for Dosage
TOPICS COVERED
1. Multiplication and division of whole numbers, fractions, and decimals
2. Addition and subtraction of fractions
3. Reading decimals
4. Changing decimals to fractions
5. Addition, subtraction, multiplication, and division of decimals
6. Clearing and rounding of decimals
7. Percents
8. Solving ratio and proportion
When a medication order differs from the fixed amount at which a drug is supplied, you must calculate the dose needed. Calculation requires knowledge of the systems of dosage measurements (see Chapter 2) and the ability to solve arithmetic. This chapter covers the common arithmetic functions needed for the safe administration of drugs.
Beginning students invariably express anxiety that they will miscalculate a dose and cause harm. Although everyone is capable of error, no one has to cause an error. The surest way to prevent a mistake is to exercise care in performing basic arithmetic operations.
For students who believe their arithmetic skills are already satisfactory, this chapter contains self-tests and a proficiency exam. Once you pass the proficiency exam, you can move on to other chapters in the book.
Students with math anxiety and those with deficiencies in performing arithmetic will want to work through this chapter page by page. Examples demonstrate how to perform calculations; the self-tests provide practice and drill. FINE POINTS boxes explain details about the calculation. After you’ve mastered the content, take the proficiency exam to verify your readiness to move on.
Since calculators are readily available, why go through all the arithmetic? For one thing, using a calculator still requires that you to know what numbers and functions to enter. In clinical situations you may encounter some problems that require a calculator’s help, but it’s good to know how to make calculations 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 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%.
1. 2 × 6 = _______________
2. 9 × 7 = _______________
3. 4 × 8 = _______________
4. 5 × 9 = _______________
5. 12 × 9 = ______________
6. 8 × 3 = _______________
7. 11 × 10 = _____________
8. 2 × 7 = _______________
9. 8 × 6 = _______________
10. 8 × 9 = ______________
11. 3 × 5 = ______________
12. 6 × 7 = ______________
13. 4 × 6 = ______________
14. 9 × 6 = ______________
15. 8 × 8 = ______________
16. 7 × 8 = ______________
17. 2 × 9 = ______________
18. 8 × 11 = _____________
19. 4 × 9 = ______________
20. 3 × 8 = ______________
21. 12 × 11 = ____________
22. 9 × 5 = ______________
23. 9 × 9 = ______________
24. 7 × 5 = ______________
25. 12 × 10 = ____________
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.
1. 63 ÷ 7 = ______________
2. 24 ÷ 6 = ______________
3. 36 ÷ 12 = ______________
4. 42 ÷ 6 = ______________
5. 35 ÷ 5 = ______________
6. 96 ÷ 12 = ______________
7. 12 ÷ 3 = ______________
8. 27 ÷ 9 = ______________
9. 49 ÷ 7 = ______________
10. 18 ÷ 3 = ______________
11. 72 ÷ 8 = ______________
12. 48 ÷ 8 = ______________
13. 28 ÷ 7 = ______________
14. 21 ÷ 7 = ______________
15. 24 ÷ 8 = ______________
16. 84 ÷ 12 = ______________
17. 81 ÷ 9 = ______________
18. 32 ÷ 8 = ______________
19. 36 ÷ 6 = ______________
20. 18 ÷ 9 = ______________
21. 21 ÷ 3 = ______________
22. 48 ÷ 4 = ______________
23. 144 ÷ 12 = ______________
24. 56 ÷ 8 = ______________
25. 60 ÷ 5 = ______________
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 
as “one divided by four.”
EXAMPLE
Types of Fractions
In a proper fraction, the numerator is smaller than the denominator.
EXAMPLE
In an improper fraction, the numerator is larger than the denominator.
EXAMPLE
A mixed number has a whole number plus a fraction.
EXAMPLE
In a complex fraction, both the numerator and the denominator are already fractions.
EXAMPLE
RULE 
REDUCING FRACTIONS
Find the largest number that can be divided evenly into the numerator and the denominator.
EXAMPLE
EXAMPLE 1:
Reduce 
EXAMPLE 2:
Reduce 
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.
