Pharmacy Calculations

Pharmacy Calculations

5

GERALD A. STORM, REBECCA S. KENTZEL

Learning Objectives

Most calculations pharmacy technicians and pharmacists use involve

After completing this chapter, the reader will be able to

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.

Review of Basic Mathematics Numerals

A numeral is a word or a sign, or a group of words or signs, that expresses a number.

Kinds of Numerals Arabic Examples: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Roman In pharmacy practice, Roman numerals are used only to denote quantities on prescriptions. ss or ss = 1/2 L o r l = 50 I or i = 1 C or c = 100 V or v = 5 D o r d = 500 X or x = 10 M or m = 1000 It 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 Roman numeral is subtracted from the larger Roman numeral. Conversely, when a smaller Roman numeral is placed after a larger one, the smaller one is added 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

1. Calcul Calculate ate conver conversio sions ns between different numbering and measuring systems. 2. Calcul Calculate ate med medicat ication ion doses from various medication dilutions. 3. Calcul Calculate ate and and defi define ne osmolarity, isotonicity, body surface area, and flow rates. 4. Calcul Calculate ate and and defi define ne arithmetic mean, median, and standard deviation.

Problem Set #1 Convert the following Roman numerals to Arabic numerals: a . i i = ___ b. DCV = ___ c. xx = ___ d. iii = ___ e. vii = ___ f. iv = ___

CHAPTER CHAPT ER 5: PHARMACY CALC ULATIONS

1

g. IX = ___ j. MXXX = ___

h. xv = ___ k. xiv = ___

i. xi = ___ l. xvi = ___

However, if the period in 0.5 were illegible, the zero would alert the reader that a period is supposed to be there.

Numbers

A number is a total quantity or amount that is made of one or more numerals.

Kinds of Numbers Whole Numbers Examples: 10, 220, 5, 19 Fractions Fractions are parts of whole numbers. Examples: 1/4, 2/7, 11/13, 3/8 Always try to express a fraction fr action in its simplest form. Examples: 2/4 = 1/2; 10/12 = 5/6; 8/12 = 2/3 The whole number above the fraction line is called the numerator, and the whole number below the fraction line is called the denominator. Mixed Numbers These numbers contain both whole numbers and fractions. Examples: 1 1/2, 13 3/4, 20 7/8, 2 1/2 Decimal Numbers Decimal numbers are actually another means of writing fractions and mixed numbers. Examples: 1/2 = 0.5, 1 3/4 = 1.75 Note that you can identify decimal numbers by the period appearing somewhere in the number. The period in a decimal number is called the decimal point. The following two points about zeros in decimal numbers are VERY important: 1. Do NOT write a whole numbe numberr in decimal decimal form. In other words, when writing a whole number, number, avoid writing a period followed by a zero (e.g., 5.0). Why? Because periods are sometimes hard to see, and errors may result from misreading the number. For example, the period in 1.0 may be overlooked, and the number could appear to be 10 instead, causing a 10-fold dosing error, which could kill someone. 2. On the contrary contrary,, when writing writing a fraction fraction in its decimal form, always write a zero before the period. Why? Once again, periods are sometimes difficult to see, and .5 may be misread as 5.

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Working with Fractions and Decimals Please note that this section is meant to be only a basic overview. A practicing pharmacy technician should already possess these fundamental skills. The problems and examples in this section should serve as building blocks 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 Functions Involving Fractions

When adding, subtracting, multiplying, or dividing fractions, you must convert all fractions to a common denominator. Also, when working with fractions, be sure that you express your answer as the smallest reduced fraction (i.e., if your answer is 6/8, reduce it to 3/4).

Addition Additi on Use the following steps to add 3/4 + 7/8 + 1/4. 1. Convert all all fractions fractions to commo commonn denominators denominators.. 3/4 × 2/2 = 6/8 1/4 × 2/2 = 2/8 2. Add the frac fractio tions. ns. 6/8 + 7/8 + 2/8 = 15/8 3. Reduce to the small smallest est fraction fraction.. 15/8 = 1 7/8 Subtraction Use the following steps to subtract 7/8 – 1/4. 1. Convert the the fractions fractions to common common denomin denominators. ators. 1/4 × 2/2 = 2/8 2. Sub Subtrac tractt the fract fraction ions. s. 7/8 – 2/8 = 5/8 Multiplicatio Multipli cationn Use the following steps to multiply 1/6 × 2/3. When multiplying and dividing fractions, you do NOT have to convert to common denominators. 1. Mul Multiply tiply the the numerators numerators:: 1 × 2 = 2 2. Mul Multiply tiply the the denominator denominators: s: 6 × 3 = 18 3. Express your answer answer as as a fraction: fraction: 2/18 4. Be sure to reduce reduce your fraction fraction:: 2/18 = 1/9

MANUAL FOR PHARMACY TECHNICIANS

Division Use the following steps to divide 1/2 by 1/4. Once again, you do NOT have to convert to common denominators. To divide two fractions, the first fraction must be multiplied by the inverse (or reciprocal) of the second fraction: 1/2 ÷ 1/4 is the same as 1/2 × 4/1 1. Multiply the fractions. 1/2 × 4/1 = 4/2 2. Reduce to lowest fraction. 4/2 = 2

Division The dividend is the number to be divided, and the divisor is the number by which the dividend is divided. When dividing decimal numbers, move the divisor’s decimal point to the right to form a whole number. Remember to move the dividend’s decimal point the same number of places to the right. When using long division, place the decimal point in the answer immediately above the dividend’s decimal point. Divide the following: 60.75 ÷ 4.5 Move the decimal points: 607.5 ÷ 45 Answer: 607.5 ÷ 45 = 13.5

Review of Basic Mathematical Functions Involving Decimals

To convert a fraction to a decimal, simply divide the numerator by the denominator. For example, 1/2 = one divided by two = 0.5

As with fractions, when adding, subtracting, multiplying, or dividing decimals, all units (or terms) must be alike.

Addition Remember to line up decimal points when adding or subtracting decimal numbers. Add the following: 0.1 + 124.7 Add the terms by lining up the decimal points: 0.1 + 124.7 124.8 Subtraction Subtract the following: 2100 – 20.5 Subtract the terms by lining up the decimal points: 2100.0 – 20.5 2079.5 Multiplication When multiplying decimal numbers, the number of decimal places in the product must equal the total number of decimal places in the numbers multiplied, as shown below. Multiply the following: 0.6 × 24 In this example, there is a total of one digit to the right of the decimal point in the numbers being multiplied, so the answer will have one digit to the right of the decimal point. Answer: 0.6 × 24 = 14.4

CHAPTER 5: PHARMACY CALC ULATIONS

Converting Fractions to Decimal Numbers

Converting Mixed Numbers to Decimal Numbers

This process involves the following two steps: 1. Write the mixed number as a fraction. Method: Multiply the whole number and the denominator of the fraction. Add the product (result) to the numerator of the fraction, keeping the same denominator. Example: 2 3/4 = two times four plus three over four = 11/4 2. Divide the numerator by the denominator. Example: 11/4 = eleven divided by four = 2.75 An alternate method involves the following three steps: 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 Mixed Numbers or Fractions

To convert decimal numbers to mixed numbers or fractions, follow these three steps: 1. Write the decimal number over one, dividing it by one. (Remember that dividing any number by one does not change the number.)

