Math Review & Basic Pharmacology Math. Posology Purpose One very important part of nursing practice...

Math Review & Math Review & Basic Basic

Pharmacology Pharmacology MathMath

PosologyPosology

PurposePurpose

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

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

ObjectivesObjectives

Identify relative value, add, subtract, multiply, Identify relative value, add, subtract, multiply, divide and reduce to the lowest terms any type divide and reduce to the lowest terms any type of fractions, decimal or mixed number.of fractions, decimal or mixed number.

Define the concept of ratio and proportion and Define the concept of ratio and proportion and use to solve equations to calculate the value of use to solve equations to calculate the value of “x”.“x”.

Explain the relationship between decimals, Explain the relationship between decimals, common fractions, ratios and percents, and common fractions, ratios and percents, and express as equivalents of each other.express as equivalents of each other.

Objectives (cont.)Objectives (cont.)

Identify specific measurements with the Identify specific measurements with the Household and Apothecaries’ system and Household and Apothecaries’ system and convert to metric equivalents.convert to metric equivalents.

Convert numbers from one unit of Convert numbers from one unit of measure to another unit, within the same measure to another unit, within the same system or between two systems (ex. system or between two systems (ex. Metric, apothecaries’).Metric, apothecaries’).

Use of this materialUse of this material

This material will help to determine your level of This material will help to determine your level of competency and what you may need to review competency and what you may need to review prior to beginning drug calculation problems.prior to beginning drug calculation problems.

This review includes some basic instructions in This review includes some basic instructions in the math that you will use and some sample the math that you will use and some sample problems for you to solve.problems for you to solve.

The more use you make of the study guide, the The more use you make of the study guide, the less problems you will have when you begin less problems you will have when you begin drug calculations.drug calculations.

IntroductionIntroduction

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

Intro (cont.)Intro (cont.)

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

Intro (cont.)Intro (cont.)

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

Measurements in the Metric and Measurements in the Metric and Apothecary SystemsApothecary Systems

MetricMetricDecimalsDecimalsGrams (Gms)Grams (Gms)Millograms (mg)Millograms (mg)Liters (L)Liters (L)Milliliters (ml)Milliliters (ml)Kilogram (Kg)Kilogram (Kg)Minims (mx)Minims (mx)Microgram (mcg)Microgram (mcg)

ApothecaryApothecaryFractionsFractionsGrains (gr)Grains (gr)Dram (Dram (℥)℥)Ounce (ℨ)Ounce (ℨ)Pound (lb)Pound (lb)Teaspoon (tsp)Teaspoon (tsp)Tablespoon (Tbsp)Tablespoon (Tbsp)Drops (gtts)Drops (gtts)

EquivalentsEquivalents

1 Gm = 1000 mg1 Gm = 1000 mg 0.5 Gm = 500 mg0.5 Gm = 500 mg gr. l = 60-64 mggr. l = 60-64 mg gr. 1 ½ = 90-100 mggr. 1 ½ = 90-100 mg gr. ½ = 30 mggr. ½ = 30 mg gr. ¼ = 15 mggr. ¼ = 15 mg gr. 1/6 = 10 mggr. 1/6 = 10 mg gr. 1/100 = 0.6 mggr. 1/100 = 0.6 mg gr. 1/150 = 0.4 mggr. 1/150 = 0.4 mg gr. 1/200 = 0.3 mggr. 1/200 = 0.3 mg

1 L = 1000 ml1 L = 1000 ml 1 mg = 1000 mcg1 mg = 1000 mcg 1 1 ℨ = 30 mlℨ = 30 ml 1 ℥ = 4 ml1 ℥ = 4 ml 1 Tbsp = 15 ml1 Tbsp = 15 ml 1 Tsp = 4-5 ml1 Tsp = 4-5 ml 1 qt. = 1000 ml = 1 L1 qt. = 1000 ml = 1 L 1 Kg = 2.2 lbs.1 Kg = 2.2 lbs. 15-16 gtts = 1 ml15-16 gtts = 1 ml 15-16 mx = 1 ml15-16 mx = 1 ml

FractionsFractions

A fraction is a part of a whole number and A fraction is a part of a whole number and consists of two parts – a consists of two parts – a numeratornumerator and a and a denominatordenominator..

