MEDICAL MEASUREMENTS MATH WORKBOOK
Community Health Aide/Practitioner Program
Author: Daniel C. Thomas, Curriculum Coordinator, Health Aide Training Center, NSHC
Computer graphics by Michael Faubion and Clifford Hunt, YKHC Media Services.
Formatting assistance from Maureen Murray, Advanced Training Program Coordinator, YKHC CHAP.
Editing suggestions from Jane Allen, Assistant Professor of Mathematics, Kuskokwim Campus, UAF
and Linda Curda, UAF/CHAP Academic Liaison, Kuskokwim Campus, UAF
1/8/01, revised 2/16/01, revised 3/29/02
Copyright 2001 by Daniel C. Thomas
All or parts of this document may be reproduced for educational purposes only.
MEDICAL MEASUREMENTS MATH WORKBOOK
Table of Contents
BP Cuff Gauge......
Oxygen Tank Gauges......
Fractions......
A 3cc Syringe in Tenths of a CC …………………………………………………………………….13
Reading a Ruler in Fractions of an Inch......
Cutting Up a Pie ………………………………………………………………………………………..19
A Clock’s Face......
A Baby Scale......
An Adult Scale......
Penicillin Syringes......
Bicillin ……………………………………………………………………………………………………………25
Wycillin
The Eighth of an Inch......
Sixteenths and Thirty-Seconds of an Inch ………………………………………………………..36
Summary of Reading a Ruler in Inches ……………………………………………………………..37
The Metric System : Centimeters and Millimeters ……………………………………………. 41
Decimals ………………………………………………………………………………………………...43
Adding Decimals and Tylenol Drops Dosages …………………………………………………….47
Dollars and Cents ……………………………………………………………………………………51
Thousandths …………………………………………………………………………………………...52
Digoxin doses …………………………………………………………………………………………..53
Thermometers …………………………………………………………………………………………57
Fifths …………………………………………………………………………………………………….57
Back to thermometers ………………………………………………………………………………...61
Reading a Hypothermia Thermometer ………………………………………………………………63
Another Hypothermia Thermometer ………………………………………………………………..64
Syringe Math …………………………………………………………………………………………...67
Volume: cc and ml ……………………………………………………………………………………...67
Concentrations …………………………………………………………………………………………69
3 cc Syringe ……………………………………………………………………………………………...73
5 cc Syringe ……………………………………………………………………………………………...77
1 cc TB Syringe ………………………………………………………………………………………….82
1 cc Epinephrine Tubex Syringe ……………………………………………………………………….92
50 Unit Insulin Syringe ………………………………………………………………………………...97
100 Unit Insulin Syringe ……………………………………………………………………………...101
Summary of Reading a Syringe ……………………………………………………………………...107
Medical Measurements
We are always measuring things as we take care of patients. To do this correctly, we must look at how measuring devices use marks to show different amounts. We will look at whole numbers first (numbers that haven’t been divided into parts smaller than 1, such as decimals or fractions). Examples of whole numbers are 1, 2, 3, 4, 5, and so on, all the numbers that we learned to count as small children.
BP Cuff Gauge
The gauge on a BP cuff has little marks to show the different blood pressure values. Only some of the marks are numbered. We have to figure out what the numbers are for the other marks. The marks that are numbered are 20, 40, 60, 80, 100, 120, and so on.
fig001
First we must note that there are two sizes of marks, big ones and little ones. There is a big mark at each of the numbers as well as halfway between each of the numbers. For instance, there is a big mark halfway between the big mark at 80 and the big mark at 100.
fig002
We need to figure out what the number is for that big mark. It would make sense that it should be the number that comes halfway between 80 and 100. To figure out what number that would be, it helps to draw a diagram called a number line, which shows the numbers as we would count them from left to right.
fig003
Looking at this diagram, we can see that the number 90 comes halfway between 80 and 100.
fig004
Therefore, 90 must be the number for the big mark that is halfway between 80 and 100 on the BP gauge.
fig005
In the same way, we can figure out each of the big marks between the numbers on the BP gauge. It turns out that each is a multiple of ten (the numbers that you get when you count by tens: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, and so on).
fig006
The next step is to figure out what the small marks are in between the big marks that we numbered above.
There are 4 small marks in between each pair of big marks. The rule to follow to figure this out is that marks like this are usually ones or twos. So we count the marks by ones and count them by twos and see which comes out right.