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Example: 3.5 = 3.5/1 2. Move the decimal point in the numerator as many places to the right as necessary to form a whole number. Then move the decimal point in the denominator the same number of places. Example: Because there is only one digit following the decimal point in 3.5, move the decimal point one place to the right in both the numerator and the denominator: 3.5/1 = 35/10. Remember that the number will remain the same as long as you do exactly the same things to the numerator and the denominator. You also have to remember that the decimal point of a whole number always follows the last digit. 3: Simplify the fraction. Example: 35/10 = 7/2 = 3 1/2

Problem Set #2 Convert the following fractions to decimal numbers: a. 1/2 b. 3/4 c. 1 d. 2/5 e. 1/3 f. 5/8 g. 50/100 h. 12/48 i. 11/2 j. 2 2/3 k. 5 1/4 l. 3 4/5 Convert the following decimal numbers to fractions or mixed numbers: m. 0.25 n. 0.4 o. 0.75 p. 0.35 q. 2.5 r. 1.6 s. 3.25 t. 0.33

Percentages Percentage means “by the hundred” or “in a hundred.” Percents (%) are just fractions, but fractions with a set denominator. The denominator is always one hundred (100). Example: “50%” means “50 in a hundred” or “50/100” or “1/2” Converting Percentages to Fractions

Write the number preceding the percent sign over 100 and simplify the resulting fraction. Example: 25% = 25/100 = 1/4 Converting Fractions to Percentages

Convert the fraction to one in which the denominator is a hundred. This is easiest when the fraction is in the form of a decimal. Follow these four steps: 1. Write the fraction in its decimal form. Example: 3/4 = three divided by four = 0.75

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2. Write the decimal over one. Example: 0.75/1 3. To obtain 100 as the denominator, move the decimal point two places to the right. To avoid changing the number, move the decimal point two places to the right in the numerator as well. Example: 0.75/1 = 75/100 4. Because you already know that “out of a hundred” or “divided by a hundred” is the same as percent, you can write 75/100 as 75%. Concentration Expressed as a Percentage

Percent weight-in-weight (w/w) is the grams of a drug in 100 grams of the product. Percent weight-in-volume (w/v) is the grams of a drug in 100 ml of the product. Percent volume-in-volume (v/v) is the milliliters of drug in 100 ml of the product. The above concentration percentages will be discussed in further detail a little later in this chapter.

Problem Set #3 Convert the following percentages to fractions (remember to simplify the fractions): a. 23% b. 67% c. 12.5% d. 50% e. 66.7% f. 75% g. 66% h. 40% i. 100% j. 15% Convert the following fractions to percentages: k. 1/2 l. 1/4 m. 2/5 n. 6/25 o. 4/100 p. 0.5 q. 0.35 r. 0.44 s. 0.57 t. 0.99

Units of Measure Metric System

The metric system is based on the decimal system, in which everything is measured in multiples or fractions of 10. Appendix 5-1 lists the conversion charts reviewed on the following pages.

Standard Measures The standard measure for length is the meter; the standard measure for weight is the gram; and the standard measure for volume is the liter. Prefixes The prefixes below are used to describe multiples or


fractions of the standard measures for length, weight, and volume. Latin Prefixes micro- (mc): 1/1,000,000 = 0.000001 milli- (m): 1/1000 = 0.001 centi- (c): 1/100 = 0.01 deci- (d): 1/10 = 0.1 Note that Latin prefixes denote fractions. Greek Prefixes deca- (da): 10 hecto- (h): 100 kilo- (k): 1000 mega- (M): 1,000,000 Note that Greek prefixes denote multiples.

dupois system (see below) to enable fine weighing of medications. Today, the apothecary system is used only for a few medications, such as aspirin, acetaminophen, and phenobarbital.

Gallons Pints

Fluid ounces

Fluid drams Minims

Prefixes with Standard Measures Length The standard measure is the meter (m). 1 kilometer (km) = 1000 meters (m) 0.001 kilometer (km) = 1 meter (m) 1 millimeter (mm) = 0.001 meter (m) 1000 millimeters (mm) = 1 meter (m) 1 centimeter (cm) = 0.01 meter (m) 100 centimeters (cm)= 1 meter (m) Volume The standard measure is the liter (L). 1 milliliter (ml) = 0.001 liter (L) 1000 milliliters (ml) = 1 liter (L) 1 microliter (mcl) = 0.000001 liter (L) 1,000,000 microliters (mcl) = 1 liter (L) 1 deciliter (dl) = 0.1 liter (L) 10 deciliters (dl) = 1 liter (L) Weight The standard measure is the gram (g). 1 kilogram (kg) = 1000 grams (g) 0.001 kilogram (kg) = 1 gram (g) 1 milligram (mg) = 0.001 gram (g) 1000 milligrams (mg) = 1 gram (g) 1 microgram (mcg) = 0.000001 gram (g) 1,000,000 micrograms (mcg) = 1 gram (g)

1=

128 = 16 = 1=

1024 = 128 = 8= 1=

Apothecary System

The apothecary system was developed after the Avoir-


Weight The standard measure for weight is the grain (gr). Pound

Ounces Drams

Scruples Grains

1=

12 = 1=

288 = 24 = 3= 1=

96 = 8= 1=

5760 480 60 20

Volume The standard measure for volume is the minim (m.). 8= 1=

61,440 7,680 480 60

Avoirdupois System

This system is used mainly in measuring the bulk medications encountered in manufacturing. Be sure to note that the pounds-to-ounces equivalent is different in the apothecary and avoirdupois systems. The avoirdupois system is most commonly used to measure weight, and the apothecary system is most commonly used to measure volume. Be sure also to note the difference in symbols used for the two systems. In addition, note that a fluid ounce, which measures volume, is often mistakenly shortened to an “ounce,” which is actually a measure of weight. Because ounces measure weight, pay close attention to the measure you are working with and convert accordingly.

Weight The standard measure for weight is the grain (gr). Pound

Ounces Grains

(lb) 1=

(oz) 16 = 1=

(gr) 7000 437.5

Household System

The household system is the most commonly used system of measuring liquids in outpatient settings. The measuring equipment usually consists of commonly used home utensils (e.g., teaspoons, tablespoons).

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1 teaspoonful (tsp) = 5 ml 1 dessertspoonful = 10 ml 1 tablespoonful (TBS) = 15 ml = 0.5 fluid ounces (fl oz) 1 wineglassful = 60 ml = 2 fl oz 1 teacupful = 120 ml = 4 fl oz 1 glassful/cupful = 240 ml = 8 fl oz 3 tsp = 1 TBS 2 TBS = 1 fl oz 8 fl oz = 1 cup 2 cups = 1 pint (pt) 2 pt = 1 quart (qt) 4 qt = 1 gallon (gal) The term drop is commonly used; however, caution should be exercised when working with this measure, especially with potent medications. The volume of a drop depends not only on the nature of the liquid but also on the size, shape, and position of the dropper. To accurately measure small amounts of liquid, use a 1-ml syringe (with milliliter markings) instead of a dropper. Eye drops are an exception to this rule; they are packaged in a manner to deliver a correctly sized droplet.

Problem Set #4 Fill in the blanks: a. 1 liter (L) = _______________ ml b. 1000 g = _______________ kg c. 1 g = _______________ mg d. 1000 mcg = _______________ mg e. 1 TBS = _______________ tsp f. 1 TBS = _______________ ml g. 240 ml = _______________ cupfuls h. 1 cup = _______________ ml i. 15 ml = _______________ TBS j. 1 tsp = _______________ ml k. 240 ml = _______________ TBS l. 1 pt = _______________ ml m. 1 fl oz = _______________ TBS n. 1 qt = _______________ pt Equivalencies Between Systems

The apothecary, avoirdupois, and household systems lack a close relationship among their units. For this reason, the preferred system of measuring is the metric

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system. The table of weights and measures below gives the approximate equivalencies used in practice.

Length Measures 1 meter (m) = 39.37 (39.4) inches (in) 1 inch (in) = 2.54 centimeters (cm) Volume Measures 1 milliliter (ml) = 16.23 minims 1 fluid ounce (fl oz) = 29.57 (30) milliliters (ml) 1 liter (L) = 33.8 fluid ounces (fl oz) 1 pint (pt) = 473.167 (480) milliliters (ml) 1 gallon (gal) = 3785.332 (3785) milliliters (ml) Weight Measures 1 kilogram (kg) = 2.2 pounds (lb) 1 pound (avoir) (lb) = 453.59 (454) grams (g) 1 ounce (avoir) (oz) = 28.35 (28) grams (g) 1 ounce (apoth) (oz) = 31.1 (31) grams (g) 1 gram (g) = 15.432 (15) grains (gr) 1 grain (gr) = 65 milligrams (mg) 1 ounce (avoir) (oz) = 437.5 grains (gr) 1 ounce (apoth) = 480 grains (gr) Temperature Conversion

Temperature is always measured in the number of degrees centigrade (°C), also known as degrees Celsius, or the number of degrees Fahrenheit (°F). The following equation shows the relationship between degrees centigrade and degrees Fahrenheit: [9(°C)] = [5(°F)] – 160° Example: Convert 110°F to °C. [9(°C)] = [5(110°F)] – 160° °C = (550 – 160)/9 °C = 43.3° Example: Convert 15°C to °F [9(15°C)] = [5(°F)] – 160° (135 + 160)/5 = °F 59° = °F Conversion Between Systems

