Post on 26-Dec-2015
Presents…
“Math for Health Care Professionals”
James J. (Jim) De Carlo, RN, MA, BSN, BA
Laboratory and Clinical InstructorNYU College of Nursing
A story pulled from the headlines…
• Actor Dennis Quaid’s newborn twins are hospitalized at a prestigious hospital in California for an infection.
• Pharmacy technicians and nurses accidentally administer 1000x too much of a anti-clotting drug Heparin.
• Quaid states that he saw blood splatter across the room as a bandage was replaced on one of his babies.
• The twins recovered, but Quaid and his wife sue the drugmaker for negligence.
Key points from this story…
• Dosing mistakes are made, even today, at the best hospitals.
• Patients are harmed by dosing errors.• Nurses are often the last stop in quality control before a
drug is administered.• Giving 10x, 100x, or 1000x too much, or too little of a
drug is a common mistake.
Steps for error prevention…
• Know “reasonable” doses for commonly prescribed drugs.
• Be comfortable with numbers and basic math.• Process a doctor’s order confidently and accurately.• Assess whether a calculated dose is “reasonable.”• Always double check with a colleague to ensure proper
dosing.
Goals for this course…
• Goal Set 1 (Slides 5-19): – To reinforce basic building blocks of arithmetic.– To introduce simple math problems commonly found in
healthcare.
• Goal Set 2(Slides 20-25):– To learn about measurements and units systems typically
encountered in the healthcare.
• Goal Set 3( Slides 26-33):– To learn how to solve dosage calculation problems with step-
by-step solutions to sample problems.
Common medical symbols/ abbreviations
Symbol Meaning Symbol Meaning
a before mEq milliequivalent@ at mg (mG) milligrama.a. of each mL (ml) milliliterac before meals mm millimeterA.M. morning (before noon) oz ouncead lib. freely as desired pc after mealsaq water per by (for each)bid twice a day P.M. afternooncaps capsule PO by mouthcc cubic centimeter prn when necessarycm centimeter pt pint
cm3 cubic centimeter qd every dayC degrees Celsius qh every hourdr dram q2h every 2 hoursF degrees Fahrenheit q3h every 3 hoursg (G, gm) gram q4h every 4 hoursgr grain qid four times a daygtt drop qod every other dayh (hr) hour qt quarths bedtime SC subcutaneousIM intramuscular stat. immediatelyIU international unit T (tbs) tablespoonIV intravenous t (tsp) teaspoonIVPB intravenous piggyback tab tabletkg (kG) kilogram tid three times a dayL (l) liter U Unitlb pound ut. dict. as directedm meter
m2 square meter
Anatomy of a “base 10” number
5,020,720WHOLE NUMBERS: Numbers used in counting: (0,1,2,3,etc) used as digits and placed in neat columns!
How many ones
How many tens
How many hundreds
Order of operations…3+5-2x2-(4-2)=oh my
Follow these simple rules:-PARENTHESES
-EXPONENTS
-MULTIPLICATION (left to right)
-DIVISION (left to right)
-ADDITION (left to right)
-SUBTRACTION (left to right)
first
last
=3+5-2x2-(4-2)
=3+5-2x2-2
Parentheses first
=3+5-4-2
Multiplication second
=8-4-2
Addition third
=4-2
Subtraction Left fourth
=2 Subtraction fifthAn acronym to remember:
Please Excuse My Dear Aunt Sally
Reversible
Reversible
Addition without a calculator…
(NaCl in one hospital drink) 61 milligrams + (NaCl in one sandwhich) 256 milligrams= ?
25661+
on
es
tens
hu
nd
reds
731
1
How many ones?
How many tens? How many hundreds?
Remember: 11 tens is the same as 1 ten and 1 hundred
A patient is not to consume greater than 300 milligrams of NaCl (salt) in a meal. Can she eat lunch from the cafeteria?
Subtraction without a calculator…
(blood gained) 525 mL - (blood lost) 256 mL =
525256-
on
es
tens
hu
nd
reds
2 How many ones?
How many tens? How many hundreds?
?
9
1
Remember: 2 tens and 5 ones is the same as 1 ten and 15 ones
1
6Remember: 5 hundreds and 1 ten is the same as 4 hundreds and 11 tens
41
A patient receives 525 milliliters of blood through a transfusion. Soon after, he accidentally cuts himself and loses approximately 256 milliliters of blood. How much total blood did he gain?
