for operations involving decimals and use acalculator to find the results
Write equivalent numbers as percents, fractions, and decimalsRepresent fractions as ratios in simplest formDifferentiate the terms solute, solvent, and solutionDifferentiate between W/V solutions and V/V solutions andexpress each as percent solutions and ratio solutionsCalculate unit price and rate of flow
Reading:
Colbert BJ et. Al. Chapter 15 DECIMALS, PERCENTS, AND RATIOS , An Integrated
Approach to Health Sciences 2e, Delmar Cengage Learning 2012: 209-19
Key Terms
https://drive.google.com/file/d/0B-tlCbPSHvfZWHdSaTE4eG9Rcm8/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZWHdSaTE4eG9Rcm8/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZWHdSaTE4eG9Rcm8/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZWHdSaTE4eG9Rcm8/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZWHdSaTE4eG9Rcm8/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZWHdSaTE4eG9Rcm8/edit?usp=sharinghttps://drive.google.com/file/d/0B-tlCbPSHvfZWHdSaTE4eG9Rcm8/edit?usp=sharing7/21/2019 BEPAA-Integrated Health Sciences (IHS) Curriculum, Fundamentals of Mathematics | Student Version
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Key Terms
decimals
intake and output (I&O)percents
percent solution
ratios
solute
solution
solvent
unit priceV/V solution
W/V solution
Decimals Ratios and Percents
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Decimals, Ratios, and Percents In the previous lesson, we learned about fractions
There are, however, other ways to represent parts of awhole These include decimals, ratios, and percents.
In the health field, these representations tendto be used more often than fractions are used.
Blood Tests and Decimals
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Blood Tests and Decimals For example, blood tests reveal information about a patients
health Tests are performed in laboratories by medical laboratory technicians and
results are analyzed by clinicians
Many of the levels are reported in decimal form. The presence of abnormal values can often be of diagnostic significance
for specific disease states An analysis of red blood cells, (carry oxygen to the tissues),
would shows a value within the range of 4.5 to 6.3 million cellsper milliliter If the number of red blood cells is too low, the patient may have a
condition called anemiawhere the bodys tissues may not get enoughoxygen.
This can be the result of a deficiency in iron or certain vitamins
Blood Tests and Decimals(2)
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Blood Tests and Decimals(2)Another type of cell found in the blood is the white blood cell
This type of cell is important for the body to fight infections. A normalwhite blood cell (WBC) count is between 4.5 and 11.0
thousand cells per milliliter (thou/mL) A level that is too high might indicate diseases such as a
bacterial infection or leukemia a level that is too low might indicate bone marrow disease.
So you can see why a thorough understanding of decimals is important
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Writing Decimals in Word Form
Given the widespread use of calculators, a complete understandingof decimals has never been more important. Using a calculator, itcan easily be seen that , which equals 3 4, can be written as0.75The decimal 0.75 is read seventy-five hundredths, which can bewritten as the fraction This, of course, simplifies to
W i i D i l i W d F (2)
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Writing Decimals in Word Form (2)
In math, the symbol {} means and so forth.Any fraction having a denominator of {10, 100,1,000, 10,000 } can easily be written as a decimal
The decimal system (showingdi l l )
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corresponding place values) In each of the preceding examples, notice that the
number of zeros in the denominator of the fraction isthe same as the number of places to the right of thedecimal point, which, in turn, corresponds to the way
the fraction is read
Decimal point read as the word and
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Another important use of the decimal point is to separatethe whole number from the portion that is less than 1 The decimal point is read as the word and
For example, 10.23 is read ten andtwenty-threehundredths
Another Example:
0.104 is read one hundred four thousandths100.004 is read one hundred andfour thousandths
Stop and Review
Write each of the following decimalsin words.a. 0.23b. 1.003c. 104.2
Decimal Operations and Estimation
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p
Because you will likely use a
calculator to perform problemsinvolving decimals, no need to spendmuch time discussing mathematicaloperations involving decimals
You should learn enough, however, todetermine whether answers obtainedvia use of a calculator arereasonable Will help you catch errors that
result from hitting wrong calculator
keys
For example, if you use acalculator to add 2.1 + 5.06and the result is 52.7, you shouldbe able to draw the conclusionthat an error has occurred You do this through
estimation
Given that 2.1 is approximately2 and 5.06 is approximately5,the sum of these two numbersshould be approximately 7 Estimation, thus, gives you
an idea of reasonableness
of an answer
Estimation
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examples(rounding off)
Stop and Review
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Stop and Reviewa. Estimate the value of each of the following, and then use a
calculator to determine the exact value.
2.01 + 3.66 + 15.1
4.02 16.03
b. Over an 8-hour period, a pediatric patients fluid intakeand urinary output (intake and output, or I&O) weremeasured. The child ingested 200.2 mL, 150.0 mL, and 25.1
mL of fluid; the childs urine output was 130.5 mLand 170.5 mLWhat was the childs fluid balance (the difference between
the I&O) for that 8-hour period?