To find out how many kilograms are in 44 lbs, follow these three steps: 1. Write down the statement of equivalency between the two units of measure, making sure that the unit corresponding with the unknown in the


question is on the right. 2.2 lbs = 1 kg 2. Write down the problem with the unknown underneath the equivalency. Equivalency: 2.2 lbs = 1 kg Problem: 44 lbs = ? kg 3: Cross multiply and divide. 1 times 44 divided by 2.2 = [(1 × 44) / 2.2] = 20 kg Determining Body Surface Area

The Square Meter Surface Area (Body Surface Area, or BSA) is a measurement that is used instead of kilograms to estimate the amount of medication a patient should receive. BSA takes into account the patient’s weight and height. BSA is always expressed in meters squared (m2) and is frequently used to dose chemotherapy agents. When using the equation below, units of weight (W) should be kilograms (kg) and height (H) should be centimeters. The following equation is used to determine BSA: BSA=(W 0.5378) × (H0.3964) × (0.024265) Now, using the formula, follow these three steps to calculate the BSA of a patient who weighs 150 pounds and is 5 feet 8 inches tall. 1. Convert weight to kilograms. 150 lb 2.2 lb/kg

= 68.2 kg

2. Convert height to centimeters. 5 feet × 12 inches/foot = 60 inches + 8 inches = 68 inches 68 inches × 2.54 cm/inch = 172.7 cm 3. Insert the converted numbers into the formula. BSA = (W 0.5378) × (H0.3964) × 0.024265 BSA = (68.20.5378) × (172.70.3964) × 0.024265 BSA = (9.69) × (7.71) × 0.024265 BSA = 1.81m2

Problem Set #5 Convert or solve the following: a. 30 ml = _____ fluid ounces b. 500 mg = _____ g c. 3 teaspoons = _____ fluid ounces d. 20 ml = _____ teaspoons


e. f. g. h. i. j. k. l. m. n. o. p. q. r. s. t. u. v. w. x. y. z. aa. bb. cc. dd. ee.

3.5 kg = _____ g 0.25 mg = _____ g 1500 ml = _____ L 48 pints = _____ gallons 6 gr = _____ mg 120 lb = _____ kg 3 fluid ounces = _____ ml 72 kg = _____ pounds 946 ml = _____ pints 800 g = _____ lb 3 tsp = _____ milliliters 2 TBS = _____ fluid ounces 2 TBS = _____ ml 2.5 cups = _____ fl oz 0.5 gr = _____ mg 0.5 L = _____ ml 325 mg = _____ gr 2 fl oz = _____ TBS 60 ml = _____ fl oz 144 lb = _____ kg 1 fl oz = _____ tsp 4 tsp = _____ ml 83°F = _____ °C –8°F = _____ °C 5°C = _____ °F 32°C = _____ °F What is the BSA of a patient who weighs 210 pounds and is 5 feet 1 inch tall?

Ratio and Proportion A ratio states a relationship between two quantities. Example: 5 g of dextrose in 100 ml of water (this solution is often abbreviated “D5W”). A proportion is two equal ratios. Example: 5 g of dextrose in 100 ml of a D5W solution equals 50 g of dextrose in 1000 ml of a D5W solution; or 5g 100 ml

=

50 g 1000 ml

A proportion consists of two unit (or term) types (e.g., kilograms and liters, or milligrams and milliliters). If you know three of the four terms, you can calculate the fourth term. 7

Problem Solving by the Ratio and Proportion Method

The ratio and proportion method is an accurate and simple way to solve some problems. To use this method, you should learn how to arrange the terms correctly, and you must know how to multiply and divide. There is more than one way to write a proportion. The most common is the following: Term #1 Term #2

=

Term #3 Term #4

This expression is read Term #1 is to Term #2 as Term #3 is to Term #4. By cross multiplying, the proportion can now be written as: (Term #1) × (Term #4) = (Term #2) × (Term #3) Example 1: How many grams of dextrose are in 10 ml of a solution containing 50 g of dextrose in 100 ml of water (D50W)? 1. Determine which is the known ratio and which is the unknown ratio. In this example, the known ratio is “50 g of dextrose in 100 ml of solution.” The unknown ratio is “ X g of dextrose in 10 ml of solution.” 2. Write the unknown ratio (Terms #1 and #2) on the left side of the proportion. Be sure the unknown term is on the top. X g

10 ml

=

Term #3 Term #4

3. Write the known ratio (Terms #3 and #4) on the right side of the proportion. The units of both ratios must be the same—the units in the numerators and the units in the denominators must match. In this case, that means grams in the numerator and milliliters in the denominator. If units of the numerators or the denominators differ, then you must convert them to the same units. X g

10 ml

=

50 g 100 ml

4. Cross multiply. X g × 100 ml = 50 g × 10 ml 5. Divide each side of the equation by the known number on the left side of the equation. This will leave only the unknown value on the left side of the equation:

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Xg=

50 g

×

10 ml

100 ml

6. Simplify the right side of the equation to solve for X grams: Answer: X g = 5 g Example 2: You need to prepare a 500-mg chloramphenicol dose in a syringe. The concentration of chloramphenicol solution is 250 mg/ml. How many milliliters should you draw up into the syringe? 1. Determine the known and unknown ratios. 1 ml

Known:

250 mg X ml

Unknown:

500 mg

2. Write the proportion. X ml

500 mg

=

1 ml 250 mg

3. Cross multiply. X ml × 250 mg = 1 ml × 500 mg 4. Divide. X ml =

1 ml

×

500 mg

250 mg

5. Simplify.

X ml = 2 ml Answer: Draw up 2 ml in the syringe to prepare a 500-mg dose of chloramphenicol.

Problem Set #6 a. How many milligrams of magnesium sulfate are in 10 ml of a 100 mg/ml magnesium sulfate solution? b. A potassium chloride (KCl) solution has a concentration of 2 mEq/ml. 1.How many milliliters contain 22 mEq? 2.How many milliequivalents (mEq) in 15 ml? c. Ampicillin is reconstituted to 250 mg/ml. How many milliliters are needed for a 1-g dose?


Concentration and Dilution Terminology ■ ■

■

5% dextrose in water is the same as D5W. 0.9% sodium chloride (NaCl) is the same as normal saline (NS). Half-normal saline is half the strength of normal saline (0.9% NaCl), or 0.45% NaCl. This may also be referred to as 0.5 NS or 1/2 NS.

Concentration Expressed as a Percentage

The concentration of one substance in another may be expressed as a percentage or a ratio strength. As stated earlier in this chapter, concentrations expressed as percentages are determined using one of the following formulas: 1. Percent weight-in-weight (w/w) is the grams of a drug in 100 grams of the product. 2. Percent weight-in-volume (w/v) is the grams of a drug in 100 ml of the product. 3. Percent volume-in-volume (v/v) is the milliliters of drug in 100 ml of the product. Example 1: 0.9% sodium chloride (w/v) = 0.9 g of sodium chloride in 100 ml of solution. Example 2: 5% dextrose in water (w/v) = 5 g of dextrose in 100 ml of solution. Example 3: How many grams of dextrose are in 1 L of D5W? Use the ratio and proportion method to solve this problem: Known ratio: D5W means Unknown ratio:

5g 100 ml

X g

1L

1. Write the proportion: X g

1L

=

5 g 100 ml

2. Are you ready to cross multiply? No. Remember, you must first convert the denominator of either term so that both are the same. Because you know that 1 L = 1000 ml, you can convert the unlike terms as follows: X g

1000 ml

=

5 g 100 ml


3. Now that the units are both placed in the same order and the units across from each other are the same, you can cross multiply. X g × 100 ml = 5 g × 1000 ml 4. Divide: Xg =

5 g × 1000 ml 100 ml

5. Simplify: X g = 50 g Answer: There are 50 g of dextrose in 1 L of D5W. Before you attempt problem sets #7 and #8, here are a few suggestions for solving concentration and dilution problems: 1. First calculate the number of grams in 100 ml of solution. That is your “known” side of the ratio. 2. Then calculate the number of grams in the volume requested in the problem by setting up a ratio. 3. Check to make sure your units are in the same order in the ratio. 4. Make sure the units that are across from each other in the ratio are the same. 5. After you have arrived at the answer, convert your answer to the requested units.

Problem Set #7 a. In 100 ml of a D5W/0.45% NaCl solution: 1. How many grams of NaCl are there? 2. How many grams of dextrose are there? b. How many grams of dextrose are in 1 L of a 10% dextrose solution? c. How many grams of NaCl are in 1 L of 1/2 NS? d. How many milligrams of neomycin are in 50 ml of a 1% neomycin solution? e. How many grams of amino acids are in 250 ml of a 10% amino acid solution? Problem Set #8 a. An order calls for 5 million units (MU) of aqueous penicillin. How many milliliters are needed if the concentration is 500,000 units/ml? b. How many milliliters are needed for a 15-MU aqueous penicillin dose if the concentration of the solution is 1 MU/ml? c. Pediatric chloramphenicol comes in a 100 mg/ml 9

concentration. How many milligrams are present in 5 ml of the solution? d. How many milliliters of a 250 mg/ml chloramphenicol solution are needed for a 4-g dose? e. Oxacillin comes in a 500 mg/1.5 ml solution. How many milliliters will be required for a 1.5-g dose? f. How many grams of ampicillin are in 6 ml of a 500 mg/1.5 ml solution? g. How many milliliters contain 3 g of cephalothin if the concentration of the solution is 1 g/4.5 ml? h. Use these concentrations to solve the following 20 problems: Hydrocortisone 250 mg/2 ml Tetracycline 250 mg/5 ml Potassium chloride (KCl) 2 mEq/ml Amoxicillin suspension 250 mg/5 ml Mannitol 12.5 g/50 ml Heparin 10,000 units/ml Heparin 1000 units/ml 1. 10 mEq KCl = _____ml 2. 125 mg amoxicillin = _____ml 3. 750 mg hydrocortisone = _____ml 4. 1 g tetracycline = _____ml 5. 25 g mannitol = _____ml 6. 20,000 units heparin = _____ml OR _____ml 7. 35 mEq KCl = _____ml 8. 6000 units heparin = _____ml OR _____ml 9. 40 mEq KCl = _____ml 10. 600 mg hydrocortisone = _____ml 11. _____mg hydrocortisone = 4 ml 12. _____mg tetracycline = 15 ml 13. _____mEq KCl = 15 ml 14. _____g mannitol = 75 ml 15. _____units or _____units heparin = 2 ml 16. _____mEq KCl = 7 ml 17. _____g tetracycline = 12.5 ml 18. _____units or _____units heparin = 10 ml 19. _____mEq KCl = 45 ml 20. _____mg amoxicillin = 10 ml i. How many grams of magnesium sulfate are in 2 ml of a 50% magnesium sulfate solution?

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j. How many grams of dextrose are in 750 ml of a D10W solution? k. How many milliliters of a D5W solution contain 7.5 grams of dextrose? l. How many grams of NaCl are in 100 ml of a NS solution? m. How many grams of NaCl are in 100 ml of a 1/2 NS solution? n. How many grams of NaCl are in 100 ml of a 1/4 NS solution? o. How many grams of NaCl are in 1 L of a 0.45% NaCl solution? p. How many grams of NaCl are in 1 L of a 0.225% NaCl solution? q. How many grams of dextrose are in 100 ml of a D5W/0.45% NaCl solution? r. How many milliliters of a 70% dextrose solution are needed to equal 100 g of dextrose? s. How many milliliters of a 50% dextrose solution are needed for a 10-g dextrose dose? t. How many grams of dextrose are in a 50 ml NS solution? Concentration Expressed as a Ratio Strength

Concentration of weak solutions is frequently expressed as ratio strength. Example: Epinephrine is available in three concentrations: 1:1000 (read one to one thousand); 1:10,000; and 1:200. A concentration of 1:1000 means there is 1 g of epinephrine in 1000 ml of solution. What does a 1:200 concentration of epinephrine mean? It means there is 1 g of epinephrine in 200 ml of solution. What does a 1:10,000 concentration of epinephrine mean? It means there is 1 g of epinephrine in 10,000 ml of solution. Now you can use this definition of ratio strength to set up the ratios needed to solve problems.

Problem Set #9 a. How many grams of potassium permanganate should be used in preparing 500 ml of a 1:2500 solution?


b. How many milligrams of mercury bichloride are needed to make 200 ml of a 1:500 solution? c. How many milligrams of atropine sulfate are needed to compound the following prescription? R /

Atropine sulfate 1:200 Dist. water qs ad 30 ml d. How many milliliters of a 1:100 solution of epinephrine will contain 300 mg of epinephrine? e. How much cocaine is needed to compound the following prescription? R /

Cocaine 1:100 Mineral oil qs ad 15 ml f. How much zinc sulfate and boric acid are needed to compound the following prescription? R /

Zinc sulfate 1% Boric acid 2:100 Distilled water qs ad 50 ml Note: “qs ad” means “sufficient quantity to make.” Dilutions Made from Stock Solutions

Stock solutions are concentrated solutions used to prepare various dilutions. To prepare a solution of a desired concentration, you must calculate the quantity of stock solution that must be mixed with diluent to prepare the final product.

CalculatingDilutions Example 1: You have a 10% NaCl stock solution available. You need to prepare 200 ml of a 0.5% NaCl solution. How many milliliters of the stock solution do you need to make this preparation? How much more water do you need to add to produce the final product? 1. How many grams of NaCl are in the requested final product? X g NaCl

=

0.5 g NaCl

200 ml soln 100 ml soln

Therefore, 200 ml of 0.5% NaCl solution contains 1 g of NaCl. 2. How many milliliters of the stock solution will contain the amount calculated in step 1 (1 g)?


Remember, 10% means the solution contains 10 g/100 ml. X ml 1g

=

100 ml 10 g

X ml = 10 ml

The first part of the answer is 10 ml of stock solution. 3. How much water will you need to finish preparing your solution? Keep in mind the following formula: (final volume) – (stock solution volume) = (volume of water) Therefore, for our problem, 200 ml – 10 ml = 190 ml of water The second part of the answer is 190 ml of water. Example 2: You have to prepare 500 ml of a 0.45% NaCl solution from a 10% NaCl stock solution. How much stock solution and water do you need? 1. How many grams of NaCl are in the requested volume? In other words, 500 ml of a 0.45% NaCl solution contains how much NaCl? X g NaCl

=

0.45 g

500 ml 100 ml X g × 100 ml = 0.45 g × 500 ml Xg=

0.45 g × 500 ml

100 ml X g = 2.25 g

2. How many milliliters of stock solution will contain the amount in step 1 (2.25 g)? X ml × 10 g = 2.25 g × 100 ml X ml × 100 g = 2.25 g × 100 ml X ml = 2.25 g × 100 ml 10 g X ml = 22.5 ml You will need 22.5 ml of stock solution. 3. How much water will you need? (Final volume) – (stock solution volume) = volume of water 500 ml – 22.5 ml = 477.5 ml water

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Answer: You will need 22.5 ml of stock solution and 477.5 ml of water to make the final product.

Problem Set #10 a. You need to prepare 1000 ml of a 1% neomycin solution for a bladder irrigation. You have only a 10% neomycin stock solution available in the pharmacy. 1.How many milliliters of the stock solution do you need to make this preparation? 2.How many milliliters of sterile water do you need to add to complete the product? b. Sorbitol is available in a 70% stock solution. You need to prepare 140 ml of a 30% solution. 1.How many milliliters of the stock solution are needed to formulate this order? 2.How much sterile water still needs to be added to complete this product? c. You need to make 1 L of dextrose solution containing 35% dextrose. 1.How many milliliters of the D70W do you use? 2.How much sterile water for injection still has to be added? d. You have a 10% amino acid solution. You need to make 500 ml of a 6% amino acid solution. 1.How many milliliters of the stock solution are needed to prepare this solution? 2.How many milliliters of sterile water for injection need to be added? e. You receive the following prescription: R /

Boric acid 300 mg Dist. water qs ad 15 ml 1.How many milliliters of a 5% boric acid solution are needed to prepare this prescription? 2.How many milliliters of distilled water do you need to add? f. You need to prepare 180 ml of a 1:200 solution of potassium permanganate (KMnO 4). A 5% stock solution of KMnO 4 is available. 1.How much stock solution is needed? 2.How much water is needed? g. How many milliliters of a 1:400 stock solution should be used to make 4 L of a 1:2000 solution?

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h. You receive the following prescription: R /

Atropine sulfate 0.05% Dist. water qs ad 10 ml You have a 1:50 stock solution of atropine sulfate available. 1.How many milliliters of stock solution are needed? 2.How many milliliters of water are needed to compound this prescription? Dosage and Flow Rate Calculations

Dosage Calculations Basic Principles 1. Always look for what is being asked: ■ Number of doses ■ Total amount of drug ■ Size of a dose If you are given any two of the above, you can solve for the third. 2. Number of doses, total amount of drug, and size of dose are related in the following way: Number of doses =

Total amount of drug Size of dose

This proportion can also be rearranged in the following two ways: Total amount of drug = (number of doses) × (size of dose) OR Size of dose =

Total amount of drug Number of doses

Calculating Number of Doses Problem Set #11 a. How many 10 mg doses are in 1 g? b. How many 5 ml doses can be made out of 2 fl oz? Calculating Total Amount of Drug Problem Set #12 a. How many milliliters of ampicillin do you have to dispense if the patient needs to take 2 teaspoonfuls four times a day for 7 days?


Note: First, calculate the total number of doses the patient needs to receive; then, multiply this number by the dose. b. How many milligrams of theophylline does a patient receive per day if the prescription indicates 300 mg tid? c. How many fluid ounces of antacid do you have to dispense if the patient is to receive 1 TBS with meals and at bedtime for 5 days?

Calculating Dose Size Problem Set #13 a. If a patient is to receive a total of 160 mg of propranolol each day, and the patient takes one dose every 6 hours, how many milligrams are in each dose? b. If a daily diphenhydramine dose of 300 mg is divided into six equal doses, how much do you have to dispense for every dose? Calculating the Correct Dose

Dosage calculations can be based on weight, body surface area, or age.

Calculating Dose Based on Weight Dose (in mg) = [Dose per unit of weight (in mg/ kg)] × [Weight of patient (in kg)] Dose/day (in mg/day) = [Dose/kg per day (in mg/kg per day)] × [Weight of patient (in kg)] To find the size of each dose, divide the total dose per day by the number of doses per day as illustrated in the following formula: Size of Dose =

Total amount of drug Number of doses

Problem Set #14 a. A patient who weighs 50 kg receives 400 mg of acyclovir q8h. The recommended dose is 5 mg/ kg every 8 hours. 1.Calculate the recommended dose for this patient. 2.Is the dose this patient is receiving greater than, less than, or equal to the recommended dose? b. The test dose of amphotericin is 0.1 mg/kg. What dose should be prepared for a patient weighing 220 lbs? (Remember, terms must be in


the same units before beginning calculations.) c. A 20-kg child receives erythromycin 25 mg q6h. The stated dosage range is 30 to 100 mg/kg per day divided into four doses. 1.What is the dosage range (in mg/day) this child should receive? 2.Is the dose this child is being given within the dosage range you have just calculated?

Calculating Dose Based on Body Surface Area As noted previously in this chapter, BSA is expressed as meters squared (m2). Problem Set #15 a. An adult with a BSA of 1.5 m2 receives acyclovir. The dose is 750 mg/m 2 per day given in three equal doses. 1.Calculate the daily dose for this patient. 2.Calculate the size of each dose for this patient. Note: These problems are done exactly like the weight problems; however, you should substitute meters squared everywhere kilograms appeared before. Calculating Dose Based on Age The following is an example of information that might be found on the label of an over-the-counter children’s medication: St. Joseph’s Cough Syrup for Children Pediatric Antitussive Syrup Active ingredient: Dextromethorphan hydrobromide 7.5 mg per 5 ml Indications: For relief of coughing associated with colds and flu for up to 8 hours Actions: Antitussive Warnings: Should not be administered to children for persistent or chronic cough such as occurs with asthma or emphysema, or when cough is accompanied by excessive secretions (except under physician’s advice) How supplied: Cherry-flavored syrup in plastic bottles of 2 and 4 fl oz Dosage: (see table below) Age

Weight

Dosage

Under2yr

below 27 lb

As directed by physician

2 to under 6 yr

27 to 45 lb

1 tsp every 6 to 8 h (not

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to exceed 4 tsp daily) 6 to under 12 yr

46 to 83 lb

2 tsp every 6 to 8 h (not to exceed 8 tsp daily)

12 yr and older

84 lb and greater

4 tsp every 6 to 8 h (not to exceed 16 tsp daily)

Problem Set #16 a. Based on the preceding dosing table: 1.What is the dose of St. Joseph’s for a 5-yearold child? 2.What should you dispense for a 12-year-old child weighing 30 kg? b. The usual dose of ampicillin is 100 mg/kg per day. The prescription for a 38-kg child is written as 1 g q6h. 1.Is this dose acceptable? 2.If not, what should it be? c. Propranolol is given as 0.5 mg/kg per day, divided every 6 h. 1.What should a 10-kg child receive per day? 2.What is the size of every dose? d. Aminophylline is given at a rate of 0.6 mg/kg per hr. 1.What daily dose will a 50-kg patient receive? 2.If an oral dose is given every 12 h, what will it be? IV Flow Rate Calculations

Using flow rates, you can calculate the volume of fluid or the amount of drug a patient will be receiving over a certain period of time. Prefilled IV bags are available in a number of volumes including: 50, 100, 250, 500, and 1000 ml.

Calculating Volume of Fluid Daily volume of fluid (in ml/day) = [Flow rate (in ml/h)] × [24 h/day] Problem Set #17 a. D5W is running at 40 ml/hr. 1.How many milliliters of D5W does the patient receive per day? 2.Which size D5W container will you dispense? b. D5W/NS is prescribed to run at 100 ml/h. 1.How much IV fluid is needed in 24 h? 2.How will you dispense this volume?

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c. A patient has two IVs running: D5W/0.5NS at 10 ml/h and a hyperalimentation solution at 70 ml/h. How much fluid is the patient receiving per day from these IVs?

Calculating Amount of Drug Problem Set #18 a. An order is written as follows: 1 g of aminophylline in 1 L D5W/0.225% NaCl to run at 50 ml/ h. How much aminophylline is the patient receiving per day? b. The dose of aminophylline in a child is 1 mg/kg per hour. 1.If the child weighs 40 kg, at what rate should the IV in question “a” be running? 2.What should the daily dose be? c. You add 2 g of aminophylline to 1 L of D5W. If this solution is to run at 10 ml/h, how much drug will the patient receive per day? d. If the dose of aminophylline should be 0.6 mg/kg per hour and the patient weighs 40 kg, at what rate should the IV in question “c” be running? Calculation of IV Flow (Drip) Rates Calculation of IV flow (drip) rates is necessary to ensure that patients are getting the amount of medication the physician ordered. For example, if an order is written as 25,000 units of heparin in 250 ml D5W to infuse at 1000 units/h, what is the correct rate of infusion (in ml/h)? Concentration of IV = Concentration of IV =

Total amount of drug Total volume 25,000 units heparin

250 ml D5W Concentration of IV = 100 units/ml of D5W

Now that you have calculated the concentration per milliliter, you can determine exactly what the rate should be by using the following formula: IV Rate = IV Rate =

Dose desired Concentration of IV (1,000 units/h) (100 units/ml)

IV Rate = 10 ml/h

Problem Set #19 a. An order is written for 2 g of lidocaine in 250 ml of D5W to infuse at 120 mg/h. What is the MANUAL FOR PHARMACY TECHNICIANS

correct rate of infusion (in ml/h)? b. An order is written for 25,000 units of heparin in 250 ml of D5W. The doctor writes to infuse 17 ml/h. How many units of heparin will the patient receive in a 12-hour period?

Moles, Equivalents, Osmolarity, Isotonicity, and pH Moles and Equivalents

A mole is one way of expressing an amount of a chemical substance or a drug. Examples: A mole of NaCl (sodium chloride) weighs 58.45 g. A mole of KCl (potassium chloride) weighs 74.55 g. An equivalent usually expresses the amount of each part (or element) of a chemical substance or a drug. Examples: One mole of NaCl contains one equivalent of Na+ (which weighs 23 g). One mole of NaCl also contains one equivalent of Cl- (which weighs 35.45 g). Remember, 1 equivalent (Eq) = 1000 milliequivalents (mEq).

Numbers to Remember If you forget these numbers, these equivalents can be found on the labels of the large-volume sodium and dextrose solutions. a. 0.9% NaCl contains 0.9 g NaCl in every 100 ml of solution, or 0.9% NaCl contains 9 g NaCl in every liter of solution. b. 0.9% NaCl contains 15.4 mEq Na+ in every 100 ml of solution, or 0.9% NaCl contains 154 mEq Na+ in every liter of solution. c. 0.45% NaCl (1/2 NS) contains 0.45 g NaCl in every 100 ml of solution, or 7.7 mEq of Na+ in every 100 ml of solution, or 0.45% NaCl contains 4.5 g NaCl in every liter of solution, or 77 mEq of Na+ in every liter of solution.


d. The table below summarizes some of the data. grams of grams of milliequivalents of NaCl/100ml NaCl/liter Na+/liter

0.9% NaCl (NS) 0.9 0.45% NaCl (1/2 NS) 0.45 0.225% NaCl (1/4 NS) 0.225

9 4.5 2.25

154 77 38.5

Problem Set #20 a. You need to make 1 L of NS. You have 1 L of SWI (Sterile Water for Injection) and a vial of NaCl (4 mEq/ml). How will you prepare this solution? b. An order calls for D10W/0.45% NaCl to run at 40 ml/h. You have 1 L of D10W and a vial of NaCl (4 mEq/ml) available. How would you prepare this bottle? c. How would you prepare 500 ml of a D10W/ 0.225% NaCl solution if you have SWI, NaCl (4 mEq/ml), and a 50% dextrose solution available? d. You have D10W and a vial of NaCl (4 mEq/ml). How would you prepare 250 ml of a D10W/NS solution? Osmolarity and Isotonicity

Osmolarity expresses the number of particles (osmols) in a certain volume of fluid. Remember, 1 osmol (Osm) = 1000 milliosmols (mOsm). The osmolarity of human plasma is about 280 to 300 mOsm/L. The osmolarity of NS is about 300 mOsm/L, and the osmolarity of D5W is about 280 mOsm/L. Solutions of the same osmolarity are called isotonic. Example: NS and plasma are isotonic. If parenteral fluids are administered that are not isotonic, the result could be irritation of veins and swelling or shrinking of red blood cells. Solutions having osmolarities higher than that of plasma are known as hypertonic solutions. Examples: Hyperalimentation solutions and D5W/ NS are hypertonic. Solutions having osmolarities lower than that of plasma are called hypotonic solutions. Example: 1/2 NS is hypotonic. pH

pH refers to the acidity or basicity of a solution.

15

The pH scale ranges from 1 to 14. pH = 7 is neutral pH < 7 is acidic pH > 7 is basic Normal human plasma has a pH of approximately 7.4. Parenteral solutions with pHs different from that of normal human plasma can be very irritating to tissue when injected. Examples are phenytoin and diazepam. Ophthalmologic preparations are buffered to maintain a pH as close to 7.4 as possible. Some drugs are not stable at a certain pH. One example is ampicillin, which is not very stable in acidic solutions. D5W solutions are slightly acidic, whereas NS solutions are more neutral. Therefore, ampicillin injection is dispensed in NS rather than D5W.

Statistics The arithmetic mean is a value that is calculated by dividing the sum of a set of numbers by the total number of number sets. This value is also referred to as an average . The following formula is used to determine the average: M = ∑ X N where ∑= sum M = mean (average) X = one value in set of data N = number of values X in data set Now, using the formula, follow these two steps to calculate the arithmetic mean age of five pharmacists whose ages are 25, 28, 33, 47, and 54 years. 1. Find the sum of all ages. 2. Divide the sum by total number of pharmacists: 25 + 28 + 33 + 47 + 54 = 187 = 37.4 years 5 5 The median is a value in an ordered set of values below and above which there are an equal number of values. When an even number of measurements are arranged according to size, the median is defined as the mean of the values of the two measurements that are nearest to the middle. Example 1: Determine the median temperature for a medication refrigerator whose temperature

16

reading was 37°F, 39°F, 40°F, 44°F, and 38°F on 5 consecutive days. 1. Arrange the temperature readings according to size, smallest to largest. 37° 38° 39° 40° 44° 2. The median is 39°. Example 2: Determine the median temperature for a medication refrigerator whose temperature reading was 45°F, 40°F, 43°F, 39°F, 46°F, and 40°F on 6 consecutive days. 1. Arrange the temperature readings according to size, smallest to largest. 39° 40° 40° 43° 45° 46° 2. The median is 40 + 43 = 83 = 41.5 2 2 Standard deviation is a statistic that tells you how tightly clustered the data points are around the mean in a set of data. Standard deviation can be calculated by using the following formula: Ω = √ ∑(X-M)2 × 1/(N-1) Where Ω = standard deviation X = one value in set of data M = the mean of all values X in your set of data N = the number of values X in your set of data Graphically, when the individual values are bunched around the mean in a set of data, the bellshaped curve is steep. Conversely, when the individual values are widely spread around the mean in a set of data, the bell-shaped curve is flat (see Figure 5-1). One standard deviation away from the mean in either direction on the X axis accounts for about 68% of the individual values in the data set. Two standard deviations away from the mean in either direction on the X axis accounts for about 95% of the individual values in the data set. Three standard deviations away from the mean in either direction on the X axis accounts for about 99% of the individual values in the

Figure 5-1. Bell Curve


data set. If the curve were flatter and more spread out, the standard deviation would be larger to account for 68% of the individual values in the data set. The standard deviation can help you explain data and evaluate various studies. Example: Using the following steps, calculate the standard deviation for body weight of normal males according to the table below: Normal Males

Body Weight (pounds)

3. Sum up all the squared values. 1004.89 + 114.49 + 561.69 + 745.29 + 28.09 + 1.69 + 767.29 + 265.69 + .49 + 1962.49 = 5452.1 4. Multiply the sum of all the squared values by 1/ (N-1) and take the square root of the resulting value to obtain the standard deviation for male body weights. (5452.1) × (1/(N-1)) = (5452.1) × (1/(10-1)) = 605.79 √ 605.79 = 24.61

1

154

2

175

3

162

4

213

Ideal Body Weight (IBW)

5

191

6

187

7

158

8

202

9

185

10

230

You need to know a patient’s IBW to determine the estimated creatinine clearance of an individual patient. The estimated creatinine clearance is needed to determine the appropriate dose of renally excreted medications, such as tobramycin. The following formula is used for calculating the IBW for a male: IBW male (kg) = 50 + (2.3 × height in inches) 5 feet The following formula is used for calculating the IBW for a female: IBW female (kg) = 45.5 + (2.3 × height in inches) 5 feet Example: Calculate the IBW for a male who is 6 feet 2 inches tall. 1. Convert the height to inches. 6 feet x 12 inches/foot = 72 inches + 2 inches = 74 inches 2. Using the formula for calculating the IBW for a male, multiply 2.3 times 74 inches. 2.3 × 74 = 170.2 3. Divide 170.2 by 5 feet. 170.2 inches/5 feet = 34.04 4. Add 50 to 34.04 to obtain the ideal body weight. 50 + 34.04 = 84.04 kg

1. Calculate the mean (average) of all body weights by adding all body weights and dividing by the total number of normal males. X = ∑ X N ∑ X = 154 + 175 + 162 + 213 + 191 + 187 + 158 + 202 + 185 + 230 = 1,857 pounds N = 10 X = 1,857 pounds = 185.7 pounds 10 2. For each of the individual male body weights, subtract the mean (average) body weight from the individual male body weights, then multiply that value by itself (also known as determining the square). (154 – 185.7)2 = 1004.89 (175 – 185.7)2 = 114.49 (162 – 185.7)2 = 561.69 (213 – 185.7)2 = 745.29 (191 – 185.7)2 = 28.09 (187 – 185.7)2 = 1.69 (158 – 185.7)2 = 767.29 (202 – 185.7)2 = 265.69 (185 – 185.7)2 = 0.49 (230 – 185.7)2 = 1962.49


Practice Calculations 1 1. 2. 3. 4. 5.

What does ss mean?_____ Write II as an Arabic numeral:_____ Write 2/5 as a decimal fraction:_____ The fraction form of 0.1 is:_____ Express 25% as a fraction:_____

17

6. Write 0.88 as a percentage:_____ 7. Express 1/4 as a percentage:_____ 8. The fraction form of 0.4 is:_____ 9. Express xiv in Arabic numerals:_____ 10. Write 1 1/2 in decimal form:_____ 11. The standard metric system measure for weight is the:_____ 12. The standard metric system measure for length is the:_____ 13. The standard metric system measure for volume is the:_____ 14. 1 km = _____ m 15. 1 L = _____ ml 16. 1 kg = _____ g 17. 1 g = _____ mg 18. 1 mg = _____ g 19. 0.01 g = _____ mg 20. 1 ml = _____ L 21. 1 tsp = _____ TBS 22. 1 cup = _____ ml 23. 1 TBS = _____ fl oz 24. 15 ml = _____ TBS 25. 1 cup = _____ fl oz 26. 1000 ml = _____ L 27. 2 kg = _____ g 28. 1 gal = _____ qt 29. 1 pt = _____ cups 30. 1 mm = _____ m 31. 1 tsp = _____ ml 32. 5 gr = _____ g 33. 3 cups = _____ ml 34. 70 kg = _____ lb 35. 45 ml = _____ fl oz 36. 80 mg = _____ gr 37. 250 mg = _____ g 38. 3 TBS = _____ tsp 39. 2 fl oz = _____ TBS 40. 3 qt = _____ pt 41. 120 ml = _____ cups 42. 10 ml = _____ tsp 43. 45 ml = _____ TBS 18

44. 50 mg = _____ mg 45. 2 gal = _____ qt 46. 0.5 pt = _____ ml 47. 20 kg = _____ lb 48. 750 mg = _____ g 49. 25 ml = _____ tsp 50. 6 tsp = _____ TBS

Practice Calculations 2 1. 6% (w/w) = _____ 10% (w/v) = _____ 0.5% (v/v) = _____ 2. A patient needs a 300-mg dose of amikacin. How many milliliters do you need to draw from a vial containing 100 mg/2 ml of amikacin? 3. A suspension of naladixic acid contains 250 mg/ 5 ml. The syringe contains 15 ml. What is the dose (in milligrams) contained in the syringe? 4. How many milligrams of neomycin are in 200 ml of a 1% neomycin solution? 5. 1/2 NS = _____ g NaCl / _____ ml solution 6. How many grams of pumpkin are in 300 ml of a 30% pumpkin juice suspension? 7. Express 4% hydrocortisone cream as a ratio. (Remember that solids, such as creams, are usually expressed as weight-in-weight.) 8. You have a solution labeled D10W/NS. a) How many grams of NaCl are in 50 ml of this solution? b)How many milliliters of this solution contain 50 g dextrose? 9. A syringe is labeled “inamrinone 5 mg/ml, 20 ml.” How many milligrams of inamrinone are in the syringe? 10.Boric acid 2:100 is written on a prescription. This is the same as _____ boric acid in _____ solution. 11.How much epinephrine do you need to prepare 20 ml of a 1:400 epinephrine solution? 12.Calculate the amounts of boric acid and zinc sulfate to fill the following prescription: R /

Zinc sulfate Boric acid

0.5% 1:50


Distilled water qs. ad 100 ml 13.Use the following concentrations to solve the problems: Gentamicin 80 mg/ml Magnesium sulfate 50% Atropine 1:200 a) 120 mg gentamicin = _____ ml b)100 mg atropine = _____ ml c) _____ g magnesium sulfate = 150 ml

Practice Calculations 3 1. You need to prepare 1 L of 0.25% acetic acid irrigation solution. The stock concentration of acetic acid is 25%. a.How many milliliters of stock solution do you need to use? b.How many milliliters of sterile water do you need to add? 2. A drug order requires 500 ml of a 2% neomycin solution. a.How much neomycin concentrate (1 g/2 ml) do you need to fill the order? b.How many milliliters of sterile water must you add to the concentrate before dispensing the drug? 3. a. Calculate the amount of atropine stock solution (concentration 0.5%) needed to compound the following prescription: R /

Atropine sulfate 1:1500 Sterile water qs ad 300 ml b.How much sterile water do you have to add to complete the order? 4. How many tablets do you have to dispense for the following prescription?

an ear infection is 50 mg/kg per day given q6h. Answer the following questions regarding this drug: a.If a child weighs 20 kg, how much erythromycin should he receive per day? b.How much drug will he receive per dose? 8. The dose of prednisone for replacement therapy is 2 mg/m2 per dose. The drug is administered twice daily. What is the daily prednisone dose for a 1.5-m2 person? 9. An aminophylline drip is running at 1 mg/kg per hour in a 10-kg child. How much aminophylline is the child receiving per day? 10. A child with an opiate overdose needs naloxone. The recommended starting dose is 0.1 mg/kg. The doctor writes for 0.3 mg naloxone stat. Answer the following questions on the basis of the child’s weight of 26.4 lbs.: a.What is the child’s weight in kg? b.On the basis of the answer to “a,” does 0.3 mg sound like a reasonable dose? 11. An IV fluid containing NS is running at 80 ml/h. a.How much fluid is the patient receiving per day? b.How many 1-L bags will be needed per day? 12. A patient has two IVs running: an aminophylline drip at 20 ml/h and saline at 30 ml/h. How much fluid is the patient receiving per day from the IVs? 13. a. You prepare a solution by adding 2 g of Bronkospaz to 1 L of NS. What is the concentration of Bronkospaz? b.If the solution of Bronkospaz you made in “a” runs at 30 ml/h, how much Bronkospaz is the patient receiving per day? c.If a 60-kg patient should receive 1 mg/kg per hour, will the dose in “b” be appropriate?

R /

Obecalp ii tablets tid for 10 days 5. How many 2 tsp doses can a patient take from a bottle containing 3 fl oz? 6. A patient is receiving a total daily dose of 2 g of acyclovir. How many milligrams of acyclovir is he receiving per dose if he takes the drug five times a day? 7. The recommended dose of erythromycin to treat


Answers to Problems Sets Problem Set #1: a. 2 b. 605 e. 7 f. 4 i. 11 j. 1030

c. 20 g. 9 k. 14

d. 3 h. 15 l. 16

Problem Set #2: a. 0.5 b. 0.75

c. 1

d. 0.4 19

f. 0.625 i. 5.5 j. 2.67 m. 1/4 n. 2/5 q. 2 1/2 r. 1 3/5

g. 0.5 k. 5.25 o. 3/4 s. 3 1/4

h. 0.25 l. 3.8 p. 7/20 t. 1/3

Problem Set #3: a. 23/100 e. 66.7/100 i. 1 m. 40% q. 35%

b. 67/100 f. 3/4 j. 3/20 n. 24% r. 44%

c. 1/8 g. 33/50 k. 50% o. 4% s. 57%

e. 667/1000

Problem Set #4: a. 1000 ml b. 1 kg c. 1000 mg d. 1 mg e. 3 tsp f. 15 ml g. 1 cup h. 240 ml i. 1 TBS j. 5 ml k. 16 TBS l. 480 ml m. 2 TBS n. 2 pt Problem Set #5: a. 1 fl oz b. 0.5 g c. 0.5 fl oz d. 4 tsp e. 3500 g f. 0.00025 g g. 1.5 L h. 6 gal i . 390 mg j. 54.5 kg k. 90 ml l. 158.4 lb m. 1.97 pt

20

d. 1/2 h. 2/5 l. 25% p. 50% t. 99%

n. 1.76 lb o. 15 ml p. 1 fl oz q. 30 ml r. 20 fl oz s. 32.5 mg t. 500 ml u. 5 gr v. 4 TBS w. 2 fl oz x. 65.5 kg y. 6 tsp z. 20 ml aa.28.3°C bb. –22.2°C cc.41°F dd. 89.6°F ee. 2.1 m2

Problem Set #6: a. 1000 mg b. 1. 11 ml 2. 30 mEq c. 4 ml Problem Set #7: a. 1. 0.45 g 2. 5 g b. 100 g c. 4.5 g d. 500 mg e. 25 g Problem Set #8: a. 10 ml b. 15 ml c. 500 mg d. 16 ml e. 4.5 ml f. 2 g g. 13.5 ml h. 1. 5 ml


2. 2.5 ml 3. 6 ml 4. 20 ml 5. 100 ml 6. a. 2 ml of heparin 10,000 units/ml or b. 20 ml of heparin 1000 units/ml 7. 17.5 ml 8. a. 0.6 ml of heparin 10,000 units/ml or b. 6 ml of heparin 1000 units/ml 9. 20 ml 10. 4.8 ml 11. 500 mg 12. 750 mg 13. 30 mEq 14. 18.75 g 15. 20,000 units or 2000 units 16. 14 mEq 17. 0.625 g 18. 100,000 units or 10,000 units 19. 90 mEq 20. 500 mg i. 1 g j. 75 g k. 150 ml l. 0.9 g m. 0.45 g n. 0.225 g o. 4.5 g p. 2.25 g q. 5 g r. 142.9 ml s. 20 ml t. 0

Problem Set #9: a. 0.2 g b. 400 mg c. 150 mg d. 30 ml e. 150 mg


f. 500 mg of zinc sulfate and 1 g of boric acid

Problem Set #10: a. 1. 100 ml of stock solution 2. 900 ml of water b. 1. 60 ml of stock solution 2. 80 ml of water c. 1. 500 ml of D70W 2. 500 ml of sterile water d. 1. 300 ml of stock solution 2. 200 ml of sterile water e. 1. 6 ml of 5% boric acid 2. 9 ml of distilled water f. 1. 18 ml of stock solution 2. 162 ml of water g. 800 ml of stock solution h. 1. 0.25 ml of stock solution 2. 9.75 ml of water Problem Set #11: a. 100 doses b. 12 doses Problem Set #12: a. 280 ml of ampicillin b. 900 mg of theophylline c. 10 fl oz Problem Set #13: a. 40 mg of propranolol b. 50 mg for every dose Problem Set #14: a. 1. 250 mg q8h 2. greater than the recommended dose b. 10 mg c. 1. 600–2000 mg/day 2. no Problem Set #15: a. 1. 1125 mg 2. 375 mg Problem Set #16: a. 1. 1 tsp every 6 to 8 h not to exceed 4 tsp daily

21

2. 2 tsp every 6 to 8 h not to exceed 8 tsp daily (Defer to weight when there is a difference between the weight-based dose and the age-based dose, since weight is a more accurate basis for dosing medications in general.) b. 1. no (Depending on your facility’s rounding conventions, this dose may be considered correct.) 2. 3800 mg/day c. 1. 5 mg/day 2. 1.25 mg per dose d. 1. 720 mg 2. 360 mg every 12 hours

Problem Set #17: a. 1. 960 ml 2. 1000 ml b. 1. 2400 ml 2. 3000 ml, three 1-L bags c. 1920 ml of fluid Problem Set #18: a. 1.2 g b. 1. 40 ml/h 2. 960 mg/day c. 480 mg/day d. 12 ml/hr Problem Set #19: a. 15 ml/h b. 20,400 units Problem Set #20: a. Add 38.5 ml NaCl to 1 L SWFI b. Add 19.25 ml NaCl to 1 L D10W c. 4.81 ml NaCl plus 100 ml D50W; qs with SWFI to 500 ml d. 9.625 ml NaCl; qs with D10W to 250 ml

Answers to Practice Calculations 1 1. 2. 3. 4. 5. 6. 22

1/2 2 0.4 1/10 1/4 88%

7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

25% 2/5 14 1.5 gram meter liter 1000 1000 1000 1000 0.001 10 0.001 1/3 240 0.5 1 8 1 2000 4 2 0.001 5 0.325 720 154 1.5 1.23 0.25 9 4 6 0.5 2 3 0.05 8 240


47. 48. 49. 50.

44 0.75 5 2

Answers to Practice Calculations 2 1. 6 g/100 g 10 g/100 ml 0.5 ml/100 ml 2. 6 ml 3. 750 mg 4. 2000 mg 5. 0.45 g/100 ml 6. 90 g 7. 4 g/100 g 8. a. 0.45 g b. 500 ml 9. 100 mg 10. 2 g in 100 ml 11. 0.05 g or 50 mg 12. 0.5 g zinc sulfate and 2 g boric acid 13. a. 1.5 ml b. 20 ml c. 75 g

b. No, the calculated dose is 1.2 mg. 11. a. 1920 ml per day b. 2 1-L bags per day 12. 1200 ml per day 13. a. 2000 mg/1000 ml or 2 mg/ml or 2 g/1000 ml or 2 g/L or 2:1000 b. 1440 mg/day c. Yes, dose is appropriate.

Answers to Practice Calculations 3 1. a. 2.5 g acetic acid = 10 ml stock solution b. 990 ml sterile water 2. a. 10 g neomycin = 20 ml stock solution b. 480 ml sterile water 3. a. 0.2 g atropine sulfate = 40 ml stock solution b. 260 ml sterile water 4. 60 tablets 5. 9 doses 6. 400 mg per dose 7. a. 1000 mg per day b. 250 mg per dose 8. 6 mg per day 9. 240 mg per day 10. a. 12 kg


23

APPENDIX 5-1 Table 1

micro-(mc): milli-(m): centi-(c): deci-(d):

Table 6

1/1,000,000 1/1000 1/100 1/10

= 0.000001 = 0.001 = 0.01 = 0.1

Ounces Drams

Scruples Grains

1=

12 =

96 =

288 =

5760

1=

8=

24 =

480

1=

3=

60

1=

20

Table 7

Table 2

deca-(da) : hecto-(h): kilo-(k): mega-(M):

Pound

10 100 1000 1,000,000

Ga llon 1=

P int s 8=

Fluid ounces 128 =

1=

16 = 1 =

Fluid drams 1024 =

Minims 61,440

128 =

7,680

8=

480

1=

60

Table 8 Table 3

1 kilometer (km)

= 1000 meters (m)

0.001 kilometer (km)

= 1 meter (m)

1 millimeter (mm)

= 0.001 meter (m)

1000 millimeters (mm)

= 1 meter (m)

1 centimeter (cm)

= 0.01 meter (m)

100 centimeters (cm)

= 1 meter (m)

Table 4

1 milliliter (ml)

= 0.001 liter (L)

1000 milliliters (ml)

= 1 liter (L)

1 microliter (mcl)

= 0.000001 liter (L)

Pound (lb)

Ounces Grains (oz) (gr)

1=

16 =

7000

1=

437.5

Table 9

1 meter (m) = 1 inch (in) = 1 micron (mc) =

39.37 (39.4) inches (in) 2.54 centimeters (cm) 0.000001 meter (m)

Table 10

1 milliliter (ml)

= 1 6.23 minims (m.)

1,000,000 microliters (mcl) = 1 liter (L)

1 fluid ounce (fl oz)

= 29.57 (30) milliliters (ml)

1 deciliter (dl)

= 0.1 liter (L)

1 liter (L)

= 33.8 fluid ounces (fl oz)

10 deciliters (dl)

= 1 liter (L)

1 pint (pt)

= 473.167 (480) milliliters (ml)

1 gallon (gal)

= 3785.332 (3785) milliliters (ml)

Table 5

1 kilogram (kg)

= 1000 grams (g)

0.001 kilogram (kg)

= 1 gram (g)

1 kilogram (kg)

= 2.2 pounds (lb)

1 milligram (mg)

= 0.001 gram (g)

1 pound avoir (lb)

= 453.59 (454) grams (g)

1000 milligrams (mg)

= 1 gram (g)

1 ounce avoir (oz)

= 28.35 (28) grams (g)

1 microgram (mcg)

= 0.000001 gram (g)

1 ounce apoth (oz)

= 31.1 (31) grams (g)

1 gram (g)

= 15.432 (15) grains (gr)

1,000,000micrograms(mcg) = 1 gram (g)

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Table 11


1 grain (gr)

= 65 milligrams (mg)

1 ounce (avoir) (oz)

= 437.5 grains (gr)

1 ounce (apoth)

= 480 grains (gr)

Table 12

grams of

grams of

milliequivalents of

NaCl/100ml NaCl/liter

Na+/liter

0.9% NaCl (NS)

0.9

9

154.5

0.45% NaCl (1/2 NS)

0.45

4.5

77

2.25

38.5

0.225% NaCl (1/4 NS) 0.225

Footnotes

This chapter was adapted with permission from the Johns HopkinsHospitalTechnicianTraining Course1991:106–38.

CHAPTER 5: PHARMACY CALCULATION S

25

Pharmacy Calculations

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