Multiplication…
Performing an important calculation by hand, in addition to a calculator, is a great way to double check your answer.
Let’s consider an example:
The doctor asks you to administer 105 milligrams of drug for every kilogram that the patient weighs. If the patient ways 80 kilograms, What is the appropriate drug dosage?
Do it again with the tens!
5 Multiply ones by ones (carry the 2)
2
2 Multiply ones by tens
5 Multiply ones by hundreds
Add it up +1575105
10515x
on
es
tens
hu
nd
reds
Division…Let’s try a similar type of problem that involves division.
60012 Division procedure
1. Divide: 12 into 60
52. Multiply: 5 x12
60 3. Subtract: 60-60 -0
4. Bring down: 0
0 Division procedure…again
0
1. Divide: 12 into 00 2. Multiply: 0 x12 03. Subtract: 0-0
-0 4. Nothing left
You are working in a pediatric ward. The doctor asks you how much drug the patient has received per kilogram of body weight. The child received 600 milligrams of drug and weighs 12 kilograms. The answer is…
mg/kg
Things you must know about fractions…
What is a fraction? A whole number divided by another whole number
Can the same fraction be written in many ways?
Can fractions be added, subtracted, multiplied, and divided?
How are fractions written?56
numeratorfraction bardenominator
Can a denominator be a number =0?
Preparing fractions for addition…
Quick add 1 6
15 42 +
Wait…
This fraction is not ready !
Both denominators must be the same before addition
Let’s multiply 1/6 x 1
Anything times 1 equals itself right?
Anything divided by itself =1
num x num
den x den=
7 42
Let’s try adding 7 42
15 42
+ =num + num
denom
Lets try multiplying 7 7
x
1 6
=
Just another way to represent 1
22 42 =
answer
Multiplication of fractions…
(numerator)
(denominator)x
(numerator)
(denominator)=
(numerator) (numerator)x
(denominator) (denominator)x
Therefore…3
5x
1
2=
3
10
3 1
5 2 =x
x
3
5x = ?
1
2
Let’s consider a problem:A patient is ordered to drink 3/5 of a can of Ensure® at each meal. The doctor asksyou to cut this dose in half. What fraction of the can should she drink now?
Division of fractions…
How to do it?
Dividing by a fraction is the same as multiplying by its reciprocal.
1
2Take its reciprocal
1
2=
2
1
So…
3
5x
2
1=
6
5
3
5= ?
1
2
3
5=
1
2
Decimal notation…Decimals are used to separate a whole number from its fractional part…
Whole number Fractional part
1 0 2 1 1
on
es
ten
s
hu
nd
red
s
tho
usa
nd
s
ten
ths
hu
nd
red
ths
tho
usa
nd
ths
ten
th
ou
san
dth
s
dec
imal
= 1021.1
1 0 2 1 1 2 = 1021.12Two more hundredths
1 0 2 1 1 2 2 = 1021.122 Two more thousandths
Converting mixed numbers to fractions…
A mixed number is a whole number plus a fraction of a whole number.
Example: A patient’s dose is 1 1/2 pills (mixed number).Real Scenario: Her doctor tells you to halve the patient’s dose
You immediately think to multiply the patient’s dose by 1/2
Convert the mixed number 1 1/2 to a fraction…here’s how:
(2x1)+1
2= 3
2
Now we can solve by multiplying two fractions:
34=3
212
x
12
1 =
Multiply
Add
Percentages…Percentages are used often in the clinic…they are worth knowing well!
“Percentage” actually means per/100.
Imagine that a sample of blood is collected from a patient. Let’s say 100 “parts” are collected (parts is an arbitrary unit).If 10 parts are alcohol, what is their blood alcohol percentage?
10 parts per 100= 10%
percentages, decimals, and fractions can be interconverted.
Mov
e de
cim
al 2
pla
ces
to th
e le
ft, re
mov
e 10
0 as
deno
min
ator
10%
10100
0.10
Remove %
sign, /100
Move decimal 2 placesTo the right, add %
Percentage Decimal
Fraction
A reminder about how decimal notation works…
Fundamentals of rounding…The doctor asks you to keep track of a patient’s temperature to the nearest degree!You’ve been given a fancy thermometer that shows temperature like this…
9 8 7 2 4o
nes
ten
s
ten
ths
hu
nd
reth
s
tho
usa
nth
s
dec
imal
All you have to do is ask…is the temperature closer to 99 or 98?
