Math Review & Basic Pharmacology Math

Posology Purpose One very important part of nursing practice is the ability to calculate correct dosages of drugs and solutions. Accurately performing mathematical calculations is essential to the nurse not only in the study of pharmacology, but also in chemistry and physiology.

The information required to correctly calculate dosages and prepare medications for administration is provided. Self-assessment tests are included. The self-assessment tests will identify strengths and weaknesses, as well as provide direction for areas that may need additional help.

Objectives Identify relative value, add, subtract, multiply, divide and reduce to the lowest terms any type of fractions, decimal or mixed number. Define the concept of ratio and proportion and use to solve equations to calculate the value of “x”. Explain the relationship between decimals, common fractions, ratios and percents, and express as equivalents of each other.

Objectives (cont.) Identify specific measurements with the Household and Apothecaries’ system and convert to metric equivalents. Convert numbers from one unit of measure to another unit, within the same system or between two systems (ex. Metric, apothecaries’).

Use of this material This material will help to determine your level of competency and what you may need to review prior to beginning drug calculation problems. This review includes some basic instructions in the math that you will use and some sample problems for you to solve. The more use you make of the study guide, the less problems you will have when you begin drug calculations.

Introduction Serious harm to a patient can result from small mathematical errors when calculating drug dosages. Therefore, the ability to accurately calculate correct dosages is of utmost importance. “This is not an option.”

Intro (cont.) Learning to calculate drug dosages need not be a source of anxiety. Following a step-by-step approach and using basic mathematical skills, you will be able to consistently calculate drug dosages accurately and confidently.

Intro (cont.) If you approach drug calculations in a logical manner, many errors can be avoided. Often, drug dosages can be calculated in your head (or at least you can obtain a rough estimate of the correct amount). Of course, you should always double-check your thinking mathematically. When calculating dosages, always consider the reasonableness of the computation. If you have a feeling that something is not right, it most likely is NOT. Let common sense prevail!

Measurements in the Metric and Apothecary Systems Decimals Grams (Gms) Millograms (mg) Liters (L) Milliliters (ml) Kilogram (Kg) Minims (mx) Microgram (mcg) Apothecary Fractions Grains (gr) Dram (℥) Ounce (ℨ) Pound (lb) Teaspoon (tsp) Tablespoon (Tbsp) Drops (gtts)

Equivalents 1 Gm = 1000 mg 0.5 Gm = 500 mg gr. l = 60-64 mg gr. 1 ½ = 90-100 mg gr. ½ = 30 mg gr. ¼ = 15 mg gr. 1/6 = 10 mg gr. 1/100 = 0.6 mg gr. 1/150 = 0.4 mg gr. 1/200 = 0.3 mg 1 L = 1000 ml 1 mg = 1000 mcg 1 ℨ = 30 ml 1 ℥ = 4 ml 1 Tbsp = 15 ml 1 Tsp = 4-5 ml 1 qt. = 1000 ml = 1 L 1 Kg = 2.2 lbs. 15-16 gtts = 1 ml 15-16 mx = 1 ml

Fractions A fraction is a part of a whole number and consists of two parts – a numerator and a denominator. Relative Value of Fractions After understanding the significance of the two parts of a fraction, one must then be able to identify the value of the fraction. When two or more fractions have the same denominator, the numerator determines the relative value of the fraction.

Fractions (cont.) Relative Value (cont.) Example: 5/6 is greater than 2/6 2/8 is less than 7/8 If two or more fractions have the same numerator, the denominator determines the relative value to the fraction. Example: 4/16 is less than 4/6 1/5 is greater than 1/8

Fractions (cont.) Type of Fractions Proper fractions – fractions that have numerators smaller than their denominator Example: ¼, 6/8, 1/150 Improper fractions – fractions that have numerators larger than or equal to their denominators. Example: 8/3, 12/4, 200/150

Fractions (cont.) Improper fractions can be changed to mixed numbers by dividing the denominator into the numerator. Example: 12/4 = 12  4 = 3 8/3 = 8  3 = 2 2/3 Mixed numbers – Numbers that have a whole number and a fraction part. Example: 4 ½, 6 4/5, 7 1/8

Fractions (cont.) Reduction of Fraction and Equivalent Fractions The purpose of changing the terms of a fraction without changing its value is called reduction. This may be done by dividing both terms of a fraction by the same number. Example: 4/16 4  4 = 1 16  4 = 4 4/16 = 1/4

Fractions (cont.) Addition and subtraction of fractions Fractions may only be added to or subtracted from each other when they have the same denominator. Example: 1/3 + 2/4 + 5/6 12 is the lowest number into which each of these denominators can be divided (least common denominator or LCD). 1/3 3 into 12 = 4 4x1 = 4 Therefore 1/3 = 4/12 3/4 4 into 12 = 3 3x3 = 9 Therefore 3/4 = 9/12 5/6 6 into 12 = 2 2x5 = 10 Therefore 5/6 = 10/12

Fractions (cont.) After determining the LCD, the numerators of the individual fractions are added together. The denominator will remain unchanged. 4/12 + 9/12 + 10/12 = 23/12 This improper fraction can then be changed to a mixed number. 23  12 = 1 11/12 The steps for subtraction of fractions are the same. First obtain the LCD, then subtract the numerators. 11/12 – 6/12 = 5/12