If we try counting by ones first, between 80 and the big mark that we know is 90, this is what we find:
fig007
We can see that this doesn’t come out right because we are counting “85” when we reach the 90 mark.
If we try counting by twos, we get this:
fig008
This comes out right, so each of the small marks is worth 2, not 1.
This is a simple way to figure out the little marks on any measuring device: count the marks by ones or twos and see which comes out right. This is true for whole numbers and decimals. Fractions are different, as we will see later.
Another way to figure out the small marks is to note that they divide the space between 80 and 90 into 5 equal parts. The space between 80 and 90 on the BP gauge is equal to 10 (BP is measured in millimeters of mercury, so this 10 actually represents 10 millimeters of mercury). The little marks divide the 10 into 5 equal parts.
fig009
If we divide 10 into 5 equal parts (10 5), we find that each of the 5 equal parts is 2 (10 5 = 2). So each little mark on the BP gauge indicates 2 millimeters of mercury.
Oxygen Tank Gauges
There are usually two gauges on an oxygen tank regulator: one shows LPM (liters per minute), the other shows psi (pounds per square inch). Liters per minute (LPM) measures how fast the oxygen is coming out of the tank (how many liters of oxygen come out in one minute). This is called the flow rate. One common type of flow gauge looks like this:
fig010
If the needle points to the 2, oxygen is leaving the tank at the rate of 2 liters per minute.
The big marks indicate the number of liters per minute. Only half of the big marks have
a number (2, 4, 6, 8, 10, 12, 14, 15). We have to figure out the numbers for the big
marks that don’t have a number.
For instance, there is a big mark halfway between 2 and 4. We know that the number 3 comes halfway between the numbers 2 and 4, so that is the correct number for the big mark between 2 and 4.
fig011
Since the rest of the gauge is arranged the same way, we can number the dial all the way around by inserting the numbers that come between the numbers listed on the gauge.
fig012
The psi (pounds per square inch) gauge measures the pressure of the oxygen in the tank. The more oxygen that is put in the tank, the more pressure it puts on the inside of the tank, measured in pounds of pressure per square inch of tank surface. A full tank has a pressure of 2,000 pounds per square inch (2,000 psi).
The psi gauge has big marks numbered 500, 1000, 1500, 2000, 2500, and so on.
fig013
We need to figure out what the little marks are between the big marks. As with the BP gauge and other measuring devices, the marks probably are worth one or two, or in this case it will be 100 or 200.
First we will try counting out the marks by 100s, starting at the 500 mark.
fig014
This comes out right, so each little mark is indeed 100 psi. If it hadn’t come out right, we would have tried counting by 200s next, like so:
fig015
This doesn’t come out right, since we are counting “1500” when we reach the number 1000 on the gauge. Therefore, each little mark cannot be 200.
Fractions
A fraction is a number that represents an amount less than 1. If you take something such as a pie or candy bar and divide it into pieces, each piece is a fraction of the whole. Fractions used frequently in everyday language include the half, third, and quarter.
12
The bottom number of a fraction states how many equal parts the whole was divided into:
12
The top number of a fraction states how many of the equal parts we have:
12
1 / we have 1 of the 2 equal parts.
2 / the candy bar is divided into 2 equal parts.
For another example, pretend this is a candy bar that has been cut into 10 equal pieces.
12
3
4
5
6
7
8
9
10
Now pretend that we have 5 of those equal pieces.
12
3
4
5
This is (five tenths) of the candy bar.
5 / we have 5 of the 10 equal parts.10 / the candy bar is divided into 10 equal parts.
(five tenths) and (one half) actually are the same amount, for 5 is half of 10.
To show this, we take the candy bar and divide it in half.
fig020
Then we divide the candy bar into 10 equal pieces.
fig021
Each half has 5 of the 10 equal pieces.
fig022
So, (one half) = (five tenths).
fig023
A 3 cc Syringe in Tenths of a cc
For a good example of halves and tenths, look at your 3 cc syringe. The 3 cc syringe has 2 sets of markings; make sure you are looking at the “cc” markings (the marks on the right side).
fig024
We see that each cc is labeled with a whole number (1, 2, and 3).
Each half of a cc is labeled also (½, 1½, 2½).
