Y khoa - Dược - Chapter 1: Numbering systems and mathematical review

McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedMath for the Pharmacy Technician: Concepts and CalculationsChapter 1: Numbering Systems and Mathematical Review Egler • Booth1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedNumbering Systems and Mathematical Review 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedLearning OutcomesIdentify and determine the values of Roman and Arabic numerals.Understand and compare the values of fractions in various formats.Accurately add, subtract, multiply, and divide fractions and decimals.Convert fractions to mixed numbers and decimals.When you have successfully completed Chapter 1, you will have mastered skills to be able to:1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedLearning Outcomes (con’t)Recognize the format of decimals and measure their relative values.Round decimals to the nearest tenth, hundredth, or thousandth.Describe the relationship among percents, ratios, decimals, and fractions.1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedIntroductionBasic math skills are building blocks for accurate dosage calculations.You must be confident in your math skills.A minor mistake can mean major errors in the patient’s medication.1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedArabic NumbersArabic numbers include all numbers used today. Numbers are written using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. You can write whole numbers, decimals, and fractions by simply combining digits. 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedArabic Numbers(con’t) The Arabic digits 2 and 5 can be combined to write: The whole number 25 The decimal 2.5 The fraction 2/5 The same two digits are used in each of the above Arabic numbers but each have different values.Example1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedRoman Numerals Are used sometimes in drug ordersYou need to understand how to change Roman numeral into Arabic numbers in order to do dosage calculationsCommonly used Roman numeralsss = ½I = 1V = 5X = 10They may be written in lower or uppercase1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedCombining Roman Numerals When reading a Roman numeral containing more than 1 letter, follow these two steps:If any letter with a smaller value appears before a letter with a larger value, subtract the smaller value from the larger value.Add the value of all the letters not affected by Step 1 to those that were combined.1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedCombining Roman Numerals(con’t)ExampleRoman numerals from 1 to 30 are the ones you are most likely to see in doctors’ orders.Be familiar with these to read orders correctly.ExampleIX = 10 –1 = 9XIV = 10 + (5-1) = 141-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedFractions and Mixed NumbersMeasure a portion or part of a whole amountWritten two ways:Common fractionsDecimals1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedCommon FractionsRepresent equal parts of a wholeConsist of two numbers and a fraction barWritten in the form:Numerator (top part of the fraction) = part of whole Denominator (bottom part of the fraction) represents the wholeone part of the whole the whole 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedCommon Fractions (con’t)Scored (marked) tablet for 2 partsYou administer 1 part of that tablet each dayYou would show this as 1 part of 2 wholes or ½Read it as “one half”1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedFraction RuleWhen the denominator is 1, the fraction equals the number in the numerator.ExampleCheck these equations by treating each fraction as a division problem.1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedMixed NumbersMixed numbers combine a whole number with a fraction.Example Fractions with a value greater than 1 are written as mixed numbers.2 (two and two-thirds) 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedIf the numerator of the fraction is less than the denominator, the fraction has a value of 1. Mixed Numbers (con’t)1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedTo convert a fraction to a mixed number:Divide the numerator by the denominator. The result will be a whole number plus a remainder.Write the remainder as the number over the original denominator.Combine the whole number and the fraction remainder. This mixed number equals the original fraction.Mixed Numbers (con’t)Only applied when the numerator is greater than the denominator1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedMixed Numbers (con’t) Convert to a mixed number. Divide the numerator by the denominator = 2 R3 (R3 means a remainder of 3)3. The result is the whole number 2 with a remainder of 34. Write the remainder over the whole ¾5. Combine the whole number and the fraction 2+ ¾ Example 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedTo convert a mixed number ( ) to a fraction: 1. Multiply the whole number (5) by the denominator (3) of the fraction ( ) 5x3 = 15 Add the product from Step 1 to the numerator of the fraction 15+1 = 16 Mixed Numbers (con’t)1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedTo convert a mixed number to a fraction:Write the sum from Step 2 over the original denominatorThe result is a fraction equal to original mixed number. Thus Mixed Numbers (con’t)1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedWhat is the numerator in ?Review and PracticeAnswer = 17Answer = 100Answer =What is the denominator in ?Twelve patients are in the hospital ward. Four have type A blood. What fraction do not have type A blood?1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedEquivalent Fractions same as same as Find equivalent fractions forExample Two fractions written differently that have the same value = equivalent fractions.1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedEquivalent Fractions (con’t) To find an equivalent fraction, multiply or divide both the numerator and denominator by the same number.Exception: The numerator and denominator cannot be multiplied or divided by zero. 