Using Fractions

A fraction is nothing more than a part of a whole thing. It is expressed as

a comparison (ratio) of the number of parts needed to the total number of

parts in a whole (e.g., one part in a total of four 1

4 ). See Figure 7-1.

EXAMPLE

An order is for 10 mg of drug. 40 mg scored tablets are available.

Amount Needed

Whole Tablet

10mg

40mg

tablet is

1/4

needed

Note: If you think of fractions as parts of a whole thing, pharmaceutical

calculations will be much easier.

Writing Fractions

The correct way to write a fraction is with the top number smaller than the

bottom (e.g., 1

2 ). If the top number is larger than the bottom number, it

is called an improper fraction, and the value is more than one. So (to be

“proper”), we should write this number as a whole number with a fraction

(e.g., 1 3

4 ). This is called a mixed fraction.

tablet

1/2

tablet

1/4

tablet

Figure 7-1: Fractions as Parts of a Whole

Adding and Subtracting Fractions

Finding the Common Denominator

The common denominator is simply a number that is divisible by the

denominators of both fractions involved. For example, if the fractions that we

wish to add are 1

4 and 1

2 , the common denominator would be 2 x 2, or 4. If

the fractions are more complex, such as 13

and 18

, the common denominator

would be 3 x 8, or 24.

• For ease of calculation, you should be careful to choose the smallest

common denominator possible. For example, when adding 38

and 5

the common denominator should be 24, rather than 48, which is obtained

by simply multiplying the 6 and 8 together.

Once the fractions have a common denominator, we can add the top

parts together, reduce, and get the answer.

EXAMPLE

¼ + 1/2

Step 1. Find the common denominator. 2 x 2 or 4, so the common

denominator is 4, and 1/2 becomes:

½ x 2/2 = 2/4. Next, convert 1/4 to have the same denominator: 1/4 x 1/1 = 1/4. Now we can add the fractions together: 2/4 + 1/4 = 3/4. To summarize, the final result of adding the fractions 1/2 and 1/4 is 3/4. In conclusion, when adding fractions, it is essential to find a common denominator to ensure accurate results. Additionally, practicing with various fractions will enhance your skills and confidence in performing these operations.

Whatever is done to

the numerator of the

fraction must also be

done to the denominator.

In this way,

the fraction stays the

same, as we are really

just multiplying by

one

Subtraction of fractions is done in a similar way, except that instead

of adding the top numbers, we subtract them.

EXAMPLE

2/3 - ½ =?

Step 1. Find the common denominator. The smallest number divisible by

both two and three is six, so the fractions now become

2/3 x 2/2 = 4/6 and 1/2 x 3/3 = 3/6. Now, we can subtract the fractions: 4/6 - 3/6 = 1/6. Therefore, the final answer to the equation 2/3 - 1/2 is 1/6.

In conclusion, when subtracting fractions, it is essential to find a common denominator to ensure accurate calculations.

Step 2. Rewrite the problem:

Identify the fractions involved: 2/3 and 1/2
Convert each fraction to have a common denominator of 6
Adjust the numerators accordingly to reflect the new denominators
Perform the subtraction using the adjusted fractions.

2/3 = 4/6

1/2 = 3/6

Now, subtract the adjusted fractions:

4/6 - 3/6 = 1/6

It is important to get in the habit of rewriting the problem at each step.

Memory is faulty, and this way the problem is organized and clear in your

mind as well as on paper. You are less likely to make a mistake.

Multiplication and Division of Fractions

In the majority of actual pharmaceutical calculations, we use fractions in

problems involving multiplication and division. These calculations are simpler,

as we do not have to have a common denominator.

Multiplying Fractions

EXAMPLE

3/8 × 1/3 =?

Rewriting the problem

with each step is a

great way to keep the

calculation organized

and decrease the possibility

of errors!

Step 1. Multiply the numerators (3 x 1 = 3)

Step 2. Multiply the denominators (8 x 3 = 24)

Step 3. Divide and reduce: ( \frac{3}{24} ) can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 3, resulting in ( \frac{1}{8} ). Step 4. Therefore, the final simplified fraction is ( \frac{1}{8} ), which represents the reduced form of the original expression. In conclusion, when multiplying fractions, it is essential to follow these steps to ensure accuracy and clarity in your calculations.

The final result is 3/24, which can be simplified to 1/8.

Division of Fractions

The easiest way to divide a fraction by a fraction is to fl ip the fraction on the

bottom over (invert) and multiply the two.

Problem:

¼ ÷ ½ = ¼ × 2 = ½. To further illustrate, if we have a problem like ( \frac{3}{5} \div \frac{2}{3} ), we would first invert the second fraction to get ( \frac{3}{5} \times \frac{3}{2} ), which simplifies to ( \frac{9}{10} ).