Each cc is also divided into 10 equal parts by the smallest marks. If we start at the “zero” (empty) line and count the marks up to 1, we will be counting “10” when we reach 1 cc.
fig025
If we divide a cc into 10 equal parts, each part is one tenth of a cc. Therefore, each little
mark is one tenth () of a cc.
fig026
We know this is correct because we are counting (ten tenths) when we reach 1 cc on the syringe. (ten tenths) means that we have all 10 of the 10 equal parts of the whole, which is one whole cc (1 cc). =1
fig027
The other clue that we have counted the marks correctly is that we are counting (five tenths) when we reach the mark on the syringe. 5 is indeed half of 10, so we know we are counting correctly. =
fig028
Reading a Ruler in Fractions of an Inch
Another example of fractions that we find in our work is reading inches on a ruler, such as the paper measuring tape:
fig029
The biggest lines are numbered, representing inches.
fig030
On the ruler below, the next biggest line (halfway between the two numbers) divides 1 inch into two equal parts. Therefore, it marks an inch.
fig031
1 / we have 1 of the 2 equal parts.2 / the inch is divided into 2 equal parts.
The 2 equal parts are both an inch long (see below).
fig032
Each of the half inch segments is divided in half by another shorter line (see below).
fig033
If we study these 3 lines between 0 and 1, we see that they divide the inch into 4 equal parts (see below).
fig034
Each of the 4 equal parts is one fourth () of an inch.
1 / we have 1 of the 4 equal parts.4 / the inch is divided into 4 equal parts.
fig035
fig036
Another name for a fourth is a quarter, so we say “one quarter of an inch”. When describing money, there are 4 quarters in a dollar: a quarter is one fourth () of a dollar.
Notice that the middle line marks of an inch as well as of an inch. =
fig037 fig038
If you divide the whole inch into 4 equal parts and take 2 of those parts, you have of the whole.
fig039
Cutting Up a Pie
Another popular way to look at fractions is by cutting up a pie.
The first cut of a knife across a pie divides it in half, into 2 equal parts. Each part is one half () of the pie.
fig040
The second cut of a knife across a pie divides each of the halves in half, giving 4 equal parts of the pie.
fig041
If a whole is divided into 4 equal parts, each of the 4 parts is one fourth of the whole.
fig042
Each half of the pie has 2 of the 4 equal parts of the pie. Each half of the pie has 2 fourths of the pie.
fig043
2 of the 4 equal parts is of the pie.
of a pie = of the pie.
One half () and two fourths ( ) are the same amount.
and and are all the same amount.
fig044
A Clock’s Face
Another use of fourths and halves that we use in everyday life is in describing parts of an hour when telling time. Remember that there are 60 minutes in one hour. If we look at the minutes of a clock’s face, it can be divided up just like a pie.
30 minutes is half an hour (30 is half of 60).
fig045
30 minutes = half an hour ()
15 minutes is a quarter of an hour (15 is one quarter of 60).
fig046
15 minutes = aquarter of an hour ()
45 minutes is three quarters ( ) of an hour (45 is three quarters of 60).
fig047
45 minutes = three quarters of an hour ()
A Baby Scale
Halves and quarters are the fractions that are encountered when reading a scale.
fig048 fig049
pounds on an adult scale ounces on an infant scale
If we look at the ounces on a baby scale, we see that each ounce is divided in half by a long mark halfway between the whole numbers.
fig050
In addition, each half of an ounce is divided in half by a shorter mark. As we know from cutting up a pie, half of a half is a fourth.
fig051
Half of a half ounce is one fourth of an ounce (½ of ½ oz = ¼ oz).
fig052
We also know that if we divide a whole into 4 equal parts, each part is one fourth () of the whole.
1 / we have 1 of the 4 equal parts.4 / the ounce is divided into 4 equal parts.
So, the shortest marks are fourths or quarters of an ounce.
Baby Scale (in ounces) fig053
Note that the mark can also be counted as the mark, for we know that = .
Also note that is the same as 1 (all 4 of the 4 equal parts equal the whole ounce)
Remember, if something is divided into 2 equal parts, each part is a half ().
If something is divided into 4 equal parts, each part is a fourth () (one quarter).
An Adult Scale
On the adult scale, the markings are almost the same. The differences are that we are talking pounds instead of ounces, and the scale is numbered every 2 pounds.