Relative Value of FractionsRelative Value of FractionsAfter understanding the significance of the two parts of a After understanding the significance of the two parts of a fraction, one must then be able to identify the value of the fraction, one must then be able to identify the value of the fraction. When two or more fractions have the same fraction. When two or more fractions have the same denominator, the numerator determines the relative value denominator, the numerator determines the relative value of the fraction.of the fraction.

Fractions (cont.)Fractions (cont.)

Relative Value (cont.)Relative Value (cont.)

Example:Example: 5/6 is greater than 2/65/6 is greater than 2/6

2/8 is less than 7/82/8 is less than 7/8

If two or more fractions have the same numerator, the If two or more fractions have the same numerator, the denominator determines the relative value to the fraction.denominator determines the relative value to the fraction.

Example:Example: 4/16 is less than 4/64/16 is less than 4/6

1/5 is greater than 1/81/5 is greater than 1/8

Fractions (cont.)Fractions (cont.) Type of FractionsType of Fractions

Proper fractions – fractions that have Proper fractions – fractions that have numerators smaller than their denominator numerators smaller than their denominator Example: ¼, 6/8, 1/150Example: ¼, 6/8, 1/150

Improper fractions – fractions that have Improper fractions – fractions that have numerators larger than or equal to their numerators larger than or equal to their denominators. Example: 8/3, 12/4, 200/150denominators. Example: 8/3, 12/4, 200/150


Improper fractions can be changed to mixed Improper fractions can be changed to mixed numbers by dividing the denominator into the numbers by dividing the denominator into the numerator.numerator.Example: Example: 12/4 = 12 12/4 = 12 4 = 3 4 = 3

8/3 = 8 8/3 = 8 3 = 2 2/3 3 = 2 2/3

Mixed numbers – Numbers that have a whole Mixed numbers – Numbers that have a whole number and a fraction part. number and a fraction part. Example: 4 ½, 6 4/5, 7 1/8Example: 4 ½, 6 4/5, 7 1/8


Reduction of Fraction and Equivalent Reduction of Fraction and Equivalent FractionsFractionsThe purpose of changing the terms of a fraction without The purpose of changing the terms of a fraction without changing its value is called reduction. This may be done changing its value is called reduction. This may be done by dividing by dividing bothboth terms of a fraction by the same number. terms of a fraction by the same number. Example: Example: 4/164/16

4 4 4 = 1 4 = 1

16 16 4 = 4 4 = 4

4/16 = 1/44/16 = 1/4


Addition and subtraction of fractionsAddition and subtraction of fractionsFractions may only be added to or subtracted from each Fractions may only be added to or subtracted from each other when they have the same denominator. other when they have the same denominator. Example:Example: 1/3 + 2/4 + 5/61/3 + 2/4 + 5/6

12 is the lowest number into which each of these denominators can be 12 is the lowest number into which each of these denominators can be divided (least common denominator or LCD).divided (least common denominator or LCD).

1/31/3 3 into 12 = 4 3 into 12 = 4 4x1 = 4 4x1 = 4 Therefore 1/3 = 4/12Therefore 1/3 = 4/12

3/4 4 into 12 = 3 3x3 = 93/4 4 into 12 = 3 3x3 = 9 Therefore 3/4 = 9/12Therefore 3/4 = 9/12

5/65/6 6 into 12 = 2 2x5 = 10 6 into 12 = 2 2x5 = 10 Therefore 5/6 = 10/12Therefore 5/6 = 10/12


After determining the LCD, the numerators of the After determining the LCD, the numerators of the individual fractions are added together. The individual fractions are added together. The denominator will remain unchanged.denominator will remain unchanged.4/12 + 9/12 + 10/12 = 23/124/12 + 9/12 + 10/12 = 23/12

