Using Ratio and Proportion

Chapter 3: Calculation of Dosages and Solution Rates Using Ratio and Proportion 4 Contact Hours By Alene Burke, MSN, RN. Alene received her Master of Science in Nursing Administration and Nursing Education from Adelphi University, and has completed coursework towards a Ph. D. Alene has been consulting on the development, design, and production of competency and educational activities since 1998. She has authored several publications including resource books and textbook chapters. She has provided continuing education for numerous medical professionals, including pharmacists. Author Disclosure: Alene Burke and Elite Professional Education, LLC. do not have any actual or potential conflicts of interest in relation to this lesson. Universal Activity Number (UAN): 0761-9999-15-159-H04-T Activity Type: Application-based Initial Release Date: June 1, 2015 Expiration Date: May 31, 2017 Target Audience: Pharmacy Technicians in a communitybased setting. To Obtain Credit: A minimum test score of 70 percent is needed to obtain a credit. Please submit your answers either by mail, fax, or online at. Learning objectives At the conclusion of this course, you should be able to accurately: Perform basic arithmetic calculations. Relate the equivalents for a household measurement system. Relate the equivalents for the apothecaries system. Relate the equivalents for the metric system. Introduction Pharmacy technologists work in a wide variety of settings. The roles and responsibilities of pharmacology technologists vary somewhat in different settings and even among those that are similar. For example, pharmacy technicians may not do intravenous admixtures in a community pharmacy department within a major retail store, but they may have to accurately add medications to intravenous solutions in an acute care hospital or medical center. Furthermore, some acute care hospitals and medical centers may only allow licensed pharmacists to prepare intravenous Basic arithmetic calculations An underlying presumption for this course is that you, the learner, have the basic ability to add, subtract, multiply and divide numbers. If you feel that you are not fully competent in terms of these basic arithmetic functions, it is recommended that you review and study these functions at this time and before continuing with this course. Questions regarding statements of credit and other customer service issues should be directed to 1-888-666-9053. This lesson is $17.00. Educational Review Systems is accredited by the Accreditation Council of Pharmacy Education (ACPE) as a provider of continuing pharmaceutical education. This program is approved for 4 hours (0.4 CEUs) of continuing pharmacy education credit. Proof of participation will be posted to your NABP CPE profile within 4 to 6 weeks to participants who have successfully completed the post-test. Participants must participate in the entire presentation and complete the course evaluation to receive continuing pharmacy education credit. Convert among the systems of measurement. Accurately calculate oral, parenteral and intravenous dosages using ratio and proportion, including for pediatric dosages that are based on body weight. admixtures; others may allow pharmacy technicians to perform this role under the supervision of a licensed pharmacist. Despite these differences, most pharmacy technicians must be thoroughly prepared and able to calculate accurate dosages of all types. There is no room for error; these dosages must be accurate and without any errors. Even the smallest error can lead to a serious medication error. This course will provide you with the knowledge, skills and abilities to provide safe, accurate pharmaceutical patient care and drug dosages without any errors whatsoever. In addition to the ability to perform basic addition, subtraction, multiplication and division, you should also be able to perform basic mathematical calculations using fractions, mixed numbers and decimals. These mathematic calculations are discussed and described below. Page 21

Fractions There are two types of fractions: 1. Proper fractions. 2. Improper fractions. Proper fractions are less than 1; improper fractions are more than 1. Fractions are indicated by a slash or a divide line, with a number above and number below the slash or divide line. The number above the slash or divide line is called the numerator, and the number below is referred to as the denominator. Here are some examples of proper fractions. All of these fractions are less than 1: 1/2 or ½ : In both figures, 1 is the numerator and 2 is the denominator. 2/5 or 2 5: In both figures, 2 is the numerator and 5 is the denominator. 