Adult Scale (in pounds) fig054
There is a long mark halfway between each of the numbered marks. For instance, there is a long mark halfway between 2 and 4. Since the number 3 is halfway between the numbers 2 and 4, that is the number for the long mark halfway between 2 and 4.
fig055
The number for each long mark between the numbers on the scale can be figured out the same way.
fig056
Once we know that, the other markings are the same as on the baby scale, except in fractions of a pound instead of fractions of an ounce. The longer mark halfway between the whole numbers is one half () of a pound.
fig057
The shortest marks divide each pound into 4 equal parts and therefore are fourths of a pound.
fig058
Note again that is also and that is also 1.
Penicillin Syringes
Injectable penicillins (Bicillin and Wycillin) come in prefilled syringes. We will look at Tubex syringes below. These types of penicillin are measured in units rather than milligrams. Reading these syringes is made easier by the fact that the CHAM dose will be either a whole syringe, three fourths of a syringe, half a syringe, or a fourth of a syringe. These are the only measurements that we need to learn on these syringes.
Bicillin
The Bicillin syringe below has 1,200,000 units of Bicillin (one million two hundred thousand units, which can also be called 1.2 million units). As with any measuring device, the first step to reading it is to figure out what the little marks mean. On these syringes there is only one mark, which is a black line across the middle of the syringe. This line across the middle of the syringe divides it in half.
fig059
Half of 1,200,000 units is 600,000 units. 1,200,000 2 = 600,000
fig060
To check our math, we can add 600,000 to 600,000 and see if it equals 1,200,000.
600,000
+ 600,000
1,200,000
So we now know how much Bicillin is in half a syringe, as indicated by the line across the middle of the syringe. Since doses on this syringe can be in fourths also, we need to figure out where fourths are on this syringe.
We learned earlier that a fourth of a pie is a half of a half.
fig061
In the same way, on this syringe, a fourth will be half of a half. We need to imagine a line dividing each half of the syringe in half.
fig062
The next step is to figure out how many units are in a fourth of the syringe and three fourths of the syringe.
If we divide each half (600,000 units) into two equal parts, each part will be a fourth of the syringe and will be 300,000 units. 600,000 2 = 300,000
fig063
To check our math, we can add 300,000 to 300,000 to see if it equals 600,000.
300,000
+ 300,000
600,000
Therefore, one fourth of 1,200,000 units is 300,000 units. If we divide 1,200,000 units into 4 equal parts, each will be 300,000 units. 1,200,000 4 = 300,000.
fig064
300,000
+ 300,000
+ 300,000
+ 300,000
1,200,000
So, 300,000 units is of the 1,200,000 unit syringe. If the order is for 900,000 units, this should be of the syringe (300,000 + 300,000 + 300,000 = 900,000).
fig065
Or we can think of it as adding 300,000 units ( of a syringe) to 600,000 units ( of a syringe) (300,000 + 600,000 = 900,000).
fig066
These are the only doses of Bicillin given in the CHAM:
300,000 units ( syringe)
600,000 units ( syringe)
900,000 units ( syringe)
1,200,000 units (1 whole syringe)
fig067
Wycillin
The 600,000 unit Wycillin Tubex syringe is similar to the Bicillin Tubex. There is a black line that divides the syringe in half:
fig068
Half of 600,000 units is 300,000 units. 600,000 2 = 300,000.
fig069
To check our math, we can add 300,000 to 300,000 and see if it equals 600,000.
300,000
+ 300,000
600,000
If we divide each half (300,000 units) into 2 equal parts, each will be 150,000 units.
fig070
To check our math, we can add 150,000 to 150,000 to see if it equals 300,000.
150,000
+ 150,000
300,000
Therefore, one fourth of 600,000 units is 150,000 units. If we divide 600,000 units into 4 equal parts, each will be 150,000 units. 600,000 4 = 150,000.
fig080
150,000
+ 150,000
+ 150,000
+ 150,000
600,000
So, 150,000 units is of the 600,000 unit syringe. If the order is for 450,000 units, this should be of the syringe (150,000 + 150,000 + 150,000 = 450,000).
fig081
Or we can think of it as adding 150,000 units ( of a syringe) to 300,000 units ( of a syringe) (150,000 + 300,000 = 450,000).
fig082
These are the only doses you are likely to give with the 600,000 unit Wycillin syringe:
150,000 units ( syringe)
300,000 units ( syringe)
450,000 units ( syringe)
600,000 units (1 whole syringe)
fig083
The Eighth of an Inch
Usually halves, fourths, and tenths are the only fractions we have to deal with in measurements. There is one more fraction that we sometimes encounter, usually when measuring in inches. That is the eighth ().