EXAMPLE
EXAMPLE 1:
Reduce 
EXAMPLE 2:
Reduce 
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.
EXAMPLE
EXAMPLE 1:
Reduce 
Try to divide both by 3 
Then use multiples 
EXAMPLE 2:
Reduce 
Try to divide both by 3 
Then divide by 9.
FINE POINTS
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.
SELF-TEST 3
Reducing Fractions
Reduce these fractions to their lowest terms. Answers are given at the end of the chapter.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
Adding Fractions
If you need to add two fractions that have the same denominator, first add the two numerators; write that sum over the denominator and, if necessary, reduce.
If the two fractions have different denominators, the process takes two steps. First convert each fraction, multiplying both of its numbers by their lowest common denominator. After you’ve converted both fractions, add their two numerators together. If necessary, reduce.
Subtracting Fractions
To subtract two fractions that have the same denominator, first subtract their numerators and then write the difference over the denominator. Reduce if necessary.
If the two fractions have different denominators, first convert each fraction using the lowest common denominator (just as you did in the adding example above). Then subtract the numerators, and reduce again if necessary.
SELF-TEST 4
Adding and Subtracting Fractions
Add and subtract these fractions. Answers are given at the end of the chapter.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
Multiplying Fractions
There are two ways to multiply fractions. Use whichever method is more comfortable for you.
First Way
Multiply the numerators across. Multiply denominators across. Reduce the answer to its lowest terms.
EXAMPLE
Second Way (When You Are Multiplying Several Fractions)
First, reduce each fraction by dividing its numerator evenly into its denominator. Multiply the remaining numerators across. Multiply the remaining denominators across. Check to see if further r eductions are possible. In Example 1, because of several fractions, you can use any numerator to divide into any of the denominators.
EXAMPLE
EXAMPLE 1:
FINE POINTS
14 ÷ 7 = 2
10 ÷ 5 = 2
12 ÷ 3 = 4
The denominators are being divided by the numerators to reduce.
If you’re multiplying mixed numbers, you first need to change each of them into an improper fraction. The process: For each fraction, multiply the whole number by the denominator; then add that total to the numerator.
EXAMPLE 2:
EXAMPLE 3:
SELF-TEST 5
Multiplying Fractions
Multiply these fractions. Answers are given at the end of the chapter.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
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.
EXAMPLE
EXAMPLE 1:
EXAMPLE 2:
EXAMPLE 3:
FINE POINTS
Complex fractions such as
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.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
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 
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 
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.
5. Complete the division.
FINE POINTS
If the answer does not have a whole number, place a zero before the decimal point:.25 is incorrect; 0.25 is correct.
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
EXAMPLE 1:
EXAMPLE 2:
EXAMPLE 3:
FINE POINTS
In the answer for Example 2, note the space between 8 and the decimal point. When such a space occurs, fill it with a zero to complete your answer.
Converting Fractions to Decimals
Divide these f ractions to produce decimals. Answers are given at the end of the chapter. Carry decimal point to three decimal places if necessary. (Do not use rounding rules yet!)
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
Decimals
Most medication orders are written in the metric system, which uses decimals.
Reading Decimals and Converting Decimals to Fractions
Start by counting how many places come after the decimal point. One space after the decimal point is the tenths place. Two spaces is the hundredths place. Three places is the thousandths place, and so on. When you read the decimal aloud, it sounds like you’re reading a fraction:
0.1 is read as “one tenth” 
0.01 is read as “one hundredth” 
0.001 is read as “one thousandth” 
Always read the number by its name first, then count off the decimal places. If a whole number precedes the decimal, read it just as you normally would.
Since decimals are parts of a whole number, you can write them as fractions:
EXAMPLE
0.56 = “fifty-six hundredths” 
0.2 = “two tenths” 
0.194 = “one hundred ninety-four thousandths” 
0.31 = “thirty-one hundredths” 
1.6 = “one and six tenths” 
17.354 = “seventeen and three hundred fifty-four thousandths” 
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