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedEquivalent Fractions (con’t) To find missing numerator in an equivalent fraction: Divide the larger denominator by the smaller one: 12 divided by 3 = 4Example Multiply the original numerator by the quotient from Step a: 2x4=81-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedEquivalent Fractions (con’t) Find 2 equivalent fractions for . Find the missing numerator . AnswersAnswer 1281-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedTo reduce a fraction to its lowest terms, find the largest whole number that divides evenly into both the numerator and denominator. When no whole number except 1 divides evenly into them, the fraction is reduced to its lowest terms.Reducing Fractions to Lowest Terms 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedBoth 10 and 15 are divisible by 5 Simplifying Fraction to Lowest Terms (con’t) Example Reduce1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedEquivalent Fractions (con’t) Reduce the following fractions: AnswerAnswer 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedCommon DenominatorsAny number that is a common multiple of all the denominators in a group of fractionsTo find the least common denominator (LCD):List the multiples of each denominator.Compare the list for common denominators.The smallest number on all lists is the LCD.1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedTo convert fractions with large denominators to equivalent fractions with a common denominator:List the denominators of all the fractions.Multiply the denominators. (The product is a common denominator.) Convert each fraction to an equivalent with the common denominator.Common Denominators (con’t) 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedCommon Denominators (con’t) Find the least common denominator:Answer 21Answer 1441-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedAdding FractionsTo add fractions:Rewrite any mixed numbers as fractions.Write equivalent fractions with common denominators. The LCD will be the denominator of your answer.Add the numerators. The sum will be the numerator of your answer. 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedTo subtract fractions:Rewrite any mixed numbers as fractions.Write equivalent fractions with common denominators. The LCD will be the denominator of your answer.Subtract the numerators. The difference will be the numerator of your answer.Subtracting Fractions 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedMultiplying Fractions To multiply fractions:Convert any mixed numbers or whole numbers to fractions.Multiply the numerators and then the denominators.Reduce the product to its lowest terms.1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedMultiplying Fractions (con’t) To multiply multiply the numerators and multiply the denominators1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedMultiplying Fractions (con’t) To cancel terms when multiplying fractions, divide both the numerator and denominator by the same number, if they can be divided evenly. Cancel terms to solve1132Answer will be1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedCAUTION!Avoid canceling too many terms.Each time you cancel a term, you must cancel it from one numerator AND one denominator.1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedMultiplying Fractions (con’t) Find the following products:AnswerAnswer1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedMultiplying Fractions (con’t) A bottle of liquid medication contains 24 doses. The hospital has 9 ¾ bottles of medication. How many doses are available?Answer 2341-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedYou have bottle of liquid medication available and you must give bottle to your patient. How many doses remain in the bottle? Dividing Fractions divided byby the reciprocal ofMultiply1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedConvert any mixed or whole number to fractions.Invert (flip) the divisor to find its reciprocal.Multiply the dividend by the reciprocal of the divisor and reduce. Dividing Fractions (con’t) 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedCAUTION!Write division problems carefully to avoid mistakes.Convert whole numbers to fractions, especially if you use complex fractions.Be sure to use the reciprocal of the divisor when convertingthe problem from division to multiplication. 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedDividing Fractions (con’t) Find the following quotients: Answerdivided byAnswerdivided by1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedDividing Fractions (con’t) A case has a total of 84 ounces of medication. Each vial in the case holds 1¾ ounce. How many vials are in the case?Answer 481-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedDecimalsDecimal system provides another way to represent whole numbers and their fractional partsPharmacy technicians use decimals dailyMetric system is decimal basedUsed in dosage calculations1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Working with DecimalsLocation of a digit relative to the decimal point determines its value. The decimal point separates the whole number from the decimal fraction.1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedTable 1-3 Decimal Place ValuesThe number 1,542.567 can be represented as follows:Whole NumberDecimal PointDecimal Fraction Working