This method can be applied consistently across various problems, allowing for a straightforward approach to handling division of fractions. This technique not only simplifies calculations but also reinforces the understanding of fraction operations, making it easier to tackle more complex mathematical concepts.

To divide, we invert the 1/2 and rewrite the problem as:

¼ / 2/1 = ¼ \times \frac{1}{2} = \frac{1}{8}, demonstrating the same principle of multiplying by the reciprocal. This consistent application of the reciprocal method ensures clarity in mathematical operations and helps students build a solid foundation in fraction manipulation, which is essential for advanced topics such as algebra and calculus. ?

Multiply the two fractions as follows:

½ x 4/1 = \frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2.

Next, we can simplify this result by recognizing that ( \frac{4}{2} ) can be reduced to its lowest terms, reinforcing the importance of simplification in fraction calculations. Now, reduce the fraction:

2/2 / 4/2 = 1/2.

and the answer is 1/2

Converting Fractions to Decimals

The modern equivalent of the fraction is the decimal. Converting fractions

to decimals can make calculation easier. Decimals are commonly used with

orders given in the metric system (see Chapter 11). To convert a fraction to a

decimal, we simply divide the top number by the bottom number and insert

a point, or period, where the whole numbers end and partial numbers begin.

EXAMPLE

Convert 1 1/2 to decimal form.

In this problem we have a whole number as well as a fraction. Place

the whole number to the left and then calculate the decimal from the

By simplifying as

much as possible

before doing a calculation,

you reduce the

possibility of errors

and make the calculation

easier!

Decimals are based on fractions of ten. A whole number such as 1 (think

of it as $1) would be expressed as $1.00 (100 pennies). 1

100 of a dollar is one

cent, or $0.01. Ten cents is 1

10 of a dollar, or $0.10. 1

1000 of $1, if there was

such a thing, would be 1

2 of a cent, or $0.001.

Decimals are based on

fractions of ten:

1/1 = 1.0

1/10 = 0.1

1/100 = 0.01

1/1000 = 0.001, etc.

Convert 1/4 to a decimal.

4/1.00 = 0.25

The 0 before the .25 is a placeholder that tells you there are no whole

numbers. The decimal point separates the whole numbers from the fraction

(decimal), which is 25 hundredths.

Properly Writing Decimals

Use of the 0 (placeholder) before the decimal is very important. If it is not used

and the number is just written .25, there can be big problems in interpretation.

The dot you see on the drug order before the 25 could be just a speck of

dirt on the order or computer screen or a fl aw in the paper! The order could

be for 25 mg instead of 0.25 mg! Depending on the drug, that’s a difference

big enough to cost a patient’s life. The placeholder makes it clear that the

number is a decimal and not a whole number.

Calculations with Decimals

Adding and Subtracting Decimals

Decimals are added and subtracted like regular numbers:

0.10 + 0.05 = 0.15

0.10 - 0.05 = 0.05

If a number is less than

one, it is important to

place a zero to the left

of the decimal point.

The placeholder

makes it clear that the

number is a decimal

and not a whole

number.

Multiplying and Dividing Decimals

To multiply or divide decimals, do the calculation as if no decimal were present,

then place the decimal back into the answer.

EXAMPLE

1.2 x 2.5

Step 1. Multiply as usual, disregarding decimal points. By multiplying

12 x 25, we obtain 300.

Step 2. Now, insert the decimal point:

Since there is one decimal place in the 1.2 and one also in the

2.5 (a total of two), we need to move the decimal place attached

to the 300 over two places from the right.

300. 3.00

The answer to the problem is then 3.00—round to 3. Trailing

zeros could be problematic also. The 3.00 could be read as 300

if the decimal was not noticed. Always round the decimals to

the whole number if they are zeros.

EXAMPLE

2.5/1.2 = ?

Step 1. Multiply both numbers by 10 to remove the decimal. Now, the

1.2 becomes 12, and the 2.5 becomes 25.

Step 2. Divide normally:

25/12 = 2.0833, which can be rounded to 2.08 for simplicity. 2 08

Division of decimals is done in a similar way. Using the previous example,

let’s divide the decimals:

Note: Remember that each time we move the decimal over one place, we

are multiplying or dividing by 10. Moving the decimal one place to the

left of the number means dividing by 10, while moving it one place to

the right means multiplying by 10.

Use of Algebra in Pharmaceutical Calculations

Knowledge of simple algebra is extremely useful in calculations, especially

when using the ratio-proportion method to calculate dosage (see

Chapters 14–17). Of particular use is cross-multiplication.