88/345 or 88 345 : In both figures, 88 is the numerator and 345 is the denominator. Here are some examples of improper fractions. All of these fractions are more than 1: Reducing fractions Both proper and improper fractions can be reduced to their lowest common denominator. Reducing fractions make them easier to work with. Reducing fractions involves recognizing a number that can be evenly divided into both the numerator and denominator. For example, if the fraction is 3/9, both the numerator (3) and the denominator (9) can be evenly divided by 3 without anything left over. When you reduce 3/9, you divide the numerator of 3 by 3, and then you divide the denominator of 9 by 3, as shown below: 3 3 = 1 9 3 = 3 Therefore, 3/9 = 1/3 Likewise, you can reduce 66/124, as shown below: 66 2 = 33 124 2 = 62 Practice problems: Reducing fractions Reduce the following fractions to their lowest common denominators: 1. 16/22 = 2. 7/77 = 3. 8/23 = Answers to practice problems: Reducing fractions 1. 16/22 = 8/11 The 16 and the 22 can both be divided by 2, therefore, 16 divided by 2 is 8; 22 divided by 2 is 11. Thus, 8/11 is the answer because 8 and 11 cannot be divided evenly by any other number. 2. 7/77 = 1/11 5/3: 5 is the numerator and 3 is the denominator. 19/4: 19 is the numerator and 4 is the denominator. 564/324: 564 is the numerator and 324 is the denominator. Did you notice that the numerators in the three above proper fractions are less than the denominators? All of these fractions are less than 1. For example, the 2/5 fraction represents that there are 5 parts in the whole and you have only 2 of the 5 parts, less than a whole, or 1. Did you also notice that the numerators in the three above improper fractions are more than the denominators? For example, the 5/3 fraction represents that there are 3 parts in the whole and you have 5 parts, which is more than the whole, or greater than 1. When the numerator and the denominator are identical, the fraction is equal to 1. For example, the fraction 6/6 is equal to 1 and the fraction 987/987 is equal to 1. The numerators and denominators are identical. Of the 6 parts in the whole, you have all 6 parts; and of the 987 parts in the whole, you have all 987 parts, therefore it is a whole, or 1. Therefore, 66/124=33/62 Now, let s try to reduce these fractions. (Hint: If there is no common denominator, the fraction cannot be reduced.) 6/9 Both 6 and 9 can be reduced by 3 to the fraction 2/3. 16/24 Both 16 and 24 can be reduced by 8 to the faction 2/3. 16/9 This fraction cannot be reduced or made smaller because there is no number you can divide into both 16 and 9. There is no common denominator. By reducing fractions to their common denominators, you are really determining their equal fractions. So 6/9 is equal or equivalent to 2/3 in the example above. 4. 12/67 = 5. 34/88 = 6. 88/880 = Now check your answers. The 7 and the 77 can both be divided by 7, therefore, 7 divided by 7 is 1; 77 divided by 7 is 11. Thus, 1/11 is the answer because 7 and 77 cannot be evenly divided by any number other than 7. 3. 8/23 cannot be reduced. Page 22

4. 12/67 also cannot be reduced. Only 1 can be evenly divided into 12 and 67. 5. 34/88 = 17/44 The 34 and the 88 can both be divided by 2, therefore, 34 divided by 2 is 17; 88 divided by 2 is 44. Thus, 17/44 is the Mixed numbers Mixed numbers are a mix of a whole number and a fraction. For example, 2 1/2 teaspoons and 4 5/8 tablespoons are both mixed numbers. As you can see from these two examples, all mixed numbers are more than 1 or more than a whole. For example, 2 1/2 teaspoons is 2 whole teaspoons plus 1/2 a teaspoon. You have to convert all mixed numbers into improper fractions, which are also more than 1, before you can perform any calculations with them. For example, you have to convert 2 1/2 into 5/2, and you have to convert 4 5/8 into 37/8 as fully discussed below. To perform this calculation, you: Multiply the denominator by the whole number. Then add the numerator to it. Then divide this number by the denominator. 2 1/2 = 2 x 2 + 1 = 5/2 2 Practice problems: Mixed numbers Now, perform these practice problems by converting the mixed numbers into improper fractions and then converting these improper fractions back to the mixed number. 1. 3 4/7 = 2. 5 3/8 = 3. 14 6/9 = 4. 11 2/11 = The solution for each of these practice problems is shown below: 1. 3 4/7 = 7 x 3 + 4 = 21 + 4 = 25/7 7 7 To check this improper fraction, do this: 25/7 (or 25 divided by 7) = 3 4/7. Decimals Decimals are another way of expressing a proper fraction, an improper fraction and mixed numbers. Deci means 10, and all decimals are based on the system of tens or the power of answer because 17 and 44 cannot be evenly divided by any other number. 6. 