1. Chapter 5: Mathematics Review and Introduction to Dosage Calculation
Pharmacology I

2. Why Do Nurses Need Math? Two of the six “rights” require calculation
Right dose
Right time
DIMS test (“does it make sense?”)
PINCH drugs
Potassium, insulin, narcotics, chemotherapy and cardiac drugs, and heparin or other anti-clotting drugs
As a nurse, you will be responsible for giving drugs to others safely and accurately.
Drugs often do not come from the pharmacy in exactly the right dose for a patient. How much to administer based on what is available on hand must be calculated.
Dosage calculations are performed in the same way whether the drug is a tablet, oral liquid, injectable liquid, or suppository.
Use the DIMS test when calculating a drug dose. Example: Answer that does not look right (e.g., 15 tablets!) is probably wrong.
PINCH drugs are considered high-risk and are very dangerous if the dose is miscalculated. Their doses must be checked by a second health care professional. 2

3. Talking About Numbers
The simplest way to think about fractions is to think about money.
Figure 5-1 shows the relationship of a fraction (part of a whole number) to a whole number.
Figure 5-1. Comparison of part of a number to a whole number. 3

4. Fractions Fractions are actually division problems
25/100 is the same as 25 ÷ 100
Top number is numerator
Bottom number is denominator
Remember NU/DE!
Whole numbers can be written as fractions
15 = 15/1
A fraction is a part of a whole number obtained by dividing one number by a larger number. 4

5. Types of Fractions Proper fractions Improper fractions
½, ¾
Improper fractions
11/5, 7/2
Mixed number fractions
1½, 2¾
Reduced fractions
2/4 = ½, 3/9 = 1/3
A proper fraction is one in which the top number is smaller than the bottom number; it is the most common type of fraction.
An improper fraction is one that has a top number (numerator) that is larger than the bottom number (denominator), meaning that the value is greater than 1.
A mixed number fraction can be changed into an improper fraction by multiplying the denominator by the whole number and then adding the numerator.
Convert 3½ into a whole number. 3 × 2 (= 6), then add 1 to the answer (= 7). The answer is 7/2.
Reduced fractions are those that have been changed to their lowest common denominator. By reducing a fraction, its value is not changed.
Which is larger - 240/120 or 2/1? They are equal! 5

6. Comparing Fractions
If all numerators are 1, smallest denominator = largest fraction
If fractions have different numerators and denominators, determine lowest common denominator
Fraction with highest numerator will be the largest
The nurse may have to compare several drugs and determine which dose is the strongest (or weakest).
A dose comes in 1/4 , 1/2 , and 1/8 strengths. How will the nurse determine which is the strongest dose? If the numerators are all the same (in this case they are all equal to 1), the smallest denominator is the strongest dose (1/2).
If the denominators AND the numerators are different, denominators must be converted to a common denominator. you can figure out exactly how they are related by comparing their numerators. 6

7. Comparing Fractions (cont’d)
Figure 5-2. Fraction sizes. 7

8. Adding and Subtracting Fractions
Fractions with common denominators
Simply add/subtract numerators
Fractions with different denominators
Convert fractions to lowest common denominator, then add/subtract numerators
Reduce to lowest terms
Subtraction rare in drug calculation
Add 2/6 + 2/6 + 4/6 + 5/6. 13/6 Subtract 2/4 – 1/4. 1/4
When the sum or difference of fractions is an improper fraction, convert it to a mixed fraction.
What is 13/6 reduced to a mixed number? 2 1/6
Add 1/2 + 4/5 + 3/4. Lowest common denominator = 20; add numerators: = 41; place over denominator 41/20; reduce to lowest terms 2 1/20. 8

9. Multiplying Fractions
Reduce fractions, if possible
Convert all mixed numbers to improper fractions
Multiply all numerators
Product is new numerator
Multiply all denominators
Product is new denominator
Reduce!
Easier than adding and subtracting fractions.
Finding a lowest common denominator is not necessary. 9

10. Dividing Fractions Invert (flip) the second fraction
Multiply fractions
Multiply numerators
Multiply denominators
Reduce
If whole number is involved, change to improper fraction 10

