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CHAPTER 1
Arithmetic Needed for Dosage 1
Multiplying Whole Numbers 2
Dividing Whole Numbers 2
Fractions 3
Decimals 9
Percents 15
Fractions, Ratio, and Proportion 18
Proficiency Test 1 Arithmetic 21
Answers to Self-Tests 22
CHAPTER 2
Metric and Household Systems of Measurement 31
Metric System 32
Household System 39
Liquid Measures 40
Other Conversions 41
Proficiency Test 1 Exercises in Equivalents and Mixed Conversions 44
Answers to Self-Tests 45
CHAPTER 3
Drug Abbreviations, Labels, and Packaging 47
Interpreting the Language of Prescriptions 47
Time of Administration of Drugs 48
Routes of Administration 52
Metric and SI Abbreviations 54
Household Abbreviations 55
Terms and Abbreviations for Drug Preparations 55
Drug Labels 57
Drug Packaging 59
4
Equipment to Measure Doses 68
Proficiency Test 1 Abbreviations/Military Time 72
Proficiency Test 2 Reading Prescriptions/Interpreting Written Prescriptions 73
Proficiency Test 3 Labels and Packaging 75
Answers to Self-Tests 79
Calculation of Oral Medications—Solids and Liquids 81
Oral Solids 82
Converting Order and Supply to the Same Weight Measure 91
Clearing Decimals 93
Special Types of Oral Solid Orders 97
Oral Liquids 98
Special Types of Oral Liquid Orders 107
Oral Solid and Liquid Problems Without Written Calculations/“Common Sense” Calculations 107
Proficiency Test 1 Calculation of Oral Doses 113
Proficiency Test 2 Calculation of Oral Doses 115
Proficiency Test 3 Calculation of Oral Doses 117
Answers to Self-Tests 118
Liquids for Injection 139
5
6
Syringes and Rounding 139
Special Types of Problems in Injections From a Liquid 153
Injections From Powders 160
Distinctive Features of Injections From Powders 162
Where to Find Information About Reconstitution of Powders 163
Insulin Injections 172
Proficiency Test 1 Calculations of Liquid Injections 187
Proficiency Test 2 Calculations of Liquid Injections and Injections From Powders 190
Proficiency Test 3 Calculations of Liquid Injections 193
Proficiency Test 4 Mental Drill in Liquids-for-Injection Problems 194
Answers to Self-Tests 195
Calculation of Basic IV Drip Rates 213
Types of IV Fluids 213
Calculating Basic IV Drip Rates 216
Adding Medications to IVs 224
Medications for Intermittent IV Administration 226
Ambulatory Infusion Device 230
Enteral Nutrition 231
Proficiency Test 1 Basic IV Problems 239
Answers to Self-Tests 241
Special Types of IV Calculations 255
Amount of Drug in a Solution 256
Medications Ordered in units/hour or mg/hour 256
Medications Ordered in mcg/minute, mcg/kg/minute, or milliunits/minute—Calculation of Rate 268
Body Surface Nomogram 276
Patient-Controlled Analgesia 279
Heparin and Insulin Protocols 281
Proficiency Test 1 Special IV Calculations 289
Answers to Self-Tests 291
Dosage Problems for Infants and Children 315
Dosage Based on mg/kg 316
Administering IV Medications 334
General Guidelines for Continuous IV Medications 347
Proficiency Test 1 Infants and Children Dosage Problems 354
Answers to Self-Tests 355
Information Basic to Administering Drugs 367
Drug Knowledge 367
Pharmacokinetics 372
Legal Considerations 376
Ethical Principles in Drug Administration 380
Proficiency Test 1 Basic Drug Information 385
Answers to Self-Tests 389
Administration Procedures 391
Three Checks and Six Rights 391
Standard Precautions Applied to Administration of Medications 400
Routes of Administration 403
Special Considerations 431
on your own. Solving the arithmetic problems yourself helps you think logically about the amount ordered and the relative dose needed. And when you can mentally calculate dosage, you increase your speed and efficiency in preparing medications. Work out the problems in this chapter with and without a calculator.
Multiplying Whole Numbers
If you need a review, study a multiplication table for the numbers 1 through 12, which can be found in a basic math book or on the internet. (Search for “multiplication table.” One site is http://www.mathisfun.com/ tables). Then do the self-tests, aiming for 100% accuracy.
