6.2 Basic Mathematical Principles Used in Pharmacy Practice

Pharmacy work often requires performing basic mathematical operations involving fractions, decimals, and percentages. A brief review of these basic mathematical operations is needed to understand the differing measuring systems and their conversions as well as pharmaceutical calculations.

Fractions

When something is divided into parts, each part is considered a fraction of the whole. For example, a pie might be divided into eight slices—each one is a fraction or $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.$ of the whole pie. A simple fraction consists of two numbers: a numerator—the number on the top or the left of the fraction—and a denominator—the number on the bottom or the right of the fraction. The denominator represents the number of equal pieces the whole pie is broken into (eight, in the case of the pie) and the numerator represents the number of those pieces which are selected (one of the pieces) (see Figure 6.1).

A fraction is also a convenient way of writing a division equation. For example, the fraction $\raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ is a shorthand way of writing 6 ÷ 3 = 2. Fractions can also be a way of writing a ratio, the comparable relationship of one element to another and the whole, such as six parts of an ingredient to three parts of the diluting substance (6:3), which will be addressed in more detail in the pharmaceutical measuring section to come.

Figure 6.1 Fractional Proportions of the Whole Pie

Adding and Subtracting Fractions

To add or subtract fractions, the numbers must have the same denominator, referred to as a common denominator. When fractions share a common denominator, they are easy to add together. For example: $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$4$}\right.=\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$. If the fractions don’t share a denominator, you need to convert them to a common denominator first. For example, to add $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.$ to $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$15$}\right.$, you need to find a number that both 8 and 15 can be divided into as a denominator. (To subtract $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.$ from $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$15$}\right.$, you have to do the same). Here is how you do it:

Simplifying Fractions

There are cases when a fraction needs to be reduced, or simplified. This requires finding numbers that are divisible by both the numerator and denominator (also called a common factor). Consider the example of simplifying the fraction $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$27$}\right.$.

The common factor of 3 and 27 is 3. Therefore:

${\displaystyle \frac{3}{27}}={\displaystyle \frac{1\times 3}{9\times 3}}={\displaystyle \frac{1}{9}}\times {\displaystyle \frac{3}{3}}={\displaystyle \frac{1}{9}}$

Another way to simplify $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$27$}\right.$ is to divide both the numerator and denominator by a common factor (in this case 3).

${\displaystyle \frac{3}{27}}={\displaystyle \frac{3÷3}{27÷3}}={\displaystyle \frac{1}{9}}$

Now simplify the fraction $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$12$}\right.$. The common factor of 2 and 12 is 2.

${\displaystyle {\displaystyle \frac{2}{12}}\times {\displaystyle \frac{1\times 2}{6\times 2}}={\displaystyle \frac{1}{6}}\times {\displaystyle \frac{2}{2}}={\displaystyle \frac{1}{6}}}$

Another way to simplify $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$12$}\right.$ is to divide both the numerator and denominator by a common factor (in this case 2).

${\displaystyle \frac{2}{12}}={\displaystyle \frac{2÷2}{12÷2}}={\displaystyle \frac{1}{6}}$

Expanding Fractions

You can also make a fraction more complex by multiplying the numerator and denominator equally by a fraction that equals one, such as $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$, or $\raisebox{1ex}{$10$}\!\left/ \!\raisebox{-1ex}{$10$}\right.$. You would do this if you were trying to get two fractions with the same denominator.

For instance, say you had two fractions $\raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$10$}\right.$ and $\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$56$}\right.$, and you wanted to work with them together to find out which one was larger or smaller. You could multiply:

${\displaystyle \frac{6}{10}}\times {\displaystyle \frac{56}{56}}={\displaystyle \frac{336}{550}}\text{and}{\displaystyle \frac{5}{56}}\times {\displaystyle \frac{10}{10}}={\displaystyle \frac{50}{560}}\text{to come up with}{\displaystyle \frac{336}{550}}\text{and}{\displaystyle \frac{50}{560}}$

Now you could subtract or add them since they share the same denominator (or have a common denominator).

Multiplying Fractions

To multiply fractions, multiply numerators by numerators and denominators by denominators. Another way to think of this is to use the fractional line as a floor. You multiply the numbers of the upper story or top level and the basement or bottom level separately. Then you simplify the fraction as a whole, two-story unit (which will be explained next).

Say you want 24 of $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$ tablets, and you want to know what the total would be. Make the whole number 24 into a fraction by placing it over the denominator 1. Multiply this fraction by $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$, multiplying each level separately.

${\displaystyle \frac{24}{1}}\times {\displaystyle \frac{1}{4}}={\displaystyle \frac{24\times 1}{1\times 4}}={\displaystyle \frac{24}{4}}$

Simplify the fraction by dividing 24 by 4, and you end up with 6 tablets.

To multiply two fractions, you simply line them up and multiply the levels separately.