Percents
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Percents are used frequently in the health fieldFor instance, drug concentrations often are represented as
percents (e.g., 2% [two percent] solutions), and percentages areoften used with burn patientsto describe how much of thepatients body has been burned (e.g., 5% of body surface)
nondepolarizing neuromuscular blocking agent
Changing Percents to Fractions
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In physiology we learned the importance of oxygen; if there were no
oxygen, we would not exist
But do you know how much of our atmosphere is composed of oxygen? The answer is approximately 21%, or 21 parts out of 100 parts
Carbon dioxide makes up 0.03% of our atmosphere, or 0.03 partsout of 100 parts
The most abundant gas is nitrogen, which composes approximately79% of our atmosphere, or 79 parts out of 100 parts
the symbol % represents the word percent, which means out of 100
-----------------------
A percent is a simple means of writing any
fraction having a denominator of 100
Changing Percents to Fractions (2)
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For example, 50% means or
50 (parts) out of 100 (parts)
To change a percent to afraction, simply write the percentas a fraction having a
denominator of 100; rememberto simplify the fraction
Stop and ReviewWrite each of the following percents
as fractions in the simplest forma. 40%b. 200%c. 75%
Changing Percents to Decimals
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C g g e ce o ec
To change a percent to a decimal, move the
decimal point two places to the left Remember, a whole number is considered to
have a decimal point to the right of it
Example:25% = 0.25 Stop and Review
Change each of the following
percents to decimals
a. 15%b. 300%c. 37.5%
Changing Decimals to Percents
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To change a decimal to a percent, move the
decimal point two places to the right, for example, 0.35= 35%
Note that it may be necessary to add zeros
for example, 0.6 = 60%
Change each of the following decimals to a percent
a. 0.15b. 0.2c. 1.25
Changing Fractions to Percents
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To change a fraction to a percent, first
change the fraction to a decimal, and thenchange the decimal to a percent Use a calculator when appropriate
QUOTES & NOTES
Practical Application Did you ever
wonder if there is an easy way to
figure out a 15% tip at a restaurant?Just move the decimal point on the bill
one place to the left to find 10%
Then take half of that 10% to arriveat the remaining 5%
Add those two numbers, and youve
got 15%!
Changing Mixed-Number
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Percents to Simple Fractions
Sometimes percents are written using mixed numbers(i.e., whole numbers along with fractions), such as
To change a mixed-number percent to a simple fraction, firstchange the mixed number to an improper fraction
Next, divide by 100 and simplify. (Remember, percentmeans out of 100)
Thus, is the same as the fraction
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One of the most common therapies is oxygen therapy
As you breathe, you are breathing room air which contains 21%oxygen
Certain disease states and environmental conditions maynecessitate increasing the amount of oxygen delivered to your
body tissues For example, at sea level the atmospheric pressure (PB ) is 760 mm Hg(millimeters of mercury)
We can find the available oxygen by multiplying the atmospheric pressureby the oxygen percent
To perform the calculation, you must convert the percentage into the decimal
formTherefore, the answer becomes:
760 mm Hg .21 (decimal form of 21% oxygen) =159.6 mm Hg of oxygen
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If we want to increase the amount of available oxygen, we can
place the patient on a higher-percentage device such as anoxygen mask set at 50%
This patient would now have more oxygen available as follows:760 mm Hg .50 = 380 mm Hg of oxygen
An environmental example is mountain climbing
As you climb above sea level, the atmospheric pressure becomesless
Lets assume you have reached a height where it has dropped to
600 mm Hg The oxygen now available is: 600 mm Hg .21 = 126 mm Hg
of oxygen
Ratios
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Ratios Another way to show the relationship between numbers is to
use ratios A ratio is simply a comparison between two numbers
For example, the fraction can be written as the ratio 1:2
Thus, the fraction is the same as the ratio 1:2, the decimal 0.50,and the percent 50% All of these are different ways of expressing the same value
As is the case with fractions, ratios can be simplified As with fractions, simply divide both numbers in the ratio by
the largest number that divides evenly into both
Equivalent expressions for fractions,
ratios, decimals, and percentages
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Ratios (2)
Example:For the ratio of 50:2, the largest number that will divideevenly into both 2 and 50 is 2Dividing both numbers in the original ratio by 2 yields a
simplified ratio of 25:1------------------Simplify each of the following ratios
a. 50:5b. 30:15c. 100:1
Ratios (3)
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Another use of ratios is to find the individual cost or unit price of anitem-that is, the price of one item
To find the unit price, set up a fraction,
Example:
A bottle of 100 aspirins sells for$7.50, while a bottle of 150aspirins of the same brand sells for$12.00. Which is the better buy?