SELF-TEST
1 Multiplication
Write the answers to these problems. Answers are given at the end of the chapter; aim for 100%.
Dividing Whole Numbers
Review division if necessary, using a division table which can be found in a basic math book or on the internet. Again, aim for 100% accuracy on the self-test.
SELF-TEST 2 Division
Write the answers to the following problems. Answers are given at the end of the chapter.
Fractions
A fraction is a portion of a whole number. The top number in a fraction is called the numerator, and the bottom number is called the denominator. The line between the numerator and the denominator is a division sign. Therefore, you can read the fraction 1 4 as “one divided by four.”
EXAMPLE 1 4 → → numerator denominator
Types of Fractions
In a proper fraction, the numerator is smaller than the denominator.
EXAMPLE 2 5 (Read as “two fifths.”)
In an improper fraction, the numerator is larger than the denominator.
EXAMPLE 5 2 (Read as “five halves.”)
A mixed number has a whole number plus a fraction.
EXAMPLE 1 2 3 (Read as “one and two thirds.”)
In a complex or common fraction, both the numerator and the denominator are already fractions.
EXAMPLE 1 2 1 4 (Read as “one half divided by one fourth.”)
RULE REDUCING FRACTIONS
Find the largest number that can be divided evenly into the numerator and the denominator.
2:
FINE POINTS
Check to see if the denominator is evenly divisible by the numerator. In this example, the number 7 can be evenly divided into 49.
Sometimes fractions are more difficult to reduce because the answer is not obvious.
FINE POINTS
In Example 1, note both the numerator and denominator can be evenly divided by 8. In Example 2 both can be evenly divided by 9.
When you need to reduce a very large fraction, it may be difficult to determine the largest number that will divide evenly into both the numerator and the denominator. You may have to reduce the fraction several times.
FINE POINTS
Try
Prime numbers cannot be reduced any further. Examples are 2, 3, 5, 7, and 11. When reducing, if the last number is even or a zero, try 2.
If the last number is a zero or 5, try 5. If the last number is odd, try 3, 7, or 11.
Try to divide both
Then divide by 9.
SELF-TEST 3 Reducing Fractions
Reduce these fractions to their lowest terms. Answers are given at the end of the chapter.
SELF-TEST 5 Multiplying
Fractions
Multiply these fractions. Reduce if necessary. Answers are given at the end of the chapter.
Dividing Fractions
To divide two fractions, first invert the fraction that is after the division sign, then change the division sign to a multiplication sign. Reduce to lowest number.
FINE POINTS
Complex fractions such as 1 4 3 8 areread as 1 4 ÷ 3 8
The vertical arrangement acts just like a division sign.
SELF-TEST 6 Dividing Fractions
Divide these fractions. Answers are given at the end of the chapter.
Changing Fractions to Decimals
To change a fraction into a decimal, begin by dividing the numerator by the denominator. Remember that the line between the numerator and the denominator is a division sign, so 1 4 can be read as 1 ÷ 4.
In a division problem, each number has a name. The number that’s being divided (your fraction’s numerator) is the dividend; the one that does the dividing (your fraction’s denominator) is the divisor: and the answer is the quotient.
1. Look at the fraction
2. Write
3. Some people find it easier to simply extend the fraction’s straight line to the right, then strike out the numerator and place that same number down into the “box.”
4. Once you’ve set up the structure for your division problem, place a decimal point immediately after the dividend. Put another decimal point on the quotient line (above), lining up that point exactly with the decimal point below. By placing your decimal points carefully, you can avoid serious dosage errors.
FINE POINTS
5.
If the answer does not have a whole number, place a zero before the decimal point: .25 is incorrect; 0.25 is correct. This is called a leading zero. The number of places to carry out the decimal will vary depending on the drug and equipment used. For these exercises, when possible, carry answers to the thousandths place (three decimal places).
EXAMPLE 2:
10 30 328 13 58 . +
To subtract decimals, stack your two decimals as you did for addition, lining up the decimal points as before. Starting at the far right of the stack, subtract the numbers; again, make sure that the decimal point in your answer aligns with those above it.
EXAMPLE 1: 29 10 13 12 54 13 00 12 54 046 . . −=
EXAMPLE 2: 416 14 56 0471456 047 14 09 . . −=
SELF-TEST 9 Addition
and
Subtraction of Decimals
Add and subtract these decimals. Answers are given at the end of the chapter.
