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CHAPTER 5: PHARMACY CALCULATIONS 1
After completing thischapter, the reader will beable to
1. Calculate conversionsbetween differentnumbering and measur-ing systems.
2. Calculate medicationdoses from variousmedication dilutions.
3. Calculate and defineosmolarity, isotonicity,body surface area, andflow rates.
4. Calculate and definearithmetic mean,median, and standarddeviation.
Review of Basic Mathematics
NumeralsA numeral is a word or a sign, or a group of words or signs, that ex-presses a number.
Kinds of NumeralsArabic
Examples: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9Roman
In pharmacy practice, Roman numerals are used only to denotequantities on prescriptions.
ss or ss = 1/2 L or l = 50I or i = 1 C or c = 100V or v = 5 D or d = 500X or x = 10 M or m = 1000It is important to note that when a smaller Roman numeral (e.g., I)
is placed before a larger Roman numeral (e.g., V), the smaller Romannumeral is subtracted from the larger Roman numeral. Conversely, when asmaller Roman numeral is placed after a larger one, the smaller one isadded to the larger.
Example 1: I = 1 and V = 5; therefore, IV = 4 because 5 1 = 4.Example 2: I = 1 and V = 5; therefore, VI = 6 because 5 + 1 = 6
Problem Set #1Convert the following Roman numerals to Arabic numerals:
a. ii = ___ b. DCV = ___ c. xx = ___d. iii = ___ e. vii = ___ f. iv = ___
Pharmacy CalculationsGERALD A. STORM, REBECCA S. KENTZEL
Most calculations pharmacy technicians and pharmacists use involve
basic math; however, basic math is easily forgotten when it is not used
routinely. This chapter reviews the fundamentals of calculations and how
those calculations are applied in pharmacy.
2 MANUAL FOR PHARMACY TECHNICIANS
g. IX = ___ h. xv = ___ i. xi = ___j. MXXX = ___ k. xiv = ___ l. xvi = ___
NumbersA number is a total quantity or amount that is made ofone or more numerals.
Kinds of NumbersWhole Numbers
Examples: 10, 220, 5, 19Fractions
Fractions are parts of whole numbers.Examples: 1/4, 2/7, 11/13, 3/8Always try to express a fraction in its simplest
form.Examples: 2/4 = 1/2; 10/12 = 5/6; 8/12 = 2/3The whole number above the fraction line is called
the numerator, and the whole number below thefraction line is called the denominator.Mixed NumbersThese numbers contain both whole numbers andfractions.
Examples: 1 1/2, 13 3/4, 20 7/8, 2 1/2Decimal NumbersDecimal numbers are actually another means of writingfractions and mixed numbers.
Examples: 1/2 = 0.5, 1 3/4 = 1.75Note that you can identify decimal numbers by the
period appearing somewhere in the number. Theperiod in a decimal number is called the decimal point.
The following two points about zeros in decimalnumbers are VERY important:
1. Do NOT write a whole number in decimalform.In other words, when writing a whole number,avoid writing a period followed by a zero (e.g.,5.0). Why? Because periods are sometimes hardto see, and errors may result from misreadingthe number. For example, the period in 1.0 maybe overlooked, and the number could appear tobe 10 instead, causing a 10-fold dosing error,which could kill someone.
2. On the contrary, when writing a fraction in itsdecimal form, always write a zero before theperiod.Why? Once again, periods are sometimesdifficult to see, and .5 may be misread as 5.
However, if the period in 0.5 were illegible, thezero would alert the reader that a period issupposed to be there.
Working with Fractions and DecimalsPlease note that this section is meant to be only a basicoverview. A practicing pharmacy technician shouldalready possess these fundamental skills. The problemsand examples in this section should serve as buildingblocks for all calculations reviewed later in this chapter.If you do not know how to do these basic functions,you need to seek further assistance.
Review of Basic Mathematical FunctionsInvolving FractionsWhen adding, subtracting, multiplying, or dividingfractions, you must convert all fractions to a commondenominator. Also, when working with fractions, besure that you express your answer as the smallestreduced fraction (i.e., if your answer is 6/8, reduce itto 3/4).
AdditionUse the following steps to add 3/4 + 7/8 + 1/4.
1. Convert all fractions to common denominators.3/4 2/2 = 6/81/4 2/2 = 2/8
2. Add the fractions.6/8 + 7/8 + 2/8 = 15/8
3. Reduce to the smallest fraction.15/8 = 1 7/8
SubtractionUse the following steps to subtract 7/8 1/4.
1. Convert the fractions to common denominators.1/4 2/2 = 2/8
2. Subtract the fractions.7/8 2/8 = 5/8
MultiplicationUse the following steps to multiply 1/6 2/3.
When multiplying and dividing fractions, you doNOT have to convert to common denominators.