Since each aspirin in the 100-tabletbottle costs less, it is the better buy
Stop and Review
a. An IV dispenses fluid at a rateof 750 mL in 5 hours. How much
liquid is dispensed in 1 hour?How much in 3 hours?
b. Which can of coffee is the
better buy, a 42-ounce can for$13.00 or a 12-ounce can for$4.00?
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114
Real Life Issues and Applications With normal breathing, it takes twice as long to exhale as it does
to inhale This can be expressed by the ratio 1:2
Certain disease states of the lungs can alter this ratio. Patient A has a 2-second inspiratory (I) time and a 4-second
expiratory (E) time
Patient B has a 2-second I time and a 1-second E time
What are their respective I:E ratios?
Which patient has a normal ratio?
CALCULATING ADULT DOSAGE
Mathematics and Dosage Calculations
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When a physician orders a medication and the dosageordered is not the same as what is available, you mayuse two additional methods to determine the correct
dosage. First is the proportional method, which may be used to
solve almost any type of dosage problem, and second isthe formula method.
Both of these methods are described in the following slides
CALCULATING ADULT DOSAGE(Alternatives to dimensional analysis)
Reference: Rice J. Unit 5 Calculating Adult Dosages: Oral and Parenteral Forms.
In Principles of Pharmacology for Medical Assisting; Cengage Learning; 5 ed. 2010:51-8
Calculating Adult Dosages Using theMathematics and Dosage Calculations
http://www.amazon.com/Principles-Pharmacology-Medical-Assisting/dp/1111131821http://www.amazon.com/Principles-Pharmacology-Medical-Assisting/dp/1111131821http://www.amazon.com/Principles-Pharmacology-Medical-Assisting/dp/1111131821http://www.amazon.com/Principles-Pharmacology-Medical-Assisting/dp/11111318217/21/2019 BEPAA-Integrated Health Sciences (IHS) Curriculum, Fundamentals of Mathematics | Student Version
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Proportional Method
The physician orders 0.2 g of meprobamate tabs. The dose onhand is 400 mg tabs
Step 1. Determine whether the medication ordered and themedication on hand are available in the same unit of measure
Step 2. If the medication ordered and the medication on hand arenot in the same unit of measure, convert so that both measures areexpressed using the same unit of measure
Conversion: To change 0.2 g to mg
1000 mg : 1 g =x mg : 0.2 gX=200 mg
ormultiply 0.2 x1000 =200
Proportional Method cont
Mathematics and Dosage Calculations
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Step 3. Now use the following proportion to calculate the dosage.
Remember, you converted 0.2 g to 200 mg
Proportional Method cont.
Step 4. Prove your answer. Remember to place your answer in theoriginal formula in thex position
Calculating Adult Dosages Using the
Mathematics and Dosage Calculations
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g g gFormula Method
The physician orders 0.2 g of meprobamate tabs. The dose onhand is 400 mg tabs
Step 1. Determine whether the medication ordered and themedication on hand are available in the same unit of measure
Step 2. If the medication ordered and the medication on hand arenot in the same unit of measure, convert so that both measures areexpressed using the same unit of measure
Conversion: To change 0.2 g to mg1000 mg : 1 g =x mg : 0.2 g
X=200 mgor
multiply 0.2 x1000 =200
Formula Method cont
Mathematics and Dosage Calculations
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Formula Method cont.
Step 3. Now use the following formula to calculate the dosage
The physician ordered 0.2 g of meprobamate tabs (0.2 gconverts to 200 mg). The dose on hand is 400 mg tabs.
MEDICATIONS MEASURED IN UNITS
Mathematics and Dosage Calculations
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MEDICATIONS MEASURED IN UNITS
Medications such as insulin, heparin, some antibiotics,hormones, vitamins, and vaccines are measured in units
These medications are standardized in units based on
their strengths
The strength varies from one medicine to another,
depending upon their sources, their conditions, and themethods by which they are obtained
When calculating medications that are ordered in units you may useHOW TO CALCULATE UNIT DOSAGES
Mathematics and Dosage Calculations
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When calculating medications that are ordered in units, you may useeither the proportional or the formula method
The Proportional MethodThe physician ordered 4000 United States Pharmacopria (USP) unitsof heparin to be administered deep subcutaneously. On hand isheparin, 5000 USP units per milliliterStep 1. Use the following proportion to calculate the dosage
You may use a tuberculin syringe to draw up 0.8 mL
HOW TO CALCULATE UNIT DOSAGES (2)
Mathematics and Dosage Calculations
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The Formula Method
The physician ordered 450,000 units of Bicillin for deep IMinjection. Available is Bicillin, 600,000 units per milliliter
Step 1. Use the following formula to calculate the dosage
Mathematics and Dosage Calculations Group Oral Exam 04-2014
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THE END BEPAA IHS AbridgedMath Review Guide
Thank You for your attention and participation ,
Dr. Cray