1. 0.9 + 0.5 =
2. 5 + 2.999 =
3. 10.56 + 357.5 =
4. 2 + 3.05 + 0.06 = 5. 15 + 0.19 + 21 =
Multiplying Decimals
6. 98.6 66.5 = 7. 0.45 0.38 = 8. 1.724 0.684 = 9. 7.066 0.2 = 10. 78.56 5.77 =
Line up the numbers on the right. Do not align the decimal points. Starting on the right, multiply each digit in the top number by each digit in the bottom number, just as you would with whole numbers. Add the products. Place the decimal point in the answer by starting at the right and moving the point the same number of places equal to the sum of the decimal places in both numbers multiplied, count the number of places that you totaled earlier. If you end up with any blank spaces, fill each one with a zero.
EXAMPLE 1:
26 0032 61 0032 .( .( ×= × decimalplace) decimaal places) decimalplacesf romthe 0 078 3 .( r right)
EXAMPLE 2:
200 003 200 0032 ×=.( .( no decimalplace) decimaal places) decimalplaces from the 60.(02 r right) or 6
FINE POINTS
Avoid the “trailing zero.” There should be no zeroes used to the right of the decimal point. The use of the trailing zero often leads to errors in writing decimals.
FINE POINTS
A quick “double-check” to determine decimal places is to count the number of decimal places in each number that is being multiplied. The answer should have the same number of decimal places.
Dividing Decimals
A reminder: The number being divided is called the dividend; the number doing the dividing is called the divisor; and the answer is called the quotient
divisor quotient dividend → → → 0 312 16 5 000 . .
Note: As soon as you write your dividend, place a decimal point immediately after it. Then place another decimal point directly above it, on the quotient line.
Divide.
Clearing the Divisor of Decimal Points
Before dividing one decimal by another, clear the divisor of decimal points. To do this, move the decimal point to the far right. Move the decimal point in the dividend the same number of places and, directly above it, insert another decimal point in the quotient.
FINE POINTS
When you’re dividing, the answer may not come out even. The dosage calculation problems give directions on how many places to carry out your answer. In Example 2, the answer is carried out to three decimal places.
SELF-TEST 10
Multiplication and Division of Decimals
Do these problems in division of decimals. The answers are given at the end of this chapter. If necessary, carry the answer to three places (unless the answer comes out “even”). (Do not use rounding rules yet!)
1. 3.14 × 0.02 = 2. 100 × 0.4 = 3. 2.76 × 0.004 =
6.3 × 7.6 =
Rounding Off Decimals
How do you determine the number of places to carry out division? This question seems to continually confuse students (and teachers sometimes!). The answer depends on the way the drug is dispensed and the equipment needed to administer the drug. Some tablets can be broken into halves or fourths. Some liquids are prepared in units of measurement: tenths, hundredths, or thousandths. Some syringes are marked to the nearest tenth, hundredth, or thousandth place. Intravenous rates are usually rounded to the nearest whole number or sometimes tenth place. Sometimes, determining when to round is dependent on the clinical setting you work in. Pediatric settings often use dosages in tenths, hundredths, and thousandths, while in adult settings, often the answer is rounded to the nearest whole number. As you become familiar with dosage, you’ll learn how far to round off your answers. To practice, first review the general rule for rounding off decimals.
RULE ROUNDING OFF DECIMALS
To round off a decimal, you simply drop the final number. Exception: If the final number is 5 or higher, drop it and then increase the adjacent number by 1.
To obtain an answer that’s rounded off to the nearest tenth, look at the number in the hundredth place and follow the above rule for rounding off.
If you want an answer that’s rounded off to the nearest hundredth, look at the number in the thousandth place and follow the above rule for rounding off.
EXAMPLE 0.664 becomes 0.66
0.148 becomes 0.15 2.375 becomes 2.38
And if you want an answer that’s rounded off to the nearest thousandth, look at the number in the ten-thousandth place and follow the same rules.
EXAMPLE 1.3758 becomes 1.376
0.0024 becomes 0.002 4.5555 becomes 4.556
SELF-TEST 11
Rounding Decimals
Round off these decimals as indicated. Answers are given at the end of the chapter.
Nearest Tenth
1. 0.25 =
2. 1.84 =
3. 3.27 =
4. 0.05 = 5. 0.63 = Nearest Hundredth
Comparing the Value of Decimals
Nearest Thousandth
1.3254 = 12. 0.0025 = 13. 0.4521 = 14. 0.7259 = 15. 0.3482 =
Understanding which decimal is larger or smaller can help you solve dosage problems. Example: “Will I need more than one tablet or less than one tablet?”