The formal way: Find the column to which you are rounding.
Look to the columns to the right of that column…
If the digit is greater than or equal to 5, round up…99 it is!
Nearest single degree Column to the right99
98
Imaginary analog thermometer
Precision is not the same thing as accuracy!Healthcare professionals are often asked to weigh patients and monitor weight changes.
Precision and Accuracy
Precise
Accurate
Precise
Accurate
11.00111.00210.99911.001
multiple measurements
Reliable measurements are critical for the patient’s health, but not all scales are perfect.Understanding how reliable a measurement is requires knowing the difference between…
“Highly Reproducible”OK for trackingsmall changes
“Close to the true value”OK for getting one value,
not good for trackingof small changes
scale #1 scale #2
Precision: How closely clustered multiple measurements are
Accuracy: How close a measurement is to the “true” value
10.3009.70010.5009.900
multiple measurements
Conceptualizing orders of magnitude…Not everything in this world comes in the same size.
If something is a lot bigger than another thing, how do you describe this difference?
Doctors and scientists assign “orders of magnitude” to objects to accurately express this difference.
Lawyer compared with janitor = 1 order of magnitude difference.
CEO compared with janitor = 2 orders of magnitude difference.
Baseball player compared with janitor = 3 orders of magnitude difference.
We’ll get into the metric system a bit later, but…
Remember that giving a patient 1 gram of a drug instead of 1 milligram of a drugis the same size difference between a Baseball player’s salary and a janitor’s!
Here is an example: Salaries!
$10 thousand $100 thousand $1 million $10 million
X 10 X 10 X 10
Janitor Lawyer CEO Baseball player
Many measurement systems are encountered in the clinic…
UNIT MASS/VOLUMEgram massliter volume
METRIC (most common)
UNIT MASS/VOLUMEdrop volume
teaspoon volumetablespoon volume
ounce massteacup volume
measuring cup volumeglass volume
HOUSEHOLDUNIT MASS/VOLUMEminim volumedram volumeounce masspint volume
quart volume
APOTHECARY
Weight and volume…
Weight / Mass)
Volume
Things typically
measured Instrument Used Typical sizes
powdered drugssaltsugar digital scale
mass scale25 mg (penicillin)
5 g (sugar)
pre-diluted drug
salinewater
nutritional supplement
measuring cup
graduated cylinder
pipette
0.5 Liter (Ensure®)
300 cc (saline)
Introduction to the Metric System…Orders
of magnitude
1 micro gram = 1 millionth of a gram
1 milligram = 1 thousandth of a gram
1 centigram = 1 tenth of a gram
1 gram = 1 gram,
1 decagram = 10 grams
1 kilogram = 1 thousand grams
1 megagram = 1 million grams
3
2
1
1
2
3
• Metric system prefixes are applied to all units of measurement. •To go from mass units to volume units, simply change grams to liters. •The logic is identical!
How many micrograms in a gram?
How many milligrams in 10 kilograms?
How many milligrams in half a centigram?
1 million
10 million
50
Brain teasers:
Cracking conversion problems (dimensional analysis)…
How many days have you been alive? Hmmm…
21 years
quantity to be converted
years
x365 days
1 year=
known ratio
days
What we need to know to answer this… and all conversion problems!
1. In what units is the answer? 2. In what ratio is given, or do we need to provide on our own?3. In what is the quantity to be converted?
DaysDays/Year
21 years
All we have to do is:
multiply (quantity to be converted) x (known ratio)
After canceling units, we should be left with our answer in the correct units;
7665 days
units in answer
Processing a doctor’s order…
Doctors will often order a patient to take a certain amount of drug, but may not specify exactly how many capsules/tablets, volume he should take.
You will be responsible for calculating this.
What are the 3 critical pieces of information?
Simple Dimensional Analysis
30 mg Prozac
Doctor’s order
x1 capsule
10 mg Prozac=
Strength of drug
3 capsules
Units of answer
The doctor’s order (given)
Example:
30 mg of Prozac
Quantity to be converted
1
Strength of drug
Example:10mg/capsule
Known ratio
2
Units the answer will have
capsules
Example:
3
Making solutions from powders…Many medicines administered by healthcare professionals are actually drugs dissolved in a liquid vehicle.