with Decimals (con’t) ThousandsHundredsTensOnes.TenthsHundredthsThousandths1,542.5671-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedDecimal Place ValuesThe number 1,542.567 is read:(1) - one thousand(5) - five hundred (42) - forty two and (5) - five hundred (67) – sixty-seven thousandthsOne thousand five hundred forty two and five hundred sixty-seven thousandths1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedWriting A Decimal NumberWrite the whole number part to the left of the decimal point.Write the decimal fraction part to the right of the decimal point. Decimal fractions are equivalent to fractions that have denominators of 10, 100, 1000, and so forth.Use zero as a placeholder to the right of the decimal point. For example, 0.203 represents 0 ones, 2 tenths, 0 hundreds, and 3 thousandths.1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedAlways write a zero to the left of the decimal point when the decimal number has no whole number part.Using zero makes the decimal point more noticeable and helps to prevent errors caused by illegible handwriting.Decimals 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved The more places a number is to the right of the decimal point the smaller the value.For example: Comparing Fractions0.3 is or three tenths0.03 is or three hundredths0.003 is or three thousandths1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedThe decimal with the greatest whole number is the greatest decimal number.If the whole numbers of two decimals are equal, compare the digits in the tenths place.If the tenths place are equal, compare the hundredths place digits.Continue moving to the right comparing digits until one is greater than the other. Comparing Fractions (con’t) 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedReview and PracticeWrite the following in decimal form:Answers = 0.2= 0.17= 0.0231-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedRounding DecimalsYou will usually round decimals to the nearest tenth or hundredth.Underline the place value to which you want to round.Look at the digit to the right of this target. If 4 or less do not change the digit, if 5 or more round up one unit.Drop all digits to the right of the target place value.1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedReview and PracticeRound to the nearest tenth:14.34Answer 14.39.293Answer 9.31-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedConverting Fractions into DecimalsTo convert a fraction to a decimal, divide the numerator by the denominator. For example:1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedConverting Decimals into FractionsWrite the number to the left of the decimal point as the whole number.Write the number to the right of the decimal point as the numerator of the fraction.Use the place value of the digit farthest to the right of the decimal point as the denominator.Reduce the fraction part to its lowest term.1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedReview and PracticeConvert decimals to fractions or mixed number:100.41.2AnswerorAnsweror1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedAdding and Subtracting DecimalsWrite the problem vertically. Align the decimal points.Add or subtract starting from the right. Include the decimal point in your answer. 2.47+0.39 2.861-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedAdding and Subtracting Decimals (con’t) Subtract 7.3 – 1.005 7.300 1.005 6.2951-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedReview and PracticeAdd or subtract the following pair of numbers:13.561 + 0.099Answer 14.625Answer 13.6616.250 – 1.6251-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedMultiplying DecimalsFirst, multiply without considering the decimal points, as if the numbers were whole numbers.Count the total number of places to the right of the decimal point in both factors.To place the decimal point in the product, start at its right end and move the decimal point to the left the same number of places.1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedMultiply 3.42 x 2.5 3.42X 2.5 1710684 8.550There are three decimal places so place the decimal point between 8 and 5 (8.55). Multiplying Decimals (con’t) 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedReview and PracticeA patient is given 7.5 milliliters of liquid medication 5 times a day. How many milliliters does she receive per day?Answer 7.5 x 5 7.5X 537.51-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedDividing DecimalsWrite the problem as a fraction.Move the decimal point to the right the same number of places in both the numerator and denominator until the denominator is a whole number. Insert zeros.Complete the division as you would with whole numbers. Align the decimal point of the quotient with the decimal point of the numerator, if needed.1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedReview and PracticeA bottle contains 32 ounces of medication. If the average dose is 0.4 ounces, how many doses does the bottle contain?Answer: 32 divided by 0.4Take 0.4 into 32Add a zero behind the 32 for each decimal place320 divided by 4 = 80 or 80 doses1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedReview and PracticeConvert the following mixed numbers to fractions: Answer Answer 1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedReview and PracticeRound to the nearest tenth:Answer 7.17.0911-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedReview and Practice7.23 + 12.38Multiply the following:12.01 x 1.005Answer 19.61Answer 12.07005Add the following:1-*McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights ReservedRemember, you control the numbers!Always ask for assistance if you are uncertain, the only bad question is the one not asked. THE END