RULE DETERMINING THE VALUE OF DECIMALS
The decimal with the higher number in the tenth place has the greater value.
EXAMPLE
Compare 0.25 with 0.5.
Since 5 is higher than 2, the greater of these two decimals is 0.5.
SELF-TEST
12 Value of Decimals
In each pair, underline the decimal with the greater value. Answers are given at the end of the chapter.
1. 0.125 and 0.25
2. 0.04 and 0.1
3. 0.5 and 0.125
4. 0.1 and 0.2
5. 0.825 and 0.44
6. 0.9 and 0.5
7. 0.25 and 0.4
8. 0.7 and 0.35
9. 0.3 and 0.225 10. 0.5 and 0.455
Fractions, ratios, and decimals also can be converted to percents. Again, remember that percent means “parts of a hundred.”
Fractions Converted to Percents
If the denominator is 100, simply write the numerator as a percent:
5 100 5 = %
If the denominator is not 100, you must convert the fraction, using 100 as the denominator. In this example, multiply the numerator and denominator, using 20 as the lowest common denominator:
Ratios Converted to Percents
If the second number in the ratio is 100, simply write the first number as a percent: 1:100 = 1%
If the second number in the ratio is not 100, you must convert the ratio, using 100 as the denominator. In this example, multiply the numerator and denominator, using 20 as the lowest common denominator: 4:5 = 80:100 = 80%
Decimals Converted to Percents
Move the decimal point two places to the right and then write the percent sign: 01..01010% ==
05 5 .. % ==
SELF-TEST 14 Fractions, Ratios, and Decimals
Change these percents to a fraction, ratio, and a decimal. Answers are given at the end of the chapter. Do not reduce.
1. 32% = 2. 8.5% = 3. 125% = 4. 64% = 5. 11.25% =
Now change these fractions, ratios, and decimals to a percent. 6. 7 10 =
Fractions, Ratio, and Proportion
Fractions show how the part (numerator) relates to the whole (denominator). Ratio indicates the relationship between two numbers. In this book, ratios are written as two numbers separated by a colon (1:10). Read this ratio as “one is to ten.” Proportion indicates a relationship between two ratios or two fractions.
EXAMPLE 2 8 10 40 = (Read the proportion like this: “two is to eight as ten is to forty.”)
5:30 : : 6:36 (Read as “five is to thirty as six is to thirty-six.”)
Proportions written with two ratios and the double colon can also be written as fractions. Thus, 5:30 : : 6:36 becomes 5 30 6 36 =
Solving Proportions With an Unknown
When one of the numbers in a proportion is unknown, the letter x substitutes for that missing number. By following three steps, you can determine the value of x in a proportion.
Step 1. Cross-multiply.
Step 2. Clear x
Step 3. Reduce.
Here’s how the three steps work.
Proportions Expressed as Two Fractions
Suppose you want to solve this proportion:
1 0 125 025 = x
Step 1. Cross-multiply the numerators and denominators.
1 0 125 025
0 125 025 = = x x
Step 2. Clear x by dividing both sides of the equation with the number that precedes x. x = 025 0 125
Step 3. Reduce the number.
Ratio and Proportion in Dosage: An Introduction
When the amount of drug prescribed by a physician or healthcare provider differs from the supply, you can solve the dosage problem with proportion, using either two ratios or two fractions.
EXAMPLE
Order: 0.5 mg of a drug
Supply: A liquid labeled 0.125 mg per 4 mL
You know that the liquid comes as 0.125 mg in 4 mL. And you know that the amount you want is 0.5 mg. You don’t know, however, what amount of liquid is needed to equal 0.5 mg. So, you need one more piece of information: the unknown, or x.
You can set up and solve this arithmetic operation as a proportion, using either two fractions or two ratios. Notice that both methods eventually become the same calculation.
So far, you’ve learned two ways to solve dosage calculation problems: the ratio method (i.e., the proportion of two ratios) and the proportion method (i.e., the proportion of two fractions). Chapter 4 introduces the simpler formula method, which is derived from ratio and proportion. Also, Chapter 4 explains another less complicated way: the dimensional analysis method. Throughout the book, self-test and proficiency test problems illustrate solutions reached by all four methods of calculation. x = 16