1. Multiply the numerators: 1 2 = 22. Multiply the denominators: 6 3 = 183. Express your answer as a fraction: 2/184. Be sure to reduce your fraction: 2/18 = 1/9
CHAPTER 5: PHARMACY CALCULATIONS 3
DivisionUse the following steps to divide 1/2 by 1/4.
Once again, you do NOT have to convert tocommon denominators.
To divide two fractions, the first fraction must bemultiplied by the inverse (or reciprocal) of the secondfraction:
1/2 1/4 is the same as 1/2 4/11. Multiply the fractions.
1/2 4/1 = 4/22. Reduce to lowest fraction.
4/2 = 2
Review of Basic Mathematical FunctionsInvolving DecimalsAs with fractions, when adding, subtracting, multiply-ing, or dividing decimals, all units (or terms) must bealike.
AdditionRemember to line up decimal points when adding orsubtracting decimal numbers.
Add the following: 0.1 + 124.7Add the terms by lining up the decimal points:
0.1+ 124.7 124.8
SubtractionSubtract the following: 2100 20.5
Subtract the terms by lining up the decimal points:2100.0 20.52079.5
MultiplicationWhen multiplying decimal numbers, the number ofdecimal places in the product must equal the totalnumber of decimal places in the numbers multiplied, asshown below.
Multiply the following: 0.6 24In this example, there is a total of one digit to the
right of the decimal point in the numbers beingmultiplied, so the answer will have one digit to theright of the decimal point.
Answer: 0.6 24 = 14.4
DivisionThe dividend is the number to be divided, and thedivisor is the number by which the dividend is divided.When dividing decimal numbers, move the divisorsdecimal point to the right to form a whole number.Remember to move the dividends decimal point thesame number of places to the right. When using longdivision, place the decimal point in the answer imme-diately above the dividends decimal point.
Divide the following: 60.75 4.5Move the decimal points: 607.5 45Answer: 607.5 45 = 13.5
Converting Fractions to Decimal NumbersTo convert a fraction to a decimal, simply divide thenumerator by the denominator.
For example, 1/2 = one divided by two = 0.5
Converting Mixed Numbers to DecimalNumbersThis process involves the following two steps:
1. Write the mixed number as a fraction.Method: Multiply the whole number and thedenominator of the fraction. Add the product(result) to the numerator of the fraction, keepingthe same denominator.Example: 2 3/4 = two times four plus three overfour = 11/4
2. Divide the numerator by the denominator.Example: 11/4 = eleven divided by four = 2.75
An alternate method involves the following threesteps:
1. Separate the whole number and the fraction.Example: 2 3/4 = 2 and 3/4
2. Convert the fraction to its decimal counterpart.Example: 3/4 = three divided by four = 0.75
3. Add the whole number to the decimal fraction.Example: 2 plus 0.75 = 2.75
Converting Decimal Numbers to MixedNumbers or FractionsTo convert decimal numbers to mixed numbers orfractions, follow these three steps:
1. Write the decimal number over one, dividing itby one. (Remember that dividing any number byone does not change the number.)
4 MANUAL FOR PHARMACY TECHNICIANS
Example: 3.5 = 3.5/12. Move the decimal point in the numerator as
many places to the right as necessary to form awhole number. Then move the decimal point inthe denominator the same number of places.Example: Because there is only one digit follow-ing the decimal point in 3.5, move the decimalpoint one place to the right in both the numera-tor and the denominator: 3.5/1 = 35/10.
Remember that the number will remain the sameas long as you do exactly the same things to thenumerator and the denominator. You also have toremember that the decimal point of a whole numberalways follows the last digit.
3: Simplify the fraction.Example: 35/10 = 7/2 = 3 1/2
Problem Set #2Convert the following fractions to decimal numbers:
a. 1/2 b. 3/4 c. 1 d. 2/5e. 1/3 f. 5/8 g. 50/100 h. 12/48i. 11/2 j. 2 2/3 k. 5 1/4 l. 3 4/5Convert the following decimal numbers to fractions
or mixed numbers:m. 0.25 n. 0.4 o. 0.75 p. 0.35q. 2.5 r. 1.6 s. 3.25 t. 0.33
PercentagesPercentage means by the hundred or in a hundred.Percents (%) are just fractions, but fractions with a setdenominator. The denominator is always one hundred(100).
Example: 50% means 50 in a hundred or50/100 or 1/2
Converting Percentages to FractionsWrite the number preceding the percent sign over 100and simplify the resulting fraction.
Example: 25% = 25/100 = 1/4
Converting Fractions to PercentagesConvert the fraction to one in which the denominatoris a hundred. This is easiest when the fraction is in theform of a decimal. Follow these four steps:
1. Write the fraction in its decimal