Problem: You are asked to make 0.5 liter of an antibiotic solution. The final concentration of the solution should be 3 grams antibiotic/1 liter of water.How much antibiotic do you need?
Think Dimensional Analysis!
In what units is the answer ? In what ratio is given?In what is the quantity to be converted ?
Grams3 g/L
0.5 L
0.5 Liter
Quantity to be converted
=x3 grams antibiotic
1 Liter H20
Known ratio
1.5 g antibiotic
Units of answer
Parenteral dosages…
Now that you can make your own drug/liquid mixtures, this next problem should be very easy!
Problem:The doctor asks you to administer 300 mg of antibiotic each day. The antibiotic comes as a liquid mixture in a strength of 150 mg/500 ml. How much liquid do you administer each day?
Think Dimensional Analysis!
What units is the answer in? What ratio is given?What is the quantity to be converted?
mL150 mg/500 mL
300 mg
300 mg antibiotic
Quantity to be converted
=
150 mg antibiotic
500 mL H20x
Known ratio
1000 mL
Units of answer
Calculating infusions…
As we know, drugs often come in liquid form. In cases in which large doses are required, not all the drug can be delivered at once.Depending on the speed at which the patient can absorb or metabolize, the drug healthcare providers must determine a suitable flow rate.A flow rate is simply how much drug is delivered at a time.Let’s consider a problem:The doctor orders a patient to receive 1500 mL of 5% dextrose in water (D5W)over 9 hours. The intravenous delivery system requires you to input a flow rateof milliliters/minute. What do you input?
The conversion is:
hour
mL
min
mL
9 hours
1500 mL
1 min
2.8 mL
Quantity to be converted Units answer
x 60 min
1 Hour=
Known ratio
Dosage that depends on surface area…The dosage of some drugs is calculated based on the body surface area of the recipient.
Surface area is measured in meters squared (m2). When the total body surface area is known, the correct drug dosage can be determined.
Let’s consider a problem:The patient is ordered to receive 5 grams of drug/ m2 of body surface area.The total surface area of the patient is 1.4 m2. What quantity of the drug should he receive?
Think Dimensional Analysis!
What units is the answer in? What ratio is given?What is the quantity to be converted?
Grams5 g/m2
1.4 m2
1.4 m2
Quantity to be converted
=
1 m2
5 gramsx
Known ratio
7 grams
Units of answer
Calculating strengths of solutions…Calculating strengths of solutions is required for knowing how to make solutions correctly and knowing how much drug is contained in a solution. Strengths of solutions are expressed in percentages.
A solution consists of two parts mixed together: Solute and Solven.t
Solute: Substance being dissolved or diluted.
Solvent: Substance dissolving or diluting the solvent.
“Concentration of liquid solvent”= Volume of solute
Total volume of solution
=10 mL of alcohol
100 mL of blood=
10% blood alcohol level
“Concentration of a solid solvent”= = Grams solute
100 mL of solution
(mass)5 grams of NaCl (salt)
100 mL of water= 5% NaCl
solution
Liquid solute: Solute and solvent are measured in same units of volume.
Solid solute: Solute is measured in mass units.
Interconverting Celsius and Fahrenheit…Two temperatures scales exist: Fahrenheit and Celsius. There may come a time when you are required to interconvert temp values.
Online converters exist, but learning to do this conversion by hand helps reinforce understanding.
Conversion:
F= (( C ) x 9/5 ) +32)
C=(F-32)x 5/9)
= (4 x 9/5) + 32 = 39 F
=(98 - 32) x 5/9 = 37 C
Celsius
0 100
Each degree covers more distance.
98 F ?C
Body temperature
Body temperature
Fahrenheit
freezing boiling
32 212
Each degree covers less distance.
4 C ?F
Refrigerator temperature
Refrigerator tempature
• Comfort with basic math is absolutely required for delivering safe and effective healthcare to patients.
• Solving a lot of problems helps make concepts “second nature”.• Double checking calculations with colleagues or by-hand helps
prevent mistakes!• Don’t be afraid to ask for help if you struggle with a concept.
Take home points…
Production Credits…
© Copyright 2007 Insight Media. All rights reserved.
content creator: Seth A. Zonies B.S.
content consultant: James J. (Jim) De Carlo, RN, MA, BSN, BA