Fractions (cont.) Multiplication and Division of Fractions To multiply a fraction by a whole number, simply multiply the numerator by the whole number. The denominator will remain unchanged. Example: 2/3 x 4 = 2 x 4 = 8 = 2 2/3 3 3

Fractions (cont.) If the whole number and the denominator have a common divisor, cancellation may be used to shorten multiplication. 2 x 9 = 18 = 6 3 3 or 3 2 x 9 = 6 = 6 3 1 1

Fractions (cont.) To multiply a fraction by a fraction, multiply the numerator of each fraction for the numerator of the product. Multiply the denominator of each fraction for the denominator of the product. Example -- 2 x 3 = 2 x 3 = 6 3 5 3 x 5 15 (6/15 can be reduced to 2/5) Again, when the denominator of one fraction and the numerator of another have a common divisor, cross cancellation can be used. 1 2 x 3 = 2 3 5 5

Fractions (cont.) Division To divide a whole number by a fraction, invert the terms in the divisor and multiply Example: 8 ÷ 3 = 8 x 4 = 32 = 10 2 4 3 3 3 To divide a fraction by a whole number, make the whole number an improper fraction (set the number over 1). For example, 4 = 4/1. Then invert the divisor and multiply. Example: 2 ÷ 4 = 2 ÷ 4 = 2 x 1 = 2 = 1 3 3 1 3 4 12 6

Decimals Decimal fractions Decimal fractions are special fractions that are often more easy to use than ordinary fractions. A decimal fraction is special because it always has 10 or some multiple of 10 as its denominator. The most commonly used measuring system for medications, the metric system, uses decimal numbers.

Decimals (cont.) To the right of the decimal point, each place represents a fraction whose denominator is 10. Each place has a name. One place to the right of the decimal point represents a denominator of 10. Two places to the right represents a denominator of 100. Three places to the right represents a denominator of 1000 and so forth. From the decimal point to the left each place has a name. Moving to the left, the places include ones, tens, hundreds, thousands and so forth. D E C I M A L HUNDREDS TENS ONES . TENTHS HUNDREDTHS THOUSANDTHS Example: 1 5 5 . 2 0 5

Decimals (cont.) Addition of Decimals When adding decimals, arrange the numbers to be added so that the decimal points line up in a column. NEXT, add the numbers as if they are whole numbers. Keep the decimal point at the same place in the answer (sum) as it is in the problem. Example: 7.439 + 32.460 (answer) 39.899

Decimals (cont.) Subtraction of Decimals When subtracting decimals, again arrange the numbers so that the decimal points line up in a column. Then subtract the numbers as if they are whole numbers. Remember to keep the decimal point at the same place in the answer as it is in the problem. 235.76 - 34.61 (answer) 201.15

Decimals (cont.) Multiplication of Decimals When multiplying decimal numbers, the numbers are multiplied in the same manner as whole numbers. The total number of places to the right of the decimals in the problem is then used to determine the decimal place in the answer. 114.2 x 2.81 (answer) 320.902

Decimals (cont.) Division of Decimals To divide one decimal by another, divide as if you are dividing whole numbers. If the divisor (the number by which you are dividing) is a decimal number, move the decimal point all the way to the right. You must also move the decimal points in the dividend (the number being divided) the same number of places to the right. Add as many zeroes as necessary to the dividend to allow correct placement of the decimal point. Example: --- 8 ÷ 0.16 50. .16 8.00. 80

Decimals (cont.) Division of Decimals (cont.) To divide a decimal or whole number by 10, by 100, or by 1000, etc., move the decimal point as many places to the left as there are zeroes in the divisor. Example -- 1 ÷ 10 = 0.1 1 ÷ 100 = 0.01 1 ÷ 1000 = 0.001

Percentages The term percentage means hundredths. Percents are merely decimal fractions with denominators of 100. For instance, “6%” means 6 in every 100 parts and may be written as 6/100, 0.06, or 6%. Example: 15% = 15  100 = 0.15

Ratio and Proportion To determine the value of an “unknown”, use a ratio and proportion formula. An unknown occurs when the amount of a medication that is to be given differs from the actual dosage of the medication you have available (on hand). Therefore, you will need to calculate how much of the actual medication you will be required to give in order to deliver the correct dosage. You will need to put to work your knowledge of “equivalents” to determine accurate dosages.

Ratio and Proportion (cont.) The Ratio and Proportion method can be accomplished by setting up a basic two-sided equation. On the left side of the equation, put what you want to determine (the amount you wish to administer). On the right side of the equation, put the information you already know (the amount of medication you have on hand). After putting the correct information in to the equation, simply solve for the unknown (X).

Ratio and Proportion (cont.) The equations can be set up using a fraction format on each side and cross multiplying or by using a ratio format with a colon and multiplying the extremes and the means to solve for the unknown. Example: You want to give 300 mg of Drug ABC. You have available 400 mg/ml. 300 mg = 400 mg x ml 1 ml 300 = 400x x = 0.75

Ratio and Proportion (cont.) 300 mg : x ml = 400 mg : 1 ml 300 = 400x x = 0.75

Roman Numerals You should know the basic numbers Examples: 1 = I 15 = XV 5 = V 16 = XVI 6 = VI 50 = L 9 = IX 100 = C 10 = X 1000 = M