88/880 = 1/10 The 88 and the 880 can both be divided by 88, therefore, 88 divided by 88 is 1; 880 divided by 88 is 10. In this example, you multiply the denominator of 2 by the whole number of 2 and then add 1, the numerator of the fraction. Finally, you divide by the denominator of 2. So, 2 x 2 = 4; 4 + 1 = 5, and 5 is then placed over the denominator of 2. As you may see, the improper fraction of 5/2 can be converted back to a mixed number as shown below: 5/2 means 5 divided by 2, which equals 2 1/2. When you turn mixed numbers back into improper fractions, you can easily check your mathematical calculation. If your original calculation gives you 5/2 and you convert this back to a mixed number, you should see the original mixed number. Now, let s convert these mixed numbers into improper fractions: 6 5/8 = 8 x 6 + 5 = 53/8 8 Next, check your answer by changing the improper fraction back to the original mixed number: 53/8 means 53 divided by 8, which equals 6 5/8. 2. 5 3/8 = 8 x 5 + 3 = 40 + 3 = 43/8 8 8 To check this improper fraction, do this: 43/8 (or 43 divided by 8) = 5 3/8. 3. 14 6/9 = 14 x 9 + 6 = 126 + 6 = 132/9 9 9 To check this improper fraction, do this: 132/9 (or 132 divided by 9) = 14 6/9. 4. 11 2/11 = 11 x 11 + 2 = 121 + 2 = 123/11 11 11 To check this improper fraction, do this: 123/11 (or 123 divided by 11) = 11 2/11. ten. For example, 0.7 is 7 tenths; 8.13 is 8 and 13 hundredths; likewise, 9.546 is 9 and 546 thousandths. Page 23

Below is a chart that shows the meaning of decimal places: Decimal places Meaning Example and equivalent 1 Tenths 2.3 = 2 and 3 tenths 2 Hundredths 21.98 = 21 and 98 hundredths 3 Thousandths 0.985 = 985 thousandths 4 Ten-thousandths 2.4444 = 2 and 4,444 ten-thousandths 5 Hundred-thousandths 0.77777 = 77,777 hundred-thousandths When the decimal point is preceded with a zero, the number is less than 1; when there is a whole number before the decimal point, the number is more than one or equal to one. Numbers with decimal points are readily converted into fractions and mixed numbers. For example: 2.3 = 2 and 3 tenths or 2 3/10. 21.98 = 21 and 98 hundredths or 21 98/(100 ). Decimal numbers are often rounded off when pharmaceutical calculations are done. When you have to round off to the nearest hundredth, you must look at the next number, or thousandth, and determine whether it is less than 5, equal to 5 or more than 5. For example, if the number in the third place after the decimal, or thousandths place, is an 8, which is greater than 5, then you round up the number in the hundredths place by 1. For example, 45.758 is rounded to the nearest hundredth, or 45.76, because the 8 in the thousandths place is greater than 5. Likewise, if you are rounding off to the nearest tenth, you look at the number in the hundredths place to see whether it is greater than 5. For example, 6.2346 is rounded off to 6.2 because the 3 in the hundredths place is not more than 5 or equal to 5. Practice problems: Rounding off decimals Round off these numbers to the nearest tenth: 1. 4.5678 = 2. 12.087 = Now, here are some numbers rounded off to the nearest tenth. Remember, if the hundredths place, or second number after the decimal, is 5 or more, the tenths place is increased by 1, and if the second number after the decimal is less than 5, the number in the tenths place remains the same. In these examples, the bolded number in the hundredths place (2 numbers after the decimal) is the one that determines whether the number in the tenths place (the first number after the decimal) moves up 1 or remains the same. 3.44 = 3.4 0.78 = 0.8 0.66 = 0.7 0.99 = 1.0 Here are some numbers rounded off to the nearest hundredth. Again, the bold numbers are the ones you have to scrutinize to see whether they are equal to or less than 5, or greater than 5: 3.456 = 3.46 0.754 = 0.75 0.766 = 0.77 1.999 = 2.00 3. 88.999 = 4. 65.123 = 5. 26.656 = Answers to practice problems: Rounding decimals to the nearest tenth 1. 4.5678 is rounded to 4.6 because the number in the hundredths place is more than 5. 2. 12.087 is rounded to 12.1 because the number in the hundredths place is more than 5. 3. 88.999 is rounded to 89.0 because the number in the hundredths place is more than 5. Now, round off these numbers to the nearest hundredth: 1. 4.5678 = 2. 12.087 = 3. 88.999 = 4. 65.123 is rounded to 65.1 because the number in the hundredths place is less than 5. 5. 26.656 is rounded to 26.7 because the number in the hundredths place is 5. 4. 65.123 = 5. 26.656 = Answers to practice problems: Rounding decimals to the nearest hundredth 1. 4.5678 is rounded to 4.57 because the number in the thousandths place is more than 5. 2. 12.087 is rounded to 12.09 because the number in the thousandths place is more than 5. 3. 88.999 is rounded to 89.00 because the number in the thousandths place is more than 5. 4. 65.125 is rounded to 65.13 because the number in the thousandths place is 5. 5. 26.656 is rounded to 26.66 because the number in the thousandths place is 5 or more. Page 24