${\displaystyle \frac{1}{4}}\times {\displaystyle \frac{2}{5}}={\displaystyle \frac{2}{20}}$

${\displaystyle \text{This is simplified to}{\displaystyle \frac{1}{10}}.}$

Dividing Fractions

To divide by a fraction, change the division sign to a multiplication sign, and invert the number to the right of the operation sign. (The inverted number is called the reciprocal.) Then multiply the fractions and reduce as necessary. You can also think of this using the floor analogy. To divide a fraction thinking of floors, invert the second fraction and then multiply the upper and basement levels as before.

Consider the following example. Divide $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ by $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$.

${\displaystyle \frac{1}{2}}÷{\displaystyle \frac{2}{3}}$

${\displaystyle \frac{1}{2}}\times {\displaystyle \frac{3}{2}}$

${\displaystyle {\displaystyle \frac{1\times 3}{2\times 2}}={\displaystyle \frac{3}{4}}}$

Another example is to divide the fraction $\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$16$}\right.$ by 2. This requires that you make the whole number 2 into a fraction first.

${\displaystyle {\displaystyle \frac{7}{16}}÷2}$

${\displaystyle {\displaystyle \frac{7}{16}}÷{\displaystyle \frac{2}{1}}}$

${\displaystyle {\displaystyle \frac{7}{16}}\times {\displaystyle \frac{1}{2}}}$

${\displaystyle {\displaystyle \frac{7\times 1}{16\times 2}}={\displaystyle \frac{7}{32}}}$

IN THE REAL WORLD

Decimals

The US money system is based on dividing things into fractions of 10 or 100. There are ten pennies in a dime ($\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$10$}\right.$) or one hundred pennies in a dollar ($\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$100$}\right.$). These amounts are commonly written as decimals ($ 0.01). Understanding decimals is crucial to understanding money as well as to preparing and compounding prescriptions.

A decimal is a number written out in proportion to fractions of “tenths” or multiples of ten. In a decimal, everything to the left of the decimal point is a whole number, and everything to the right is a portion of a whole number in powers of ten, such as three one-hundredths—0.03. You can see this relationship clearly in terms of US money, since one dollar ($1.00) is equal to 100 cents (100 × 0.01). The amount of five dollars and twenty cents is written as the decimal $5.20, which is equal to five full dollar units and $\raisebox{1ex}{$20$}\!\left/ \!\raisebox{-1ex}{$100$}\right.$ called twenty cents (or two dimes that are each worth $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$10$}\right.$ of a $1.00 bill).

In prescriptions, you often go into decimals with fractions of 1,000 or 10,000 because you need to deal with very small drug quantities. The decimal 0.056 means 56 one-thousandths or $\raisebox{1ex}{$56$}\!\left/ \!\raisebox{-1ex}{$\mathrm{1,000}$}\right.$ while 0.0184 would be 184 ten-thousandths or $\raisebox{1ex}{$184$}\!\left/ \!\raisebox{-1ex}{$\mathrm{10,000}$}\right.$ (see Figure 6.2). Most prescriptions and hospital medication orders are written using decimals as shown below:

If a decimal number is less than one, place a zero (0) before the decimal point, as in 0.36. This zero is called a leading zero and is used to prevent confusion and potential medication errors when reading decimals. For example, the decimal 0.131313 is less than one and has a leading zero placed to the left of the decimal point. When the calculations are done, this leading zero should be changed to an X when writing down the final answer, prescription, or compounded amount. This ensures that the correct decimal place is seen: X.131313.

Figure 6.2 Decimal Units and Values

Converting Decimals to Fractions

Any decimal number can be expressed as a fraction by using the appropriate power of 10 as the denominator. To do this, remove the decimal point and use the resulting number as the numerator. Then count the number of places to the right of the decimal point in the original decimal number to designate the correct power of 10, with 10 for one place, 100 for two places, and so on. For instance, 0.24 = $\raisebox{1ex}{$24$}\!\left/ \!\raisebox{-1ex}{$100$}\right.$. Use Table 6.2 to find the corresponding power of 10 to put in the denominator.

${\displaystyle 2.33={\displaystyle \frac{233}{100}}}$

${\displaystyle 0.1234={\displaystyle \frac{1,234}{10,000}}}$

${\displaystyle 0.00367={\displaystyle \frac{367}{100,000}}}$

Simplifying Decimal Fractions

Some fractions use decimals. When this happens, you will want to simplify them to get rid of the decimals in the fractions. You do this by multiplying the fraction by the sufficient power of ten to transform the decimal into a whole number. This is the same as moving the decimal point to the right the same number of zeros you add to the denominator. Consider the following examples with $\raisebox{1ex}{$0.2$}\!\left/ \!\raisebox{-1ex}{$10$}\right.$, which becomes $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$100$}\right.$.