11. Decimals Based on multiples of 10
A decimal divides a whole number and part of a number (fraction)
Whole number (.) tenths, hundredths, thousandths
Always include a zero to the left of a decimal less than one
Never put an extra zero to the right of a decimal!
A decimal, like a fraction, describes parts of a whole.
Decimals can be written as fractions and fractions can be written as decimals.
Any number to the left of a decimal point is a whole number; any number to the right of a decimal point is a part of a whole number.
How should “five tenths” be written numerically? 0.5
If you calculate 0.250, how should you write this? 0.25 11

12. Adding and Subtracting Decimals
Align decimal points
Add or subtract as with whole numbers
When adding , keep the decimal points of all the numbers to be added in the same position. 12

13. Multiplying Decimals Multiply as with whole numbers
Count number of decimal spaces to the right of decimal points in the problem
Starting at the far right of the answer, count same number of decimal spaces and place the decimal
In multiplying decimals, the answer must contain the right number of decimal places for it to make sense. 13

14. Dividing Decimals
Quotient
Parts:
Divisor
Dividend 14

15. Dividing Decimals (cont’d)
If divisor is whole number, keep decimal in quotient in the same place as dividend
If divisor is decimal, convert to whole number
Do to the divisor what is done to the dividend
Check the math! Multiplying divisor by quotient should equal the dividend!
Keep in mind that you must always divide by a whole number.
If the divisor is a decimal, move the decimal point all the way to the right to make it a whole number. Whatever is done to the divisor must also be done to the dividend--move the decimal point in the dividend the same number of decimal spaces it was moved in the divisor to make it a whole number.
For drug calculations, a decimal problem will not need to be worked through to more than the thousandths (third) place. 15

16. Fractions and Decimals
Fraction to decimal:
Divide numerator by denominator
Add 0 to left of decimal point if answer is less than 1
Decimal to fraction:
Keep numbers to left of decimal point as whole numbers
Drop decimal point and place numbers over the place value (e.g., 0.25 = 25/100)
Reduce to lowest terms (e.g., 0.25 = ¼)
Decimals and fractions are related because they are both parts of a whole. 16

17. Rounding Parts of Numbers
Liquid doses usually rounded to nearest tenth
Most syringes calibrated in tenths
Answers ending below 0.05 are rounded down
Answers equaling or ending above 0.05 are rounded up
Tablets usually rounded to nearest whole
Exception – some tablets can be cut in half!
Exception: heparin is measured in TB syringes that are calibrated in hundredths.
The key concept in rounding decimals is to remember the number 5!
If an answer is 2.82 mL, give 2.8 mL (rounded down). If an answer is 2.17 mL, give 2.2 mL (rounded up). 17

18. Percents Express number as part of a hundred
Used to calculate drug doses and strength of solutions
Example: 5% dextrose in water (D5W)
The word percent literally means “for each hundred.” 18

19. Percents and Decimals Convert percent to decimal:
Drop % sign, multiply by 0.01 (or move decimal 2 places left)
Example: 9% = 0.09
Convert decimal to percent:
Add % sign, multiply by 100 (or move decimal 2 places right)
Example 0.05 = 5%
Moving decimal the wrong way can result in serious error! Work it out or call the pharmacist if you are not sure. 19

20. Percents and Fractions
Percent to fraction:
Convert percent to decimal
Convert decimal to fraction
Fraction to percent:
Convert fraction to decimal
Multiply by 100 (or move decimal 2 places right)
Add % sign
Be sure to move the decimal in the correct direction.
Ask another health care practitioner to check your work if you are not sure. 20

21. Percentage of a Number Convert percent to decimal Multiply
Problem: 25% of 200
Convert: 25% = 0.25
Multiply: 200 × 0.25 = 50
Be careful with decimal points!
Example: 50% of 84 is 42. To get that answer, either multiply 84 by 0.50, or divide 84 by 2.
Remember to count the total number of decimal places and put them in your answer. 21

22. Introduction to Dosage and Calculation
Most drugs dispensed from pharmacy in correct dose
Nurse is last check in the system
Double-check decimal points and zeros!
Label all numbers in calculations
Plug numbers into formula, do the math
Do the DIMS test!
The nurse is responsible for making sure that the patient not only gets the right drug, but also gets the right dose.
Formulas can be used only when the drug dose on hand is the same measurement unit (e.g., mg and mg) as the drug dose to be given. 22

