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Presents…
“Math for Health Care Professionals”
James J. (Jim) De Carlo, RN, MA, BSN, BA
Laboratory and Clinical Instructor
NYU 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
a
@
a.a.
ac
A.M.
ad lib.
aq
bid
caps
cc
cm
cm3
C
dr
F
g (G, gm)
gr
gtt
h (hr)
hs
IM
IU
IV
IVPB
kg (kG)
L (l)
lb
m
m2
Meaning
before
at
of each
before meals
morning (before noon)
freely as desired
water
twice a day
capsule
cubic centimeter
centimeter
cubic centimeter
degrees Celsius
dram
degrees Fahrenheit
gram
grain
drop
hour
bedtime
intramuscular
international unit
intravenous
intravenous piggyback
kilogram
liter
pound
meter
square meter
Symbol
mEq
mg (mG)
mL (ml)
mm
oz
pc
per
P.M.
PO
prn
pt
qd
qh
q2h
q3h
q4h
qid
qod
qt
SC
stat.
T (tbs)
t (tsp)
tab
tid
U
ut. dict.
Meaning
milliequivalent
milligram
milliliter
millimeter
ounce
after meals
by (for each)
afternoon
by mouth
when necessary
pint
every day
every hour
every 2 hours
every 3 hours
every 4 hours
four times a day
every other day
quart
subcutaneous
immediately
tablespoon
teaspoon
tablet
three times a day
Unit
as directed
Anatomy of a “base 10” number
How many ones
How many hundreds
5,020,720
WHOLE NUMBERS:
Numbers used in counting: (0,1,2,3,etc)
used as digits and placed in neat columns!
How many tens
Order of operations…
3+5-2x2-(4-2)=oh my
first
Follow these simple rules:
=3+5-2x2-(4-2)
Parentheses first
-PARENTHESES
-EXPONENTS
Reversible
-MULTIPLICATION (left to right)
=3+5-2x2-2
Multiplication second
=3+5-4-2
-DIVISION (left to right)
last Reversible
-ADDITION (left to right)
Addition third
=8-4-2
Subtraction Left fourth
-SUBTRACTION (left to right)
=4-2
An acronym to remember:
Please Excuse My Dear Aunt Sally
Subtraction fifth
=2
Addition without a calculator…
A patient is not to consume greater than 300 milligrams of NaCl (salt) in a
meal. Can she eat lunch from the cafeteria?
(NaCl in one hospital drink) 61 milligrams + (NaCl in one sandwhich) 256
milligrams= ?
ones
tens
hundreds
1
256
+ 61
3 17
How many hundreds?
How many ones?
How many tens?
Remember: 11 tens is the same as
1 ten and 1 hundred
Subtraction without a calculator…
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?
(blood gained) 525 mL - (blood lost) 256 mL =
?
ones
tens
hundreds
1
4 1 1
525
- 256
2 69
Remember: 2 tens and 5 ones is
the same as 1 ten and 15 ones
How many ones?
How many hundreds?
How many tens?
Remember: 5 hundreds and 1 ten is
the same as 4 hundreds and 11 tens
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?
ones
tens
hundreds
2
Add it up
Do it again
with the tens!
105
x 15
5 25
+ 105
1575
Multiply ones by ones
(carry the 2)
Multiply ones by tens
Multiply ones by hundreds
Division…
Let’s try a similar type of problem that involves division.
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…
50
12 600
- 60
00
- 0
0
mg/kg Division procedure
1. Divide: 12 into 60
2. Multiply: 5 x12
3. Subtract: 60-60
4. Bring down: 0
Division procedure…again
1. Divide: 12 into 00
2. Multiply: 0 x12
3. Subtract: 0-0
4. Nothing left
Things you must know about fractions…
What is a fraction?
How are fractions written?
Can a denominator be a
number =0?
Can the same fraction be
written in many ways?
Can fractions be added,
subtracted, multiplied, and
divided?
A whole number
divided by another
whole number
5
6
numerator
fraction bar
denominator
Preparing fractions for addition…
1
Quick add 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?
Just another way to represent 1
Anything divided by
itself =1
1
7
Lets try multiplying 6 x 7
=
num x num
den x den
=
7
42
answer
Let’s try adding 7
+ 15
42
42
num + num
=
denom
22
= 42
Multiplication of fractions…
Let’s consider a problem:
A patient is ordered to drink 3/5 of a can of Ensure® at each meal. The doctor asks
you to cut this dose in half. What fraction of the can should she drink now?
3
x
5
(numerator)
x
(denominator)
(numerator)
(denominator)
3
5
x
1
2
= ?
(numerator) x (numerator)
=
(denominator) x (denominator)
Therefore…
1
3x 1
=
=
5
x
2
2
3
10
Division of fractions…
How to do it?
1
2
3
5
= ?
Dividing by a fraction is the same as multiplying by its reciprocal.
1
2
1
2
Take its reciprocal
2
=
1
So…
3
5
1
2
=
3
5
x
2
1
=
6
5
Decimal notation…
Decimals are used to separate a whole number
from its fractional part…
1 0 2 1 1
ten thousandths
thousandths
hundredths
tenths
decimal
Fractional part
ones
tens
hundreds
thousands
Whole number
= 1021.1
Two more hundredths
1 0 2 1 1 2
= 1021.12
Two more thousandths
1 0 2 1 1 2 2
= 1021.122
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:
Add
1
1
2
(2x1)+1
=
2
=
3
2
Multiply
Now we can solve by multiplying two fractions:
3 x
2
1
2
=
3
4
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.