The household system of measurement In pharmacology, there are three systems of measurement. These systems include the household measurement system, the metric system and the apothecary system. The household measurement system is more often used in the outpatient setting, like a local pharmacy, rather than within medical settings. It is the least precise and exact of all the measurement systems. The household system is the system that most of us use at home, usually in the kitchen. The household system uses measurements for drops, teaspoons, tablespoons, ounces, cups, pints, quart, gallons, and pounds. There are some similarities between the household measurements and the apothecary system. For example, a fluid ounce is the same in both systems. There are some differences as well, for example, the ounce that is used to determine weight is different in these systems. In the apothecary system, there are 12 ounces in a pound, whereas 16 ounces makes up a household system pound. The household measurement system table displays household units of measurement and their approximate equivalents. Unit of measurement The household measurement system Approximate equivalents 1 teaspoon 3 teaspoons = 1 tablespoon 60 drops 5 ml 1 tablespoon 1 tablespoon = 3 teaspoons 15 ml 1 liquid ounce 1 fluid ounce = 2 tablespoons 30 ml 1 ounce (weight) 16 ounces = 1 pound 30 g 1 cup 8 ounces 16 tablespoons 240 ml 1 pint 2 cups 480 ml 1 quart 2 pints 4 cups 1 liter 1 gallon 4 quarts 8 pints 3,785 ml 1 pound 16 ounces 480 g The apothecaries system of measurement The apothecary system of measurement is one of the oldest forms of measurement. In the 1970s, the United States practically abolished this system, but some physicians still use it, so it is important to be educated on this system even though it is rare and less commonly used than the metric system. In the apothecary system, the basic measurement of weight is the grain (gr). The other forms of measurement for weight in this system include the scruple, the dram, the ounce and the pound. For volume, the basic unit of measurement is the minim (m). This is equivalent to the amount of water in a drop, which is also equal to 1 grain. Other measurements for volume include a fluid dram, a fluid ounce, a pint, a quart and a gallon. Lowercase Roman numerals are used in this system of measurement, and these Roman numerals follow the unit of measurement. For example, 4 grains is written as 4 iv. The apothecary system table shows the weight and volume apothecary system measures and their approximate equivalents. The apothecary system Weight Approximate equivalents Volume Approximate equivalents 1 grain (gr) Weight of a grain of wheat 60 mg 1 minim Quantity of water in a drop or 1 grain 1 scruple 20 grains (gr xx) 1 fluid dram 60 minims 1 dram 3 scruples 1 fluid ounce 8 fluid drams 1 ounce 8 drams 1 pint 16 fluid ounces 1 pound 12 ounces 1 quart 2 pints 1 gallon 4 quarts The metric system of measurement The metric system is the most commonly used measurement system in pharmacology. It is used all over the world. Understanding the metric system and being able to convert between units of measurements is critical when working in the pharmacy. Volume measurements are liters (L), cubic milliliters (ml) and cubic centimeters (cc). This system s volume measurements are used for oral medications, such as cough syrups, and with parenteral drug dosages used intramuscularly, subcutaneously and intravenously. Page 25

The units of weight in this system include kilograms (kg), grams (g), milligrams (mg) and micrograms (mcg). The metric system The metric system table displays the metric length, volume and weight measurements and their equivalents. Length Equivalent Volume Equivalent Weight Equivalent 1 millimeter (mm) 0.001 meter 1 milliliter (ml) 0.001 liter 1 milligram (mg) 0.001 gram (g) 1 centimeter (cm) 0.01 meter 1 centiliter (cl) 0.01 liter 1 centigram (cg) 0.001 gram (g) 1 decimeter (dm) 0.1 meter 1 deciliter (dl) 0.1 liter 1 decigram (dm) 0.1 gram (g) 1 kilometer (km) 1,000 meters 1 kiloliter (kl) 1,000 liters 1 kilogram (kg) 1,000 grams (g) 1,000 milliliters 1 liter 1 kilogram (kg) 2.2 pounds (lbs) (ml) 1 milliliter (ml) cubic centimeter (cc) 1 pound (lb) 43,592 milligrams (kg) 10 millimeters (mm) 1 centimeter (cm) 10 milliliters (ml) 1 centiliter (cl) 1 pound (lb) 45,359.237 centigrams (cm) 10 centimeters (cm) 1 decimeter (dm) 10 centiliters (cl) 1 deciliter (dl) 1 pound (lb) 4,535.9237 decigrams (dg) 10,000 decimeters (dm) 1 kilometer (km) 10,000 deciliters (dc) 1 kiloliter (kl) Converting between measurement systems Pharmacy technicians often have to convert from one measurement system to another. For example, if the doctor has ordered a medication in terms of grains (gr) and you have the Conversions among the systems of measurement Metric Apothecary Household medication but it is measured in terms of milligrams (mg), you will have to mathematically convert the grains into milligrams. The table below shows conversion equivalents among the metric, apothecary and household measurement systems. 