23. Oral Drugs Formula #1: Dry pill, tablets, etc.
= Number of tablets to give
Formula #2: Liquid medication
× LIQUID = Amount of liquid to give
Formula #1 example: A dose of diazepam (Valium) ordered is 15 mg; on hand is Valium 10-mg tablets. 15 mg/10 mg = 1 1/2 tablets.
Label both the numerator and denominator of the formula to ensure that the two dosage measurements are the same. 23

24. Drugs Given by Injection
Three types
Intramuscular (IM)
Subcutaneous
Intradermal (ID)
Available in single- or multi-dose packaging
Use formula #2, same as other liquid drugs
All three types are parenteral forms of drug delivery that do not go through the GI tract. 24

25. Proportion Equal mathematical relationship between two sets of numbers
Example: ½ = 2/4
Example: 3 boats/6 sails = 9 boats/18 sails
Always label components
Right side must be set up in same order as left (e.g., boats “as to” sails on each side)
Another way of thinking of proportion is that one side of the equation is equally related to the other side.
Proportion can be used to solve for x, the unknown, as an alternate approach to drug calculations.
When writing a proportion as a set of fractions, be careful to label each piece of the equation. For example: 1 case/12 bottles = 3 cases/36 bottles. 25

26. Using Proportion to Solve for X
Calculate dose using proportion
Order: Give 500 mg of primidone by mouth (orally)
On hand: Primidone 250 mg per 1 caplet
Question: How many caplets equal 500 mg?
Set up problem: 250 mg/1 caplet = 500 mg/ X caplets
Calculate: Cross-multiply and solve for X – 500/250 = X
Answer: 2 caplets
When the prescriber orders a drug strength that is different from the one you have on hand, a piece of the proportion is missing. To figure out how many drug tablets you have on hand will be equal to the strength that is ordered, set up a proportion to solve for the missing piece.
X is what you need to give. By figuring out X correctly, both sides of the proportion will be equal.
That leaves the word “caplets” as the missing part of the proportion. Therefore, the answer must be the number of caplets needed.
Remember to label the problem correctly or you might not know what the answer actually means! 26

27. KEY POINTS The top number of a fraction is the numerator.
The bottom number of a fraction is the denominator.
If the numerator of a fraction is 1, then the larger the denominator, the smaller a part of the whole it is.
A whole number is turned into a fraction by making the whole number the numerator and making “1” the denominator.
Reducing fractions to their lowest terms (1 is the only number that can be evenly divided into the numerator and the denominator) helps simplify working with fractions.
Adding fractions that have the same denominator involves only adding the numerators; the denominator remains the same.
Adding fractions that have different denominators requires calculating the lowest common denominator, changing the numerators to proportionately match their new denominators, and adding the numerators.

28. KEY POINTS
Subtracting fractions that have the same denominator involves only subtracting the smaller numerator from the larger numerator; the denominator remains the same.
Subtracting fractions that have different denominators requires calculating the lowest common denominator, changing the numerators to proportionately match their new denominators, and subtracting the smaller numerator from the larger numerator.
Multiplying fractions involves multiplying the numerators with one another and then multiplying the denominators with one another.
When dividing fractions, the second fraction is inverted and multiplied by the first fraction.
When dividing a fraction with both the top number and the bottom number ending in one or more zeros, the same number of zeros can be “chopped off” both numbers.

29. KEY POINTS Decimal places are always written in multiples of 10.
The places to the right or left of the decimal determine the value of a number.
Place a zero before the decimal point of any proper fraction written as a decimal.
Never put a meaningless zero at the end of a decimal.
To change a fraction into a decimal, always divide the bottom number into the top number.
To check division of a decimal, multiply the divisor by your answer. If the division was performed correctly, you will get the dividend.
Moving the decimal point in error to the right will make a drug dose too high and may cause serious or even lethal side effects.
Moving a decimal point in error to the left will make a drug dose too small to be effective.

30. KEY POINTS
If your drug calculation for tablets results in a decimal number below 0.5, round down to the next lowest whole number. If the calculation results in a decimal number above 0.5, round up to the next highest whole number.
Do not attempt to cut a tablet that is not scored, a capsule, a gelcap, a drug that is enteric coated, or one that is long acting.
Urge a patient to drink a full glass of water whenever he or she takes a tablet or capsule unless fluids must be restricted because of another medical problem.
When reading a drug order, check and double check carefully for decimal points and zeros.
Remember to label proportion problems so you know the correct units for your final answer.