PassAssured's Pharmacy Technician Training Systems. Calculations. Fractions, Decimals, and Percents

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1 Calculations Fractions, Decimals, and Percents Pharmacy Technician Training Systems Passassured, LLC p1

2 Calculations, Fractions, Decimals, and Percents Click Here for Glossary Index! Click Here to Print Topic Help File,.pdf (Internet Access is Required for this Feature) Common Fractions Denominator Numerator Proper Fractions Rules for Decimals when Writing Drug Doses Adding Fractions Subtracting Fractions Multiplying Fractions Dividing Fractions Improper Fractions Operations on Numerators Operations on Denominators Percents General Principles with Fractions Multiplication Division of Mixed Numbers Decimal Fractions Mathematical Operations with Decimals Pass Assured, LLC, Pharmacy Technician Training Systems Copyright Pass Assured, LLC, Web Site - -o- p2

3 Calculations, Fractions, Decimals, and Percents Click Here for Glossary Index! Click Here to Print Topic Help File,.pdf (Internet Access is Required for this Feature) Common Fractions Consists of two numbers separated by either a horizontal or diagonal line. Denominator Bottom Number: Indicates the number of parts into which one (1) is divided. Numerator Top Number: Specifies the number of those parts with which we are concerned. For example, the fraction one-half, written as 1/2, indicates that one has been divided into two parts and we are concerned with one of those two parts. The fraction 3/4 indicates that one has been divided into four parts and we are concerned with three of those four parts. Proper Fractions Numerator is smaller than the denominator. 1 = numerator 2 = denominator 3 = numerator 4 = denominator Improper Fractions Numerator is larger than the denominator. Occurs when the fraction has a value greater than 1. 6 = numerator 5 = denominator p3

4 Here the fraction has a greater value than 1. When handling fractions, it is helpful to take note of some general principles concerning what happens when you perform certain mathematical operations on numerators and denominators. (Notice thay we are careful to specify that the multiplier or divisor is greater than (>1). In each case, if this number is less than one (<1), we get the opposite result.) Operations on Numerators If you multiply the numerator by a number >1, the value of the fraction is increased. If you divide the numerator by a number >1, the value of the fraction is decreased. Operations on Denominators If you multiply the denominator by a number >1, the value of the fraction is decreased. If you divide the denominator by number >1, the value of the fraction is increased. Operations on Both the Numerator and Denominator General Principles with Fractions If you multiply both the numerator and denominator by the same number, the value of the fraction remains the same. If you divide both the numerator and denominator by the same number, the value of the fraction remains the same. This information comes in handy when working with fractions because it enables you to perform certain mathematical operations with fractions when you are required to change the form of the fraction. Various examples given below will illustrate this point. In pharmacy, we are often required to perform mathematical operations with fractions. The most common operations are illustrated below. Adding Fractions Add the numbers, the denominators stay the same. Example: Adding Fractions ( ) p4

5 In order for the denominator to stay the same, it must be the same for both the fractions being added. What if the denominators are different? Example: Adding Different Denominators ( ) In this case, if you multiply both the numerator and denominator by the same number, the value of the fraction remains the same. The fraction 1/4 can be changed to a fraction with the same value but with a denominator of 8. This is achieved by multiplying both the numerator and denominator of the faction by the number 2. Make the denominators the same by multiplying both the numerator and denominator by the same number (see General Principles). Example: Common Denominator - 1 ( ) Substitute this new fraction, then add. Example: Common Denominator - 2 ( ) Sometimes the forms of both fractions must be changed. In this case, the closest common denominator is 12, therefore the numerator and denominator of the fraction 1/4 must be multiplied by 3 to get 3/12. The numerator and denominator of the fraction 2/3 must be multiplied by 4 to get 8/12. Then the addition can be performed. Example: Adding Fractions ( ) p5

6 Subtracting Fractions Subtraction of fractions is very similar to addition of fractions. Subtract the numerators, the denominators stay the same. Example: Subtracting Fractions ( ) As with addition, if the denominators are different, change one or both fractions so they are the same, then subtract. Example: Subtraction of Different Denominators ( ) As in the example above, the fraction 2/3 is changed to 8/12 and the fraction 1/4 is changed to 3/12. The indicated mathematical operation, subtraction, is performed. Example: Subtracting Fractions ( ) p6

7 Multiplying Fractions Multiply the numerators together = the new numerator. Multiply the denominators together = the new denominator. Example: Multiplying Fractions ( ) Multiplication is easy because the numerators are multiplied together. Unlike addition and subtraction there is no need for a common number in the denominator. Dividing Fractions Invert the divisor and perform multiplication. Example: Dividing Fractions -1 ( ) Invert the 3/4 and multiply. Dividing Fractions - 2 ( ) p7

8 Reducing Common Fractions to the Lowest Terms Fractions are usually expressed in the simplest form possible, reducing the fraction to lowest terms. Example: Reducing Common Fractions ( ) Divide the numerator and denominator by 3. This fraction is expressed by the equivalent, but more simple fraction of 4/5. This more simple expression is obtained by dividing both the numerator and denominator by the same number. From the General Principles you will remember that when you do this, you get a different fraction, but one with the same value. The numerator and denominator of the fraction are examined to determine if a common factor exists in each. In the fraction 12/15, the number 3 is a common factor since both 12 and 15 are divisible by 3. This number is divided into both the numerator and the denominator. Example: Reducing Fractions ( ) Reducing Improper Fractions to the Lowest Terms An improper fraction (one in which the numerator is greater than the denominator) is reduced to lowest terms by changing the fraction to a mixed number. Divide the numerator by the denominator and express the remainder as a proper fraction. Example #1: Improper Fractions - 1 ( ) p8