Using Cross-Multiplication

If we know the relationship between two quantities (e.g., milliliters and teaspoons),

we can calculate another quantity that we might need. For example,

say we have 100 mL of cough syrup and want to know how many oneteaspoon

doses are in the bottle. We set up a problem where we set the two

relationships equal and go from there:

Problem: You have 100 mL of cough syrup. The order is for one teaspoon

BID. Given that there are 5 mL in one teaspoon, how many doses

are in the bottle?

Solution: 1. First, recognize that the question is really asking how many

5 mL doses are in the 100 mL bottle. All other information is

irrelevant to the calculation.

2. Set up the formula:

5 mL/100 mL = 1 tsp/? tsp doses

3. Cross-multiply and set the two multiplication products equal:

use “X” for the amount to be calculated. This gives us the

following equation:

5 mL /? tsp dose = X

100 mL/1tsp = X

4. This gives us the following equation:

5 mL x X tsp doses = 100 mL x 1 tsp

5. Divide both sides of the equation by 5 mL to get “X” by itself

on one side:

X tsp doses = 100 mL x 1 tsp \ 5 mL

then cancel:

X tsp doses = 100/ 5 = 20 doses. tsp

Therefore, we can conclude that a total of 20 teaspoon doses can be administered from a 100 mL solution. This dosage calculation is essential for ensuring accurate medication administration and patient safety. Proper documentation of these calculations is crucial for maintaining a clear medication administration record and preventing dosing errors. Additionally, it is important to regularly review and update these calculations as needed, especially when there are changes in patient conditions or medication formulations.

Rounding Numbers

When a calculation produces a number that is very long and cumbersome, it

is necessary to round off the number. This becomes particularly important

when a calculated dose comes out to be more precise than the calibrations

on your measuring device. To round a number accurately, fi rst locate the fi rst

digit following the number where it should be rounded (e.g., 3.58). If the

digit is greater than fi ve, round up. If it is less than 5, round down. If the

number is equal to 5, then we must look at the number immediately to

the left of the 5 to determine which way to round (see second example

below). If this number is even (e.g., 2, 4, 6, 8), round the number down; if

it is odd, round up.

Using Roman Numerals

A knowledge of Roman numerals, both uppercase and lowercase, is necessary

for the exam. In modern use, prescribers may use lowercase Roman numerals

to specify the number of units of medication per dose on a written prescription.

In addition, specifi cation of amounts of drugs measured in apothecary

units also requires the use of Roman numerals.

EXAMPLE

You calculate the amount to be drawn up for an injection to be

1.1265 mL. The syringe is only accurate to 0.01 mL, so you need to

round to two decimal places.

Looking at the amount to be rounded, 1.1265, fi nd the second

decimal place (the 2) and look at the numbers immediately following it.

In this case, it is 6. Since 6 is greater than 5, you will round the number

up from 1.12 to 1.13.

EXAMPLE

The amount to be drawn up is calculated to be 1.1250 mL. In this

case, the last digit is equal to 5. Now, the decision to round up or

down is determined by the number that precedes the last digit. This

is 2, which is an even number. Thus, we round down to 1.12. If the

number had been an odd number (for example, 3), we would have

rounded up.

Calculating with Roman Numerals

Roman numerals are based on a series of letters. A number, in this system,

is made up of individual Roman numerals that are written from left to

right in descending order of value (e.g., MCVII). To convert the number

into the traditional Arabic numbering system, the individual values of the

numerals are added. See Table 7-1 for commonly used Roman numerals

and their values.

The order (placement) of the individual numbers is important— numerals

written from left to right in order of descending value are added, whereas

a smaller numeral placed to the left of a larger numeral is subtracted. For

example, MC = 1,000 + 100, but CM = 1,000 – 100!

Problem: Convert MCXXIII to an Arabic number.

Solution: The individual numerals should fi rst be separated, then added:

M = 1,000

C = 100

X = 10

X= 10

III = 3 (1 1 1)

Table 7-1 Commonly Used Roman Numerals and Their Values

ss = I/2

I (i) = 1

V (v) = 5

X (x) = 10

L (l) = 50

C (c) = 100

D (d) = 500

M (m) = 1,000

Note: Numerals may be expressed as capital letters or lowercase letters. The values do not change

whether lower- or uppercase.

Convert the number MCMXCVII to an Arabic number.

Solution: First, separate the numbers:

M = 1,000

CM = 900 (1000 - 100)

XC = 90 (100 - 10)

VII = 5 +1 + 1 = 7

Then add the numerical values to obtain the value of the number:

Total: 1,000 + 900 + 90 + 7 = 1,997