Percentage
10%
Move decimal 2 places
To the right, add %
Fraction
10
100
Decimal
0.10
Fundamentals of rounding…
Imaginary
analog thermometer
99
thousanths
hundreths
tenths
ones
tens
decimal
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 4
Nearest single degree
Column to the right
A reminder about how decimal notation works…
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…
98
If the digit is greater than or equal to 5, round up…99 it is!
Precision is not the same thing as accuracy!
Healthcare professionals are often asked to weigh patients and monitor weight changes.
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…
Precision and Accuracy
Precision: How closely clustered multiple measurements are
Accuracy: How close a measurement is to the “true” value
scale #1
multiple measurements
“Highly Reproducible”
OK for tracking
small changes
11.001
11.002
10.999
11.001
Precise
Accurate
scale #2
multiple measurements
10.300
9.700 “Close to the true value”
OK for getting one value,
10.500
not good for tracking
9.900
of small changes
Precise
Accurate
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.
Here is an example: Salaries!
Janitor
Lawyer
$10 thousand
CEO
$100 thousand
X 10
$1 million
X 10
Baseball player
$10 million
X 10
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 drug
is the same size difference between a Baseball player’s salary and a janitor’s!
Many measurement systems are
encountered in the clinic…
METRIC (most common)
UNIT
gram
liter
HOUSEHOLD
UNIT
drop
teaspoon
tablespoon
ounce
teacup
measuring cup
glass
MASS/VOLUME
volume
volume
volume
mass
volume
volume
volume
MASS/VOLUME
mass
volume
APOTHECARY
UNIT
minim
dram
ounce
pint
quart
MASS/VOLUME
volume
volume
mass
volume
volume
Weight and volume…
Things
typically
measured
Weight / Mass)
powdered drugs
salt
sugar
pre-diluted drug
Volume
saline
water
nutritional supplement
Instrument Used
mass scale
digital scale
measuring cup
graduated cylinder
pipette
Typical sizes
25 mg (penicillin)
5 g (sugar)
0.5 Liter (Ensure®)
300 cc (saline)
Introduction to the Metric System…
Orders
of magnitude
1 micro gram = 1 millionth of a gram
3
1 milligram = 1 thousandth of a gram
2
1 centigram = 1 tenth of a gram
1
1 gram = 1 gram,
1
1 decagram = 10 grams
2
• 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!
Brain teasers:
How many micrograms in a gram?
1 million
How many milligrams in 10 kilograms?
1 kilogram = 1 thousand grams
10 million
How many milligrams in half a centigram?
3
1 megagram = 1 million grams
50
Cracking conversion problems (dimensional analysis)…
How many days have you been alive? Hmmm…
What we need to know to answer this… and all conversion problems!
1. In what units is the answer? Days
2. In what ratio is given, or do we need to provide on our own? Days/Year
3. In what is the quantity to be converted? 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;
years
21 years
quantity to
be converted
days
365 days
x
= 7665 days
1 year
known ratio
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?
1
2
The doctor’s order (given)
Quantity to be converted
3
Strength of drug
Known ratio
Units the answer will have
Example:
10mg/capsule
Example:
capsules
Example:
30 mg of Prozac
Simple Dimensional Analysis
30 mg Prozac
1 capsule
x
Doctor’s order
=
3 capsules
10 mg Prozac
Strength of drug
Units of answer
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 ? Grams
In what ratio is given? 3 g/L
In what is the quantity to be converted ? 0.5 L
0.5 Liter
Quantity to
be converted
3 grams antibiotic
x
=
1.5 g antibiotic
1 Liter H20
Known ratio
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? mL
What ratio is given? 150 mg/500 mL
What is the quantity to be converted? 300 mg
300 mg antibiotic
500 mL H20
x
= 1000 mL
150 mg antibiotic
Quantity to
be converted
Known ratio
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 rate
of milliliters/minute. What do you input?
Quantity to be converted
Units answer
The conversion is:
mL
mL
hour
min
1500 mL
1 Hour
=
x
9 hours
2.8 mL
60 min
Known ratio
1 min
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? Grams
What ratio is given? 5 g/m2
What is the quantity to be converted? 1.4 m2
1.4 m2
5 grams
x
= 7 grams
1 m2
Quantity to
be converted
Known ratio
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.
Liquid solute: Solute and solvent are measured in same units of volume.
“Concentration of liquid solvent”=
Volume of solute 10 mL of alcohol
=
=
Total volume
100 mL of blood
of solution
10% blood
alcohol level
Solid solute: Solute is measured in mass units.
(mass)
“Concentration of a solid solvent”=
Grams solute
100 mL of
solution
5 grams of NaCl (salt)
=
=
100 mL of water
5% NaCl
solution
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.
freezing
boiling
Refrigerator tempature Body temperature
Fahrenheit
Each degree covers less distance.
32
212
Each degree covers more distance.
Celsius
0
Refrigerator temperature
100
Conversion:
4C
?F F= (( C ) x 9/5 ) +32) = (4 x 9/5) + 32 = 39 F
98 F
?C C=(F-32)x 5/9)
Body temperature
=(98 - 32) x 5/9 = 37 C
Take home points…
• 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.
Production Credits…
content creator: Seth A. Zonies B.S.
content consultant: James J. (Jim) De Carlo, RN, MA, BSN, BA
© Copyright 2007 Insight Media. All rights reserved.