1 milliliter 15-16 minims 15-16 drops 4-5 milliliters 1 fluid dram 1 teaspoon or 60 drops 15-16 milliliters 4 fluid drams 1 tablespoon or 3-4 teaspoons 30 milliliters 8 fluid drams or 1 fluid ounce 2 tablespoons 240-250 milliliters 8 fluid ounces or 1/2 pint 1 glass or cup 500 milliliters 1 pint 2 glasses or 2 cups 1 liter 32 fluid ounces or 1 quart 4 glasses, 4 cups or 1 quart 1 milligram 1/60 grain 60 milligrams 1 grain 300-325 milligrams 5 grains 1 gram 15-16 grains 1 kilogram 2.2 pounds The 10 most frequently used conversions The most frequently used conversions are shown below. It is suggested that you memorize these. If at any point you are not sure of a conversion factor, look it up. Do NOT under any circumstances dispense or prepare a medication that you are not certain about. Accuracy is required. 1 Kg = 1,000 g 1 Kg = 2.2 lbs 1 L = 1,000 ml 1 g = 1,000 mg 1 mg = 1,000 mcg 1 gr = 60 mg 1 oz. = 30 g or 30 ml 1 tsp = 5 ml 1 lb = 454 g 1 tbsp = 15 ml Page 26

Ratio and proportion The ratio and proportion method is the most popular method to calculate dosages and solutions. Other methods include the memorization of a number of rules, which are often forgotten, and a simple, no-rules method called dimensional analysis. The remainder of this course will teach you about ratio and proportion and ways to precisely calculate all types of dosages. A ratio is two or more pairs of numbers that are compared in terms of size, weight or volume. For example, the ratio of women less than 20 years of age compared to those over 20 years of age who attend a specific college can be 6 to 1. This means that there are six times as many women less than 20 years old as there are women over 20 years of age. There are a couple of different ways that ratios can be written. These different ways are listed below. 1/6 1:6 1 to 6 Calculating proportions Proportions are used to calculate how one part is equal to another part or to the whole. For these calculations, you cross multiply the known numbers and then divide this product of the multiplication by the remaining number to get the unknown number. For example: 2/4 =?/12 Method 1: 12 x 2 = 24 24/4 or 24 4 = 6 Answer: 6 is the unknown, so the final equation will look like this: 2/4 = 6/12 Method 2: 1. 2/4 =?/12 You can reduce the first fraction by 2 to make the calculation a little easier. 2. 1/2 =?/12 12 x 1 = 12 12 2 = 6 Calculating oral dosages with ratio and proportion Oral dosages are calculated in a number of different ways, including using ratio and proportion. Using the same techniques that you just learned for ratio and proportions, you will now learn how to accurately calculate oral dosages using this method. Doctor s order: 500 mg of medication once a day Medication label: 1 tablet = 250 mg How many tablets should be administered daily? In this problem, you have to determine how many tablets the patient will take if the doctor orders 500 mg a day and the tablets are manufactured in tablets of 250 mg. This problem can be set up and calculated like this. 500 mg: X tablets = 250 mg: 1 tablet or Page 27 When comparing ratios, they should be written as fractions. The fractions must be equal. If they are not equal, they are not considered a ratio. For example, the ratios 2/8 and 4/16 are equal and equivalent. To prove they are equal, simply write down the ratios and cross multiply both the numerators and the denominators. The answer for both of these would be the same. For example, you can cross multiply the 2 and 16 as well as the 4 and 8 with this ratio of 2/8 and 4/16: 2 x 16 = 32 8 x 4 = 32 Because both multiplication calculations yielded 32, this is a ratio. On the other hand, 2/5 and 8/11 do not make a ratio because 8 x 5 (40) is not equal to 11 x 2 (22). Answer: 6 is the unknown, so the final equation will look like this: 2/4 = 6/12 Practice problems: Ratio and proportion 1. 5/15 = 20/? 2. 8/? = 7/22 3. 66/36 =?/12 4.?/6 = 43/53 5. 4/3 = 9/? Answers to practice problems: Ratio and proportion 1. 5/15 = 20/? 15 x 20 = 300; 300 5 = 60 2. 8/? = 7/22 8 x 22 = 176; 176 7 = 25.14 3. 66/36 =?/12 66 x 12 = 792; 792 36 = 22 4.?/6 = 43/53 43 x 6 = 258; 258 53 = 4.87 5. 4/3 = 9/? 9 x 3 = 27; 27 divided by 4 = 6.75 500 mg = 250 mg X 1 tablet Then you cross multiply: 500 mg x 1 = 500 mg 250 X = 500 mg X = 500 mg/250 mg 500 250 = 2 tablets Answer: 2 tablets

