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