Math for the Pharmacy Technician: Concepts and Calculations Egler • Booth Chapter 1: Numbering Systems and Mathematical Review McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Numbering Systems and Mathematical Review McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Learning Outcomes When you have successfully completed Chapter 1, you will have mastered skills to be able to: Identify 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. .pc McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Learning 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Introduction Basic 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Arabic Numbers Arabic 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Arabic 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. Example McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Roman Numerals Commonly used Roman numerals ss = ½ I = 1 V = 5 X = 10 They may be written in lower or uppercase Are used sometimes in drug orders You need to understand how to change Roman numeral into Arabic numbers in order to do dosage calculations McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Roman Numerals Symbol Value I 1 (unus) V 5 (quinque) X 10 (decem) L 50 (quinquaginta) C 100 (centum) D 500 (quingenti) M 1,000 (mille)

Combining 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Combining Roman Numerals (con’t) Example Example Roman 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. IX = 10 –1 = 9 XIV = 10 + (5-1) = 14 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Fractions and Mixed Numbers Measure a portion or part of a whole amount Written two ways: Common fractions Decimals McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Common Fractions Represent equal parts of a whole Consist of two numbers and a fraction bar Written in the form: Numerator (top part of the fraction) = part of whole Denominator (bottom part of the fraction) represents the whole one part of the whole the whole McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Common Fractions (con’t) Scored (marked) tablet for 2 parts You administer 1 part of that tablet each day You would show this as 1 part of 2 wholes or ½ Read it as “one half” McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Check these equations by treating each fraction as a division problem. Fraction Rule When the denominator is 1, the fraction equals the number in the numerator. Example Check these equations by treating each fraction as a division problem. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Mixed Numbers Example 2 (two and two-thirds) Mixed numbers combine a whole number with a fraction. Example 2 (two and two-thirds) Fractions with a value greater than 1 are written as mixed numbers. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Mixed Numbers (con’t) If the numerator of the fraction is less than the denominator, the fraction has a value of < 1. If the numerator of the fraction is equal to the denominator, the fraction has a value =1. If the numerator of the fraction is greater than the denominator, the fraction has a value > 1. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Only applied when the numerator is greater Mixed Numbers (con’t) To 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. Only applied when the numerator is greater than the denominator McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Mixed Numbers (con’t) Example 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 3 4. Write the remainder over the whole ¾ 5. Combine the whole number and the fraction 2+ ¾ McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Mixed Numbers (con’t) 5x3 = 15 To 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 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Mixed Numbers (con’t) To convert a mixed number to a fraction: Write the sum from Step 2 over the original denominator The result is a fraction equal to original mixed number. Thus McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

What is the denominator in ? Review and Practice What is the numerator in ? Answer = 17 What is the denominator in ? Answer = 100 Twelve patients are in the hospital ward. Four have type A blood. What fraction do not have type A blood? Answer = McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Equivalent Fractions Two fractions written differently that have the same value = equivalent fractions. Example same as same as Find equivalent fractions for McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Equivalent 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Equivalent Fractions (con’t) To find missing numerator in an equivalent fraction: Example Divide the larger denominator by the smaller one: 12 divided by 3 = 4 Multiply the original numerator by the quotient from Step a: 2x4=8 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Equivalent Fractions (con’t) Find 2 equivalent fractions for . Answers Find the missing numerator . Answer 128 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Reducing Fractions to Lowest Terms To 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Simplifying Fraction to Lowest Terms (con’t) Example Reduce Both 10 and 15 are divisible by 5 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Equivalent Fractions (con’t) Reduce the following fractions: Answer Answer McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Common Denominators Any number that is a common multiple of all the denominators in a group of fractions To 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Common Denominators (con’t) To 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Common Denominators (con’t) Find the least common denominator: Answer 21 Answer 144 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Adding Fractions To 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Subtracting Fractions To 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Multiplying 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Multiplying Fractions (con’t) To multiply multiply the numerators and multiply the denominators McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Multiplying 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 solve 1 1 3 2 Answer will be McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

CAUTION! Avoid canceling too many terms. Each time you cancel a term, you must cancel it from one numerator AND one denominator. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Multiplying Fractions (con’t) Find the following products: Answer Answer McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Multiplying 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 234 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Dividing Fractions divided by by the reciprocal of Multiply You have bottle of liquid medication available and you must give bottle to your patient. How many doses remain in the bottle? divided by by the reciprocal of Multiply McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Dividing Fractions (con’t) Convert 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

CAUTION! 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 converting the problem from division to multiplication. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Dividing Fractions (con’t) Find the following quotients: divided by Answer divided by Answer McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Dividing 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 48 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Decimals Decimal system provides another way to represent whole numbers and their fractional parts Pharmacy technicians use decimals daily Metric system is decimal based Used in dosage calculations McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Working with Decimals Location of a digit relative to the decimal point determines its value. The decimal point separates the whole number from the decimal fraction. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Working with Decimals (con’t) Table 1-3 Decimal Place Values The number 1,542.567 can be represented as follows: Whole Number Decimal Point Decimal Fraction Thousands Hundreds Tens Ones . Tenths Hundredths Thousandths 1, 5 4 2 6 7 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Decimal Place Values The number 1,542.567 is read: (1) - one thousand (5) - five hundred (42) - forty two and (67) – sixty-seven thousandths One thousand five hundred forty two and five hundred sixty-seven thousandths McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Writing A Decimal Number Write 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Decimals Always 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

0.003 is or three thousandths Comparing Fractions The more places a number is to the right of the decimal point the smaller the value. For example: 0.3 is or three tenths 0.03 is or three hundredths 0.003 is or three thousandths McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Comparing Fractions (con’t) The 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Review and Practice Write the following in decimal form: = 0.2 Answers = 0.17 = 0.023 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Rounding Decimals You 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Review and Practice Answer 14.3 9.293 Answer 9.3 Round to the nearest tenth: 14.34 Answer 14.3 9.293 Answer 9.3 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Converting Fractions into Decimals To convert a fraction to a decimal, divide the numerator by the denominator. For example: McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Converting Decimals into Fractions Write 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Review and Practice Answer 100.4 Answer Convert decimals to fractions or mixed number: 1.2 Answer or 100.4 Answer or McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Adding and Subtracting Decimals Write 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 2.86 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Adding and Subtracting Decimals (con’t) 7.300 1.005 6.295 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Review and Practice Add or subtract the following pair of numbers: 13.561 + 0.099 Answer 13.66 16.250 – 1.625 Answer 14.625 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Multiplying Decimals First, 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Multiplying Decimals (con’t) Multiply 3.42 x 2.5 3.42 X 2.5 1710 684 8.550 There are three decimal places so place the decimal point between 8 and 5 (8.55). McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Review and Practice Answer 7.5 x 5 7.5 X 5 37.5 A patient is given 7.5 milliliters of liquid medication 5 times a day. How may milliliters does she receive per day? Answer 7.5 x 5 7.5 X 5 37.5 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Dividing Decimals Write 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. McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Review and Practice Answer: 32 divided by 0.4 Take 0.4 into 32 A 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 0.4 Take 0.4 into 32 Add a zero behind the 32 for each decimal place 320 divided by 4 = 80 or 80 doses McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Review and Practice Answer Answer Convert the following mixed numbers to fractions: Answer Answer McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Review and Practice Answer 7.1 7.091 Round to the nearest tenth: McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Review and Practice Add the following: 7.23 + 12.38 Answer 19.61 Multiply the following: 12.01 x 1.005 Answer 12.07005 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Remember, you control the numbers! Always ask for assistance if you are uncertain, the only bad question is the one not asked. THE END McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved