Introduction

In the previous chapter you learned the basic measurements found in the household, metric, and apothecary systems of measurements. Each of these is used today in prescribing and administering medications. Another measurement system, avoirdupois, is also available, but this system is not normally found in calculations or conversions. In this system, 16 ounces = one pound (7000 grains = one pound). Two other measures are used in dosage calculations—milliequivalents and units—that will be discussed in Chapter 6.

The household measures are those that use common measuring tools found in most homes. Therefore for the person taking medication to be able to use utensils available, you must convert any orders written in apothecary or metric systems to the household measurements for ease of drug administration. If a physician writes an order in the apothecary system and the medication is available in a metric dose, you must convert the apothecary dose to an approximate metric equivalent to ensure the patient receives the correct amount of medication. As the metric system becomes more widely accepted, the need for conversion will decrease, although the need to calculate a dose in the household measure will remain as long as the United States is measuring in household equivalents.

Conversions among the systems are only necessary if the two factors needed for calculation are not found within the same measurement system. If both factors are already in the metric system, then moving within the system is all that is necessary, as seen in Chapter 4. This is likewise true for the apothecary and household measurements. However, if one factor is in one system, such as the metric system, and the other factor is in another system, such as household, proportional equations are used to find the unknown approximation.

Either ratio and proportion or an advanced system of ratio and proportion, called dimensional analysis, may be used.

Both systems are presented in this chapter so you can decide which method is easier for your personal use. Find the system with which you feel comfortable and use it regularly to perfect its use. Only through practice will this become a skill that is easy for you to use for making pharmaceutical calculations.

One further hint is necessary before beginning the process of converting measurements: Because the measurement systems are not identical, any conversions among them are approximate in the conversion. You cannot find an exact equivalent in most conversions; rather the number will be an approximate amount. An example is that a dram is often used as a symbol for a teaspoon when a dram is actually 4 mL and a teaspoon may be either 4 or 5 mL depending on the utensil. This conversion therefore becomes an approximate amount.

Using Ratio and Proportion for Conversion of Units

Ratio is the relationship of one quantity to another, whereas proportion is the relationship between two equal ratios. In ratio and proportion, an unknown is solved using “x” as the unknown. This provides a logical and systematic means of finding the equivalent unknown when knowns are used for calculations. Two equal ratio sets (or the relationship between two equivalents, such as a [extreme] : b [mean] :: c [mean] : d [extreme]) is indicated for calculating proportional equivalents. When one of the equivalents is unknown, an “x” is used to indicate the unknown. (See Chapter 2 if you need to review this material.)

Example 5-1

A physician orders an antibiotic that is available as 250 mg/5 mL. The household measurement to be used is a teaspoon. To be able to change the metric amount of 5 mL to the household teaspoon, the proportion would appear as follows:

In review, when calculating using ratio and proportion, a good rule of thumb is to place the known conversion ratio on the left side of the equation. The equal measurements must be placed in the same position in the both ratios. (If “x” and “y” are being used to identify specific units of a ratio, the “x” and “y” must remain in the same order in all ratios of the proportions. In other words, if the “x” is on the left of the first ratio, “x” must be on the left in the second ratio.)

< ?xml:namespace prefix = "mml" />KnownUnknownx:y::x:y

When the ratios have been properly aligned into proportion(s), multiply the two inside numbers (means, or “insies”) and the two outside numbers (extremes, or “outsies”) and then solve for “x.” Remember that what is done to numbers on one side of the equation must be done to numbers on the other side of the equation. If you need more review for ratio and proportion, refer to Chapter 2.

Using Dimensional Analysis for Converting Among Units

Dimensional analysis is actually ratio and proportion in multiple sections expressed as fractional forms written across one fractional equation.

Example 5-2

To use dimensional analysis, take the example of a prescription to be prepared for 250 mg of an antibiotic. The available medication is 125 mg/5 mL. How many teaspoons of antibiotic should be administered? The formula appears asfollows:

To use dimensional analysis, the system uses the multiplication of a series of fractions in which the numerator and denominator contain related conversion factors. Each factor has a number and unit of measurement. By using dimensional analysis, remembering multiple formulas to solve drug calculations is not necessary for drug dosages to be accurate and safe.

Using dimensional analysis involves using a series of ratios or factors that are arranged as a fractional equation. Each factor is written as a fraction, and the factors must be related to each other and to the problem that is being solved. As with all fractions, each factor must have a numerator and a denominator. Each problem requires the use of only one equation to determine the answer. If the units are not in the same measurement system, the conversion to the system becomes part of the equation. If both quantities are in the same measurement system, the conversion among systems is not indicated.

Conversion factors are the equivalents between two measurements whether in the same system or not. Each conversion factor includes a value (numerical value) and a label (units of measurement). For example, 12 in = 1 ft. This is a conversion between feet and inches. It can be used to convert measurements for length within the U.S. system but not for direct analysis if the measurement is given in meters or centimeters found in the metric system. In that case, the factor for the metric system must be placed within the equation for the answer to be correct.

Example 5-3

4′=____________cm

The equation for dimensional analysis would be on the conversion 2.54 cm = 1″.

To perform dimensional analysis, always start with what you are looking for, or the unknown (x). Place the x to the left of the equation so that you will not forget what you are solving for. As you work the problem, all matching symbols in the equation must be removed by identifying and canceling those identifying symbols/abbreviations that are alike in the numerators and denominators as the equation evolves.

There are six distinct steps in setting up a fractional analysis, or the fractional equation used in dimensional analysis. To illustrate these steps, the following example used the familiar household measurement system. A recipe calls for cup of an ingredient for a cake. At the time of baking, the only available measurement device is a tablespoon. How can you help the person baking the cake know the proper amount of the ingredient to be used when only a tablespoon is available? ( c = __________ tbsp)

1. Find the known, or given, quantity— c.

2. What is the desired, or wanted, amount?—x tablespoons.

3. What are the conversion factors that are needed to make the necessary calculation? In this case, we need to know that 2 tablespoons equals 1 oz and 8 oz equals 1 c.

4. Set up the problem with factors that are available for conversion factors. You want to cancel out like factors by placing them in a numerator of the first fraction followed by the denominator in the next fraction as follows:

xtbsp=1/4c1×8oz1c×2tbsp1oz

5. Cross out the unwanted units. Just as with any other mathematical equation, if units appear in both the numerator and the denominator, you may cancel them to find the unit that is desired. In this case, the remaining unit is a tablespoon, which is the unit you are seeking.

6. Multiply the numerators, multiply the denominators, and divide the product of the numerators by the product of the denominators. This will give you the desired quantity that was originally your unknown factor.

xtbsp=1/4×8×2tbsp1×1×1or16/4tbsp1or4tbsp1orx=4tbsp

So, after completing the problem:

1/4c=4tbsp of the ingredient needed in the cake

Now let’s look at other examples found within the same measurement systems.

Example 5-4

Example 5-5

Conversions Between Household and Apothecary Systems

The apothecary system has only a few significant measurements that, although not commonly used, do convert into household measurements. Of importance are minims to drops, drams to teaspoons, and ounces to tablespoons (Figure 5-1). Cups, pints, quarts, and gallons are used by both systems. Length is not found in the apothecary system, so the conversions are not included in Table 5-1. As the chapter unfolds, the metric system will be added to combined tables for conversions among measurement systems.

TABLE 5.1

Conversions between Apothecary and Household Systems of Measurement

PARAMETER	APOTHECARY UNITS	HOUSEHOLD UNITS	METRIC UNITS
Volume	i (1 minim)	1 gtt (1 drop)	—
	i (1 dram, 60)	1 tsp (1 teaspoon)	—
	( ounce)	1 tbsp (1 tablespoon, 3 tsp)	—
	i (1 oz)	2 tbsp (2 tbsp, 6 tsp)	—
	viii	1 c (8 oz, 1 cup, pint)	—
	1 pt (1 pint, 2 c, xvi)	1 pt (1 pint, 2 c, 16 oz)	—
	1 qt (1 quart, 2 pt, 4 c, 32)	1 qt (1 quart, 2 pt, 4 c, 32 oz)	—
	1 gal (4 qt, 8 pt, 16 c, 128)	1 gal (4 qt, 8 pt, 16 c, 128 oz)	—
Mass/Weight	i (1 oz, 60 gr)*	1 oz	—
	1# (1 lb, xii)	1# or 1 lb (16 oz)*	—
Length	N/A	N/A	—

^*Note that an apothecary pound contains only 12 oz, while a household pound contains 16 oz.

Example 5-6

Ratio and Proportion	Dimensional Analysis

Example 5-7

4oz=_______________tsp

1xtsp=6×4tsp x=24tspor4oz=24tsp

(1oz=2tbsp and1tbsp=3tsp)

xtsp=(4×2×3)tsp1

4oz=24tsp

Conversions Between Household and Metric Units

Now that the household system has been compared with the apothecary system, it is time to compare the household system with the metric system, which is more frequently used in drug measurements (Table 5-2). Recall that the metric system is formed on the base 10 for ease of conversion. Most conversions from metric to household, such as milliliters to teaspoons, or milliliters to ounces, are volume conversions, although length from inches to centimeters is used in the medical field. Also notice that conversions between household and metric measurements are most commonly found in volume, but length and weight are found minimally when converting between these two systems for pharmaceutical purposes (Figure 5-2).

TABLE 5.2

Conversions between Household and Metric Systems of Measurement

PARAMETER	APOTHECARY UNITS	HOUSEHOLD UNITS	METRIC UNITS
Volume	—	1 gtt (1 drop)	N/A
	—	15 or 16 gtts*	1 mL
	—	1 tsp (1 teaspoon)	5 mL or cc
	—	1 tbsp (1 tablespoon, 3 tsp)	15 mL
	—	2 tbsp (2 tablespoons)	30 mL
	—	1 oz (2 tbsp)	30 mL
	—	1 c (8 oz, 1 cup, pint)	240 mL
	—	1 pt (1 pint, 2 c, 16 oz)	480 mL
	—	1 qt (1 quart, 2 pt, 4 c, 32 oz)	960 mL
	—	1 gal (4 qt, 8 pt, 16 c, 128 oz)	3840 mL
Mass/Weight	—	2.2#	1 kg
Length	—	1″	2.54 cm

^*Remember that the equivalents are approximate and not exact when doing conversions.

Example 5-8

4tsp=______________mL

Ratio and Proportion	Dimensional Analysis
1x=5mL×4 x=20mLor4tsp=20mL

xmL=(4×5)mL1x

4tsp=20mL

Conversions Between Metric and Apothecary Systems

Remember that the metric system uses grams, liters, and meters, whereas the apothecary system frequently uses grains, minims, drams, and ounces. Also remember that in converting between two systems of measure, the equivalents are approximate. This is important when converting between apothecary and metric systems. If the answer is a portion of a minim, you should round to the closest full amount because minims cannot be divided (e.g., if the calculation is for minims, the amount should be rounded to 5 minims; if the calculation is minims, the amount should be rounded to 4 minims). Portions of grains, milligrams, grams, and other measures can be expressed with fractional units. Again, with these conversions, ratio and proportion or dimensional analysis may be used, depending on the method that is most comfortable for you. Table 5-3 shows the equivalents between the apothecary system and the metric system.

TABLE 5.3

Conversions between Apothecary and Metric Systems of Measurement

PARAMETER	APOTHECARY UNITS	HOUSEHOLD UNITS	METRIC UNITS
Volume	xv (15) (or xvi [16])*	—	1 mL
	i	—	4 or 5 mL *
		—	15 mL
	i	—	30 mL
	viii	—	240 mL
	xvi or 1 pt	—	480 mL
	32 or 1 qt	—	960 mL
	128 or 1 gal	—	3840 mL
Mass/Weight	gr i	—	60 mg
	gr xv	—	1 g

^*Remember that all equivalents are approximates when converting between measurement systems.

The important conversions to remember are as follows and these are approximations as stated early in the chapter:

MASS	VOLUME
gr i = 60 mg (or 65 mg)	1 mL = xv or xvi (use the figure that will give you as close to a whole number as possible)
1 g = gr xv	i = 4 or 5 mL
	i = 30 mL

A quick way to remember conversions between grains and milligrams is to use a clock as the basis for conversions. There are 60 minutes in an hour just as there are 60 mg to a gr. Figure 5-3 shows how this can be used.

If you use the clock as a basis for conversions, gr or hour is 15 milligrams or minutes; gr is 30 milligrams, while hour is 30 minutes; gr or hour equals 45 milligrams or hours; and finally gr i equals 60 milligrams, while 1 hour equals 60 minutes. This same clock can be used for 1/6, 1/3, and 2/3 grains or hours just as easily. See Figure 5-3, A and B.

The clock can also be used to convert drams to minims as shown in Figure 5-3, C. As with the other clock examples, drams can be converted to minims by keeping the minim on the outside of the clock and the drams on the inside.

As with previous conversions, ratio and proportion or dimensional analysis may be used for conversions as desired.

Example 5-10

120mg=gr______

Ratio and Proportion	Dimensional Analysis
60×x=120×1 60x=120or120÷60 x=2 120mg=gr ii

xgr=120×gr160×1orx=120gr60orx=gr ii

Putting It All Together—Metric, Household, and Apothecary Measurements and Their Conversions

Throughout this chapter you have converted between two measurement systems at a time. Now you need to put all of the conversions together. Table 5-4 is a combination of all tables previously shown in this chapter. Please remember that all conversions are approximations and could be changed depending on the conversion used for calculation, such as the number of mL in a dram. The conversion should always be made on the basis of a measurable dose.

TABLE 5.4

Conversions among Metric, Apothecary, and Household Systems of Measurement

PARAMETER	APOTHECARY UNITS	HOUSEHOLD UNITS	METRIC UNITS
Volume	i	1 gtt	N/A
	xv (or xvi)*	15 or 16 gtts	1 mL
	i	1 tsp	4 or 5 mL
		1 tbsp or 3 tsp	15 mL
	i	2 tbsp or 6 tsp	30 mL
	viii	1 c (8 oz, pt)	240 mL
	xvi or 1 pt	1 pt (2 c, 16 oz)	480 mL
	32 or 1 qt	1 qt (2 pt, 4 c, 32 oz)	960 mL
	128 or 1 gal	1 gal (4 qt, 8 pt, 16 c, 128 oz)	3840 mL
Mass/Weight	gr i	N/A	60 mg
	gr xv	N/A	1 g
	i (60 gr)	1 oz	4 g
	N/A	2.2#	1 kg
Length	N/A	1″	2.54 cm

^*Remember that all equivalents are approximates when converting among measurements.

Now add one last conversion to the clock (see Figure 5-3, D). Remember that you are now adding a liquid measure that should not be confused with the weight or solid measure.

Using the clock, the calculation of minims to drams can easily be accomplished by using 60 minims equals 1 fl dram. Again, remember to place all symbols or abbreviations ending in “r” in the center of the clock and all beginning with “m” on the outside of the clock. The fractional parts as seen with the grains and hours will again apply to fluid drams, such as 10 minims is 1/6 fluid dram.

Review

In this chapter the concepts of conversion among the metric, apothecary, and household systems have been shown. The conversions may be accomplished using either ratio and proportion, dimensional analysis, or in some cases the use of the clock. As a student, you should find one method with which you are the most comfortable and use it regularly. Although conversions are made, the fact that the conversion is not exact but is an approximation is an important concept to remember. Conversions are necessary only when more than one measurement system is presented in the problem. Remember that the more you use conversions and approximate equivalencies, more comfortable you will be with this concept.