Now, if you were working in a medical center that uses unit dosage, you would deliver two tablets of this medication each day. If, however, the doctor wants the outpatient to take 500 mg per day for 30 days, you would dispense 60 tablets, as shown below. Daily dosage = 2 tablets x 30 days = 60 tablets Doctor s order: Tetracycline syrup 300 mg po once daily Medication label: Tetracycline syrup 50 mg/ml How many ml should be administered per day? For this oral dosage problem, you have to find out how many ml of tetracycline the patient will get when the doctor has ordered 300 mg and the syrup has 50 mg/ml. This problem is set up and calculated as shown below. 300 mg: X ml = 50 mg: 1 ml or 300mg = 50mg X ml 1 ml 50 X = 300 X = 300/50 300 50 = 6 ml Answer: 6 ml And if, for example, the doctor had ordered this dosage two times a day for 10 days for an outpatient, in addition to the above calculation, you also would perform the following calculation: 6 ml (per dose) x 2 (times per day) = 12 ml each day. This client would have a total of 12 ml per day in two equal doses of 6 ml each. Because the doctor ordered 300 mg po two times a day for 10 days, you would additionally perform the below calculation: 12 ml per day x 10 days = 120 ml You would dispense 120 ml of the medication with instructions that the client take 6 ml two times per day for 10 days. Practice problems: Oral dosages using ratio and proportion 1 Patient s weight: 40 kg Medication label: Trimethoprin 80 mg/tablet How many tablets should be administered? Doctor s order: Gantrisin 500 mg po Medication label: Gantrisin 1 g/tablet How many tablets should be administered? 3 2 Doctor s order: Trimethoprin 2.5 mg/kg po Doctor s order: Nystatin 6 mg/kg po Patient s weight: 230 lbs Medication label: Nystatin 200 mg/tablet How many tablets should be administered? Answers to practice problems: Oral dosages using ratio and proportion Problem 1: Doctor s order: 500 mg po Medication label: Gantrisin 1 g/tablet How many tablets should be administered? This is a two-step problem. The first step calculates the number of grams equal to the 500 mg dose. This is done because the drug label is written in grams, and the doctor s order is written in mg. What you do know, however, is that there are 1,000 mg in 1 g. Below is how you set this up to find out how many grams are equal to 500 mg. 500 mg = 1000 mg g 1g When you cross multiply, you will get 1000 X = 500 x 1, and then you set this up: X = 500/1000 X = 1/2 g 1/2 g = 0.50 g is equal to 500 mg The next step is to calculate what dose, in grams, should be administered. X = 500/1000 = 1/2 Answer: 1/2 tablet Problem 2: Doctor s order: Trimethoprin 2.5 mg/kg po Patient s weight: 40 kg Medication label: Trimethoprin 80 mg/tablet How many tablets should be administered? This is another two-part question. First you calculate the number of milligrams to be administered based on the weight of the patient. You know that the doctor ordered 2.5 mg/kg po and the patient weighs 40 kg. 40 kg = 1kg X mg 2.5 mg X = 40 2.5 X = 100 mg will be given In the next step, you have to calculate the number of tablets that the patient will be given based on the patient s weight and that the doctor has ordered 2.5 mg for each kg of body weight. 100 mg = 80 mg X tablet 1 tablet 80 X = 100 X = 100/80 100 80 = 1.25 tablets Answer: 11/4 tablets Page 28

Problem 3: Doctor s order: Nystatin 6 mg/kg po Patient s weight: 230 lbs Medication label: Nystatin 200 mg/tablet This problem has three steps. In the first step, you calculate the patient s weight in kilograms. There are 2.2 pounds in each kilogram, so this problem is set up as below. 230 lbs = 2.2 lbs X kg 1 kg 2.2 X = 230 kg X = 230/2.2 230 2.2 = 104.55 kg Round off the patient s weight to 105 kg because the number in the tenths place is 5 or more. You then have to calculate the dosage of the medication, based on the patient s weight of 105 kg. 105 kg = 1 kg X mg 6 mg 105 6 = 630 mg In the last step, you have to calculate how many tablets will be administered when each tablet is 200 milligrams. X tablets = 1 tablet 630 mg 200 mg 200 X = 630 X = 630/200 630 200 = 3.1 tablets; rounded off to 3 tablets Answer: 3 tablets Calculating intramuscular and subcutaneous dosages with ratio and proportion The process for calculating intramuscular and subcutaneous 105 10 = 1050 mg dosages is practically identical to that of calculating oral dosages using ratio and proportion. Doctor s order: Meperidine 20 mg IM q4h prn for pain Medication label: Meperidine 60 mg/ml How many ml or cc will you give? Using ratio and proportion, this problem is set up and solved as shown below. 20 mg = 60 mg X 1 ml 60 X = 20 x 1 x = 20/60 20 60 = 0.33 ml, which rounded off to the nearest tenth is 0.3 ml Answer: 0.3 ml Here is another example: Doctor s order: Amikacin 10 mg/kg IM tid Patient s weight: 230 lbs Medication label: 250 mg/1 ml How many milliliters need to be administered? For the first step, you calculate the patient s weight in kilograms. 239 lbs = 2.2 lbs X kg 1 kg 2.2 X = 230 X = 230/2.2 230 2.2 = 104.54 kg The patient s weight can be rounded off to 105 kg because the tenths place (5) is equal to or more than 5. The next step is to figure out how many milliliters the patient will get in each of the three doses per day. 10 mg = X mg 1 kg 105 kg 1 X = 105 10 In the final step, you will need to calculate how many milliliters are needed to administer the ordered number of milligrams. 