${\displaystyle {\displaystyle \frac{0.2}{10}}\times {\displaystyle \frac{10}{10}}={\displaystyle \frac{2.0}{100}}\text{which simplifies to}{\displaystyle \frac{1}{50}}\text{So}{\displaystyle \frac{0.2}{10}}\text{is the same as}{\displaystyle \frac{1}{50}}}$

Here’s another example with $\raisebox{1ex}{$0.45$}\!\left/ \!\raisebox{-1ex}{$90$}\right.$ equals $\raisebox{1ex}{$45$}\!\left/ \!\raisebox{-1ex}{$\mathrm{9,000}$}\right.$, or $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$200$}\right.$:

${\displaystyle {\displaystyle \frac{0.45}{90}}\times {\displaystyle \frac{100}{100}}={\displaystyle \frac{45}{9,000}}={\displaystyle \frac{1}{200}}}$

Converting Fractions to Decimals

To do pharmaceutical measurements, you often have to convert a fraction to a decimal. Doing this can be a little trickier if the denominators are not in tenths, hundredths, or thousandths. To find the correct decimal number, divide the denominator into the numerator. For instance, the fraction $\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$8$}\right.$ is 7 divided by 8, which equals the decimal value 0.875. Here are some other examples:

${\displaystyle {\displaystyle \frac{1}{2}}=1÷2=0.5}$

${\displaystyle {\displaystyle \frac{1}{3}}=1÷3=0.33333}$

${\displaystyle {\displaystyle \frac{438}{64}}=438÷64=6.84375}$

Rounding Off Decimals

When decimal numbers get too long, with too small fractional amounts at the end, the end amounts need to be “rounded off,” or simplified. To do this, simplify to the nearest tenth place by either adding a little or taking a little away. For instance, if you have a number that goes into the nearest hundredth, like 2.68, and you want a simpler number, you round to the nearest tenth, which would be 2.7. You do this by checking the number in the hundredths place (the “8” in this case). If the number in place is 5 or greater, add 1 to the tenths-place number. If the number in the hundredths place is less than 5, simply drop off the digit in the hundredths place (2.64 to 2.6). Here are some other examples:

5.65 becomes 5.7 (you added 0.05 to the 5.65)

4.24 becomes 4.2 (you dropped 0.04)

The same procedure may be used when rounding to the nearest hundredths or thousandths place.

3.8421 is rounded down to 3.84 (hundredths place)

41.2674 is rounded up to 41.27 (hundredths place)

0.3928 is rounded up to 0.393 (thousandths place)

4.1111 is rounded down to 4.111 (thousandths place)

Rounding down or up, however, can depend on the requirements of the measuring device and the situation. For example, scales all have different degrees of measurement where rounding is required, and the more powerful or toxic the substance, the greater the degree of exactitude you need.

The calculated dose needed is 0.08752 g

Rounded to nearest tenth: 0.1 g

Rounded to nearest hundredth: 0.09 g

Rounded to nearest thousandth: 0.088 g

Rounding to the nearest tenth place is typical in dosages for adults. However, with large amounts, this rounding off can make huge differences. For instance, you will see 1 pint listed as 473 mL but also rounded up as 480 mL. Also, when you are dealing with especially vulnerable populations, such as people with low body weight, you may tend to round down rather than up, even if the numeral in the rounding place is 5 or more.

Because children are so much smaller than adults, not just in weight but in bone density, and because their organs and muscles are not fully formed, you always want to lean towards caution. It is better to underdose than overdose, as you can always administer more if needed. That is why in doses for children, rounding to the nearest hundredth or thousandth may be more appropriate, and you almost always round down for children, especially for premature babies and infants.

Calculating Percents

A percentage is a specific portion related to a whole of 100 units. For instance, 3% means 3 out of 100. So if 12 patients out of 100 request to speak with the pharmacist, that would mean 12% of the patients seek counseling.

Percentages are very important for mixing dilutions and compounds. Percents are often used to communicate the portion, or percentage, of the active ingredient in the whole drug product. The whole drug product is considered to be 100%, and every ingredient in it has a lower percentage of this whole. Percentages of an active ingredient are usually expressed as the specific weight or volume of the ingredient as compared to the weight or volume of the whole of 100 units, such as 1% hydrocortisone cream. Percents can be expressed in many ways, and all of the following expressions are equivalent:

• a percent (e.g., 3%, or 3 percent)

• a fraction with 100 as the denominator (e.g., 3/100)

• a decimal (e.g., 0.03)

• a ratio (e.g., 3:100, or 3 to 100)

A pharmacy technician must be able to describe a piece of a tablet as a ratio (1:2), a fraction ($\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$), and a percentage (50%).

Converting Percents to Decimals

Since 3% equals $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$100$}\right.$, it can easily be written as a decimal (0.03) as shown above. To convert any percent to a decimal, simply drop the percent symbol and divide the number by 100. Dividing a number by 100 is equivalent to moving the decimal two places to the left and inserting zeros if necessary.

200% = 200 ÷ 100 = 2.00

4% = 4 ÷ 100 = 0.04

0.75% = 0.75 ÷ 100 = 0.0075