This improper fraction can then be changed to a mixed This improper fraction can then be changed to a mixed number.number.23 23 12 = 1 11/12 12 = 1 11/12

The steps for subtraction of fractions are the same. First The steps for subtraction of fractions are the same. First obtain the LCD, then subtract the numerators.obtain the LCD, then subtract the numerators.11/12 – 6/12 = 5/1211/12 – 6/12 = 5/12


Multiplication and Division of FractionsMultiplication and Division of Fractions MultiplicationMultiplication

To multiply a fraction by a whole number, To multiply a fraction by a whole number, simply multiply the numerator by the whole simply multiply the numerator by the whole number. The denominator will remain number. The denominator will remain unchanged.unchanged.

Example:Example: 2/3 x 4 = 2/3 x 4 = 2 x 4 2 x 4 = = 88 = 2 2/3 = 2 2/3

33 3 3

If the whole number and the denominator have If the whole number and the denominator have a common divisor, cancellation may be used to a common divisor, cancellation may be used to shorten multiplication.shorten multiplication.

22 x 9 = x 9 = 1818 = 6 = 6

3 33 3

oror 33

22 x 9 = x 9 = 66 = 6 = 6

3 13 1

11


To multiply a fraction by a fraction, multiply the numerator of To multiply a fraction by a fraction, multiply the numerator of each fraction for the numerator of the product. Multiply the each fraction for the numerator of the product. Multiply the denominator of each fraction for the denominator of the product.denominator of each fraction for the denominator of the product.

Example -- Example -- 22 x x 3 3 = = 2 x 3 2 x 3 = = 663 5 3 x 5 153 5 3 x 5 15(6/15 can be reduced to 2/5)(6/15 can be reduced to 2/5)

Again, when the denominator of one fraction and the numerator Again, when the denominator of one fraction and the numerator of another have a common divisor, cross cancellation can be of another have a common divisor, cross cancellation can be used.used.

1122 x x 33 = = 223 53 5 5 511


Fractions (cont.)Fractions (cont.) DivisionDivision

To divide a whole number by a fraction, invert the terms in the To divide a whole number by a fraction, invert the terms in the divisor and multiplydivisor and multiply

Example:Example: 8 ÷ 8 ÷ 33 = 8 x = 8 x 44 = = 3232 = 10 = 10 22 44 3 3 3 3 3 3

To divide a fraction by a whole number, make the whole To divide a fraction by a whole number, make the whole number an improper fraction (set the number over 1). For number an improper fraction (set the number over 1). For example, 4 = 4/1. Then invert the divisor and multiply. example, 4 = 4/1. Then invert the divisor and multiply.

Example:Example: 22 ÷ 4 = ÷ 4 = 22 ÷ ÷ 44 = = 22 x x 11 = = 22 = = 113 3 1 3 4 12 63 3 1 3 4 12 6

DecimalsDecimals

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

Decimals (cont.)Decimals (cont.) To the right of the decimal point, each place represents a fraction To the right of the decimal point, each place represents a fraction

whose denominator is 10. Each place has a name. One place to whose denominator is 10. Each place has a name. One place to the right of the decimal point represents a denominator of 10. Two the right of the decimal point represents a denominator of 10. Two places to the right represents a denominator of 100. Three places places to the right represents a denominator of 100. Three places to the right represents a denominator of 1000 and so forth. From 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 the decimal point to the left each place has a name. Moving to the left, the places include ones, tens, hundreds, thousands and so left, the places include ones, tens, hundreds, thousands and so forth.forth.

DDEECCIIMMAALL

HUNDREDSHUNDREDS TENS TENS ONESONES . . TENTHSTENTHS HUNDREDTHS HUNDREDTHS THOUSANDTHSTHOUSANDTHS

Example: 1 5 5 . 2 0 5Example: 1 5 5 . 2 0 5

Decimals (cont.)Decimals (cont.)