250 mg = 1050 mg 1 ml X ml 250 X = 1050 = 1050/250 1050 250 = 4.2 ml Answer: 4.2 ml Now, let s do this one together: Doctor s order: Heparin 2,500 units subcutaneously Medication label: 5,000 units/ml How many milliliters will be administered for this patient? X ml = 1 ml 2,500 Units 5,000 units 5,000 X = 2,500 X = 2,500/5,000 2,500 5,000 = 0.5 Answer: 0.5 ml And one more: Doctor s order: Ticarcillin 300 mg IM Medication label: Ticarcillin reconstituted with 2 ml of sterile water to yield 1 g of Ticarcillin in 2.6 ml of solution How many milliliters need to be administered? For these kinds of problems, the information about how much sterile water is added for reconstitution is not used in the calculation. What is, however, used in the calculation is how many grams are yielded after the sterile water has been added. In this case, 1 g is contained in every 2.6 ml. You will be doing these types of calculations when you are adding medications to intravenous fluids (admixtures), particularly when you are doing IV piggybacks in an acute care setting. The first step is to find out how many g there are in 300 mg: 300 mg = 1,000 mg X g 1 g Page 29

1000 X = 300 X = 300/1000 300 1000 = 0.3 g The next step is to determine how many ml will be administered when the ordered dosage is 300 mg or 0.3 g 0.3 g = 1 g X ml 2.6 ml X = 0.3 2.6 0.3 2.6 = 0.78 ml Rounded off to: 0.8 ml Answer: 0.8 ml Practice problems: Intramuscular and subcutaneous dosages using ratio and proportion 1 4 Doctor s order: Neomycin 40 mg/kg/day IM in 3 divided doses Patient s weight: 160 lbs Medication label: Neomycin 250 mg/ml How many milliliters are needed for each of the three daily doses? 2 Doctor s order: Heparin 2000 units subcutaneously Medication label: 3500 units/ml How many milliliters would this patient need to have administered? 3 Doctor s order: Ceruroxime 250 mg IM Medication label: The addition of 3.2 ml of sterile water yields a suspension of 750 mg in 4.2 ml How many milliliters need to be administered in this case? Doctor s order: Cephalothin 200 mg IM Medication label: The addition of 4 ml of sterile water yields 0.5 g in 2.2 ml of suspension How many milliliters would you administer in this situation? 5 Doctor s order: Neomycin 40 mg/kg/day IM in 3 doses Patient s weight: 240 lbs Medication label: Neomycin in 500 mg/ml How many milliliters should be given? 6 Doctor s order: 250,000 Units of Ampicillin Medication Label: 50,000 units/ml How many milliliters should be administered to this patient? Answers to the practice problems: Intramuscular and subcutaneous dosages using ratio and proportion Problem 1: Doctor s order: Neomycin 40 mg/kg/day IM in 3 divided doses Patient s weight: 160 lbs Medication label: Neomycin 250 mg/ml How many milliliters are needed for each of the three daily doses? This, again, is a three-step problem. In the first step, you will find out the patient s weight in kilograms. 160 lbs = 2.2 lbs X kg 1 kg 2.2 X = 160 X = 160/2.2 160 2.2 = 72.72 kg Patient s weight can be rounded off to 73 kg. In step two, you will figure out how many mg there are in the total daily dosage. 40 mg = X mg 1 kg 73 kg X = 73 40 73 40 = 2920 mg The medication label states that there are 250 mg/ml, so you will now calculate how many ml the patient will get. 250 X = 2920 X = 2920/250 2920 250 = 11.68 ml The final step for this problem is to calculate each dosage based on a total of 11.68 ml in three divided doses. 11.68 3 = 3.89 The dosage can be rounded to 3.9 ml per dose, and the patient will get this dose three times per day Answer: 3.9 ml per dose Problem 2: Doctor s order: Heparin 2000 units subcutaneously Medication label: 3500 units/ml How many milliliters would this patient need to have administered? 2,000 units = 3,500 X ml 1 ml Page 30

3500 X = 2000 X = 2000/3500 2000 3500 = 0.57 ml This dosage can be rounded off to 0.60 ml or 0.6 ml Answer: 0.6 ml Problem 3: Doctor s order: Ceruroxime 250 mg IM Medication label: The addition of 3.2 ml of sterile water yields a suspension of 750 mg in 4.2 ml How many milliliters should be dispensed? 250 mg = 750 mg XmL 4.2 ml 750 X = 250 4.2 750 X = 1050 X = 1050/750 1050 750 = 1.4 ml Answer: 1.4 ml Problem 4: Doctor s order: Cephalothin 200 mg IM Medication label: The addition of 4 ml of sterile water yields 0.5 g in 2.2 ml of suspension How many milliliters would you administer in this situation? 200 mg = 1,000 mg X g 1 g 1000 X = 200 X = 200/1000 200 1000 = 0.2 g 0.2 g = 0.5 g X ml 2.2 ml 0.5 X = 0.2 2.2 ml 0.5 X = 0.44 X = 0.44/(0.5 ) 0.44 0.5 = 0.88, which is rounded off to 0.9 ml Answer: 0.9 ml Problem 5: Doctor s order: Neomycin 40 mg/kg/day IM in 3 doses Patient s weight: 240 lbs Medication label: Neomycin in 500 mg/ml How many milliliters should be given? 