9 Example #2: Improper Fractions - 2 ( ) Multiplication and Division of Mixed Numbers Example: Multiplication of Mixed Fractions ( ) Multiplication and Division Cannot be Performed Using Mixed Numbers If a mixed number is to be multiplied or divided, it must first be changed to an improper fraction. To convert the mixed number to a fraction, multiply the whole number by the denominator and add this to the numerator. Then write these results as the new numerator over the original denominator. Example: Mixed Numbers ( ) p9

10 Substitute this improper fraction for the mixed number, then multiply the fraction. Example: Substituting Improper Fractions ( ) Express the answer, 110/24, as a mixed number reduced to lowest terms. Example: Reducing Mixed Numbers ( ) Decimal Fractions A fraction in which the denominator is 10, or a multiple of 10, is called a decimal fraction, or a decimal. However, instead of writing the fraction as two numbers separated by a line, a decimal point is used. With decimals, the placement of a decimal point is used to indicate the multiple of ten in the denominator. Only the numbers of the numerator are actually written. Example: Fractions as Decimals Table ( ) p10

11 When performing any mathematical operation, all terms must be in the same system. Therefore, if some numbers are given as common fractions and some as decimal fractions, you must first convert all the numbers to one common form. Conversion between common and decimal fractions: Decimal = Common Fractions = 11/10,000 Use the number in the decimal (11) as the numerator. The denominator is a multiple of ten determined by the position of the decimal point (one zero for each place). Common fraction = Decimal 5/8 = Divide the numerator of the fraction by the denominator. For the above example, divide 5 by 8 which equals /8 = Divide the numerator of the fraction by the denominator. For the above example, divide 3 by 8 which equals Note that for some fractions the division does not come out even and equals a repeating number. For example, 1/3 equals Mathematical Operations with Decimals Most often when we perform mathematical operations with decimals we use a calculator and it automatically gives an answer with the decimal in the proper place. If you were to perform these operations manually there are a few rules to remember. When adding or subtracting decimals, you must first line up the numbers so that the decimal points are directly under each other. Then add or subtract as you would whole numbers. To add : Adding Decimals ( ) p11

12 To add : Adding Decimals ( ) To subtract 0.56 from 0.70: Subtracting Decimals ( ) To subtract 0.44 from 0.80: Subtracting Decimals ( ) p12

13 To multiply decimals, multiply the numbers just as you would with whole numbers. To determine the position of the decimal point in the answer: Count the number of places to the right of the decimal point for each of the multipliers. Add these together. Count off this number of places in the answer, starting at the far right number. Example: If multiplying by 0.62, there is a combined five places to the right of the decimal in the two multipliers. Therefore, in determining the decimal place in the product 52638, count five places to the left of the far right number (8 in this case). Example: Multiplying Decimals ( ) If the answer does not have enough numbers, use zeros as place keepers. In the example above, if the number 0.62 was and we needed six decimal places, the answer would be Example: If multiplying by 0.42, there is a combined five places to the right of the decimal in the two multipliers. Therefore, in determining the decimal place in the product 30828, count five places to the left of the far right number (8 in this case). Example: Multiplying Decimals ( ) To divide decimals, position the numbers for long division as you would for whole numbers. If the divisor has a decimal place, move the decimal point all the way to the right of its numbers. Then move the decimal point of the dividend (the number being divided) the same number of places to the right. If the dividend is a whole number, you will need to add a zero to the right of the last dividend number for each decimal place needed. For example, to divide 16.8 by 0.12, move the decimal point of 0.12 to the right two places to get 12; likewise move the decimal point for p13

14 the dividend, 16.8, two places to the right. Then perform the division as usual. Example: Dividing Decimals ( ) Rules for Decimals When Writing Drug Doses Decimal points must be clearly written when writing drug doses since a missed or misread decimal point can cause a ten-fold error in dose. Never add a "trailing zero" when writing whole numbers. Use 25 mg, not 25.0 mg. Never write a "naked decimal" when writing a decimal fraction for a number less than one (1). Use 0.25 mg, not.25 mg. Percents Percents are merely a specialized type of fraction. The term percent (symbolized %) means per hundred. Example: 50% means 50 per 100 Always convert percents to common or decimal fractions before performing any mathematical operation. Percent = Decimal Move the decimal point two places to the left and drop the percent sign. Example: 50% = 0.50 or simply 0.5 Example: 12.5% = Decimal = Percent Move the decimal point two places to the right and add the percent sign. Example: 0.5 = 50% Example: = 12.5% Percent = Proper Fraction p14

15 Use the number as the numerator and 100 as the denominator. Example: 50% = 50/100 (or 1/2 when reduced to lowest terms) Example: 12.5% First write it as a common fraction,12.5/100. Since we do not usually mix decimals in common fractions this fraction should be converted to an equivalent non decimal fraction by multiplying both the numerator and denominator by 10, which would equal 125/1000. This fraction would be reduced to its lowest terms by dividing both the numerator and denominator by the common factor 125 to equal 1/ % = 12.5./100 = 125/1000 = 1/8 (when reduced to its lowest terms) Proper Fraction = Percent Convert the fraction to a decimal, then convert the decimal to a percent. Example: 1/4 = 0.25 = 25% Taken by themselves, percents, like common and decimal fractions, are just numbers without any units or dimensions. However, certain conventions have been established for the use of percents when describing the concentrations of drug products. These are discussed in the section, Expressions of Quantity, Concentration, Dose and Dosage Regimen. Pass Assured, LLC, Pharmacy Technician Training Systems Copyright Pass Assured, LLC, Web Site - -o- p15