Fundamentals in Pharmaceutical Calculations

Pharmaceutical Calculations

Pharmaceutical calculations is an area of study that applied the basic principles of mathematics to the preparation and safe and effective use of pharmaceuticals.

Common Fractions, Decimal Fractions, and Percentages

Common Fractions

Common fractions are portions of a whole and is used rarely in pharmaceutical calculations nowadays.

Below is an example of a common fraction:

$\frac{2}{5}$

Note that common fractions have a numerator on top and a denominator in the bottom; the numerator represents the number of parts we have and the denominator represents the total number of parts in a whole.

Addition and Subtraction of Fractions

When adding fractions, their denominators must be the same.

$\frac{1}{4}+\frac{2}{4}=\frac{3}{4}$

If their denominators are not the same, you may multiply in a way that changes the denominators to their least common multiple.

$\frac{1}{2}+\frac{1}{3}\to \frac{3}{6}+\frac{2}{6}=\frac{5}{6}$

Multiplying Fractions

When multiplying fractions, simply multiply the numerators together and multiply the denominators together.

$\frac{1}{2}\times \frac{3}{4}=\frac{3}{8}$

Dividing Fractions

When dividing fractions, simply multiply your dividend by the reciprocal of your divisor; or you could just cross multiply ( multiply you dividend’s numerator by your divisor’s denominator and vice versa ).

$\frac{1}{2}÷\frac{2}{3}\to \frac{1}{2}\times \frac{3}{2}=\frac{3}{4}$

Decimal Fractions

Decimal fractions are fractions with a denominator of 10 or any power of 10.

An example of a decimal fraction is the following:

$0.500$

Percentages

Percent literally means “in a hundred”. Percentages are number values that are followed by a percent sign ( $\%$ ). $100\%$ is equivalent to 1 whole.

An example of a percentage is the following:

$50\%$

Conversion of Common Fractions, Decimal Fracttions, and Percentages

To convert from a common fraction to a decimal fraction, just simply divide the numerator by the denominator, either by hand or through a calculator.

$\frac{1}{2}\to 1÷2=0.5$ To convert from a decimal fraction to a percentage, simply multiply your decimal fraction by a hundred and append a percent sign ( $\%$ ) at the end.

$0.5\to 0.5\times 100\%=50\%$

Exponential Notation

Exponential notation–– otherwise known as scientific notation ––is an alternative method of writing very large or very small numbers, which is often necessary in the subject of pharmaceutical calculations.

Below is an example of writing a number in exponential notation.

$250000\to 2.5\times 10^5$ Exponential notation has two parts, the coefficient and the exponential factor.

The first term you encounter is the coefficient; it is always written to have a value below 10 or with only one figure to the left of the decimal point.

The second term you encounter is the exponential factor, and is always a power of 10; the exponent denotes how many places to the left or to the right of a decimal point will the real value take.

Multiplication of Exponential Notation

When multiplying values in exponential notation, keep in mind the following:

• Exponents on your exponential factors are added.

$10^2\times 10^4=10^6$

• The coefficients are multiplied.

$2.5\times 2.5=6.25$

With the rules above, below would be our full equation.

$(2.5\times 10^2)\times (2.5\times 10^4)=6.25\times 10^6$

Division of Exponential Notation

When dividing values in exponential notation, the same rules as multiplication applies but with their respective inverse operations. The rules are the following:

• Exponents on you exponential factors are subtracted.

$10^5÷10^3=10^2$

• The coefficients are divided.

$7.5÷2.5=3.0$

With the rules above, below would be our full equation.

$(7.5\times 10^5)\times (2.5\times 10^3)=3.0\times 10^2$

Addition and Subtraction of Exponential Notation

When adding or subtracting values in exponential notation, keep in mind the following:

• The expressions must be changed to forms having any common power of 10.
• Then the coefficients are added to subtracted.
• Then the result should be expressed with only one figure to the left of the decimal point.

$(1.4\times 10^4)+(5.1\times 10^3)\to (1.4\times 10^4)+(0.51\times 10^4)$

$(1.4\times 10^4)+(0.51\times 10^4)=1.91\times 10^4=1.9\times 10^4$

Ratio, Proportion, Variation, and Dimensional Analysis

Ratio

A ratio is a relative magnitude of two quantities. Ratios are often expressed as common fractions and the same rules apply.

$\frac{1}{2}=1:2=\text{one is to two}$

Proportion

A proportion is the expression of the equality of two ratios.

$a:b=c:d$

$a:b::c:d$

$\frac{a}{b}=\frac{c}{d}$

$a\text{ is to }b\text{ as }c\text{ is to }d$

Proportions are useful because we can use it to find unknown values.

$\text{if}\frac{a}{b}=\frac{c}{d},\text{then}$

$a=\frac{bc}{d},b=\frac{ad}{c},c=\frac{ad}{b},d=\frac{bc}{a}$

Variation

In most pharmaceutical calculations deal with simple, direct relationships: twice the cause, double the effect, and so on or inverse relationships: twice the cause, half the effect.

Dimensional Analysis

Also known as four factor analysis or factor-label method or unit-factor method. This method involves the logical sequencing and placement of a series of ratios (called factors) in an equation.

To perform a dimensional analysis, simply multiply the given value by the conversion factors necessary to obtain your desired value and unit.

Conversion factors are ratios of one unit to another, with the denominator’s unit being the unit of your given value.

Below is an example of using dimensional analysis to convert from liters ( $\text{L}$ ) to fluid ounces ($\text{fl.oz.}$):

Note: 1 fluid ounce is equal to 29.57 milliliters, and 1 liter is equal to 1000 milliliters.

$2.5\cancel{\text{L}}\left(\frac{1000\cancel{\text{mL}}}{1\cancel{\text{L}}}\right)\left(\frac{1\text{fl.oz}}{29.57\cancel{\text{mL}}}\right)=84.55\text{fl.oz}$

Significant Figures

Significant Figures are consecutive figures that express the value of a denominate number accurately enough for a given purpose.

The accuracy varies with the number of significant figures, which are all about in value except the last, and this is properly called uncertain.

If we have the following value as an example: $325\text{g}$

• $3$ represents a value of $300$
• 2 represents a value of $20$
• 5 represents a value of $5$

Rules of Significant Figures

1. Digits other than zero are significant.

Value	# of Significant Figures
$12.5$	$3$
$1.256$	$4$

2. A zero between digits is significant

Value	# of Significant Figures
$102.56$	$5$

3. Final zeroes after a decimal point are significant.

Value	# of Significant Figures
$1.0$	$2$
$7.90$	$3$

4. Zeroes used only to show the location of the decimal point are not significant.

Value	# of Significant Figures
$0.5$	$1$
$0.05$	$1$
$0.65$	$2$

Decimal Places vs Significant Figures

The number of decimal places indicates the degree of precision with which the measurement has been made.

The number of significant figures indicates the degree of accuracy that is sufficient for a given purpose

if we use the following value as an example: $27.625918$

Rounding the value to five decimal places would give us $27.62592$.

Whereas rounding the value to five significant figures would give us $27.626$.