240 lbs = 2.2 lbs X g 1 kg 2.2 X = 240 240/2.2 240 2.2 = 109.09 kg Rounded off to 109 kg 109 kg = 1 kg X mg 40 mg X = 109 x 40 1 109 40 = 4360 4360 = 500 mg X ml 1 ml 500 X = 4360 4360/500 4360 500 = 8.72 Rounded off to 8.7 ml 8.7 3 doses = 2.9 ml for each of the three doses Answer: 2.9 ml Problem 6: Doctor s order: 250,000 units of Ampicillin Medication label: 50,000 units/ml How many milliliters should be administered to this patient? 250,000 units = 50,000 units X ml X ml 50,000 X = 250,000 X = 250,000/50,000 250,000 50,000 = 5 ml Answer: 5 ml Calculating intravenous flow rates with ratio and proportion The rule for intravenous flow rates is: Total number of ml gtts/min = Total number of minutes Drip or drip factor Now, here is how it is set up and calculated: Doctor s order: 0.9% MaCl solution at 50 ml per hour How many gtts per minute should be administered if the tube delivers 20 gtt/ml? X gtts per min = 50 20 = 1000/60 = 16.6 gtt 60 Rounded off to 17 gtt/min Here s another example: Doctor s order: 500 ml of 5% D 0.45 normal saline solution to infuse over 3 hours How many gtt per minute should be given if the tubing delivers 10 gtt/ml? X gtts per minute = 500 10 = 5000/180 180 5000 180 = 27.7 gtt Rounded off to 28 gtts per minute Answer: 28 gtts/min And now, here is one more: Doctor s order: 15 ml/h of 5% DO 0.45 normal saline solution delivers 60 gtt/ml? X gtts per minute = 15 60 = 900/60 = 15 gtt 60 Answer: 15 gtts/min Page 31

Practice problems: Intravenous medications Problem 1: Doctor s order: 50 ml/h is 60 gtt/ml? Problem 2: Doctor s order: 75 ml/h is 10 gtt/ml? Problem 3: Doctor s order: 300 ml/h is 20 gtt/ml? Problem 4: Doctor s order: 40 ml/h is 60 gtt/ml? Problem 5: Doctor s order: 300 ml/h is 30 gtt/ml? Answers to the practice problems: Intravenous medications Problem 1: Doctor s order: 50 ml/h is 60 gtt/ml? X gtts per min = 50 60 = 50 gtt (This fraction was reduced by 60) 60 Answer: 50 gtt/min Problem 2: Doctor s order: 75 ml/h is 10 gtt/ml? Conclusion Pharmacology is a precise science. Medical errors, including medication errors, can lead to disastrous results. Medication errors can occur at any point of this complex multidisciplinary process. For example, medication errors can occur as the result of an incorrect or illegible doctor s order; they can occur during the preparation and dispensing of the medication; and they can also occur at the point of administration. It is the professional responsibility of the pharmacology technician to insure that NO errors occur in the preparation and dispensing of medications. Pharmacy technicians must check the doctor s order for completeness and correctness; they must validate that the patient is not allergic to the ordered X gtts per min = 75 10 = 75/6 60 75 6 = 12.5, which is rounded off to 13 gtt because the tenths place is 5 or more Answer: Rounded to: 13 gtt/min Problem 3: Doctor s order: 300 ml/h is 20 gtt/ml? 300 20 = 6000/60 60 6000 60 = 100 gtt Answer: 100 gtt/min Problem 4: Doctor s order: 40 ml/h is 60 gtt/ml? X gtts per min = 40 60 = 2400/60 60 2400 60 = 40 Answer: 40 gtt/min Problem 5: Doctor s order: 300 ml/h is 10 gtt/ml? X gtts per min = 300 10 60 3000 60 = 50 Answer: 50 gtt/min medication; they must determine whether there are any drug interactions; and they must also ensure that the dosage is accurately prepared, labeled and dispensed. This course has provided you with the knowledge, skills and abilities to prepare and dispense oral, intramuscular, subcutaneous, and intravenous medications and solutions in an accurate and precise manner using the ratio and proportion method. As always, check and double-check your calculations, and consult resources, both human and written, whenever you are not sure and certain about the accuracy of your mathematical calculations. Page 32

CAlculation of dosages and solution RATES USING RATIO AND PROPORTION Final Examination Questions Choose the best answer for questions 1 through 5 and mark your answers online at 1. Round 23.6547 to the nearest hundredth. a. 23.7. b. 23.66. c. 2.36. d. 23.65. 2. Which of the following is equivalent to 1 fluid dram? a. 60 drops. b. 2 teaspoons. c. 10 ml. d. 20 ml. 4. The doctor has ordered an oral tablet of a medication. The order states that the patient will get 3 mg/kg and the patient weighs 130 pounds. How many total tablets will be delivered to the nursing unit into the patient s unit dose cassette for 24 hours if this dosage is given 3 times per day and the label on the medication states that there are 120 mg per tablet? a. 4. b. 4.5. c. 5. d. 5.5. 3. Select the accurate conversion. a. 2.2 lbs = 1 Kg. b. 1 g = 1,000 mg. c. 60 gr = 1 mg. d. 1 oz. = 20 ml. 5. The doctor has ordered 500 mg of an antibiotic to run over one hour. You have mixed the antibiotic into 250 ml of normal saline. How many drops should be administered per minute if the nurses use a 10 gtt/ml intravenous set? a. 83 gtt/min. b. 20 gtt/min. c. 42 gtt/min. d. 40 gtt/min. RPTFL04CDE14 Page 33