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

Example:Example: 7.439 7.439

+ + 32.46032.460

(answer) 39.899(answer) 39.899


Subtraction of DecimalsSubtraction of DecimalsWhen subtracting decimals, again arrange the When subtracting decimals, again arrange the numbers so that the decimal points line up in a numbers so that the decimal points line up in a column. Then subtract the numbers as if they column. Then subtract the numbers as if they are whole numbers. Remember to keep the are whole numbers. Remember to keep the decimal point at the same place in the answer as decimal point at the same place in the answer as it is in the problem.it is in the problem.

235.76235.76 - - 34.6134.61

(answer) 201.15(answer) 201.15


Multiplication of DecimalsMultiplication of DecimalsWhen multiplying decimal numbers, the numbers When multiplying decimal numbers, the numbers are multiplied in the same manner as whole are multiplied in the same manner as whole numbers. The total number of places to the right numbers. The total number of places to the right of the decimals in the problem is then used to of the decimals in the problem is then used to determine the decimal place in the answer.determine the decimal place in the answer.

114.2114.2 x 2.81x 2.81

(answer) 320.902(answer) 320.902

Division of DecimalsDivision of DecimalsTo divide one decimal by another, divide as if you are dividing whole 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 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. decimal number, move the decimal point all the way to the right. You must also move the decimal points in the dividend (the number 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 being divided) the same number of places to the right. Add as many zeroes as necessary to the dividend to allow correct placement of zeroes as necessary to the dividend to allow correct placement of the decimal point.the decimal point.

Example: --- Example: --- 8 ÷ 0.16 8 ÷ 0.16

50.50..16 8.00..16 8.00. 8080

00



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

Example --Example -- 1 ÷ 10 = 0.11 ÷ 10 = 0.1

1 ÷ 100 = 0.011 ÷ 100 = 0.01

1 ÷ 1000 = 0.0011 ÷ 1000 = 0.001

PercentagesPercentages

The term The term percentagepercentage means hundredths. means hundredths. Percents are merely decimal fractions with Percents are merely decimal fractions with denominators of 100. For instance, “6%” denominators of 100. For instance, “6%” means 6 in every 100 parts and may be means 6 in every 100 parts and may be written as 6/100, 0.06, or 6%.written as 6/100, 0.06, or 6%.

Example: 15% = 15 Example: 15% = 15 100 = 0.15 100 = 0.15

Ratio and ProportionRatio and Proportion

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

Ratio and Proportion (cont.)Ratio and Proportion (cont.)

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

Ratio and Proportion (cont.)Ratio and Proportion (cont.) The equations can be set up using a fraction format on The equations can be set up using a fraction format on

each side and cross multiplying or by using a ratio each side and cross multiplying or by using a ratio format with a colon and multiplying the extremes and format with a colon and multiplying the extremes and the means to solve for the unknown.the means to solve for the unknown.

Example: You want to give 300 mg of Drug ABC. Example: You want to give 300 mg of Drug ABC. You have available 400 mg/ml.You have available 400 mg/ml.

300 mg300 mg = = 400 mg400 mg x mlx ml 1 ml 1 ml

300 300 == 400x400x

xx == 0.750.75

Ratio and Proportion (cont.)Ratio and Proportion (cont.)

OR:OR:

300 mg : x ml300 mg : x ml == 400 mg : 1 ml400 mg : 1 ml

300300 == 400x400x

xx == 0.750.75

Roman NumeralsRoman Numerals

You should know the basic numbersYou should know the basic numbers

Examples:Examples:1 = I1 = I 15 = XV15 = XV5 = V5 = V 16 = XVI16 = XVI6 = VI6 = VI 50 = L50 = L9 = IX9 = IX 100 = C100 = C10 = X10 = X 1000 = M1000 = M

Math Review & Basic Pharmacology Math. Posology Purpose One very important part of nursing practice...

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