1 Foundations of Pharmacy Math

Reliable pharmacy calculations

Accurate calculations begin with reliable arithmetic and clear unit notation. Fractions, decimals, and percentages are different ways to represent quantities, while units identify what those quantities measure.

Keep units attached throughout a calculation, avoid rounding intermediate results, and check whether the final value is reasonable. Dimensional analysis and equivalent ratios can help organize calculations and verify that units are handled correctly.

Fractions and their operations

A fraction has a numerator, the top number, and a denominator, the bottom number. The denominator cannot be zero.

Operations with fractions

• Simplify: divide the numerator and denominator by the same factor. For example, $\frac{6}{8}=\frac{3}{4}$.

• Add or subtract: use a common denominator. For example, $\frac{1}{4}+\frac{1}{3}=\frac{3}{12}+\frac{4}{12}=\frac{7}{12}$.

• Multiply: multiply the numerators and multiply the denominators. For example, $\frac{2}{3}\times\frac{3}{5}=\frac{6}{15}=\frac{2}{5}$.

• Divide: multiply by the reciprocal of the second fraction. For example, $\frac{2}{3}\div\frac{4}{5}=\frac{2}{3}\times\frac{5}{4}=\frac{10}{12}=\frac{5}{6}$.

A fraction can be converted to a decimal by dividing its numerator by its denominator: $\frac{3}{4}=3\div4=0.75$.

Decimal calculations and notation

Decimal place value is based on powers of ten, including tenths, hundredths, and thousandths. For addition and subtraction, line up the decimal points. For example, $2.35+0.8=2.35+0.80=3.15$.

For multiplication, multiply as though the factors were whole numbers, then place the decimal using the total number of decimal places in the factors: $1.2\times0.35=0.420=0.42$.

For division, move the decimal point in both numbers the same number of places so the divisor becomes a whole number. For example, $2.4\div0.6=24\div6=4$.

In pharmacy work, write a leading zero before a value less than one, as in $0.5$, not $.5$. Avoid a trailing zero after a decimal in a dose expression: write $5\text{ mg}$, not $5.0\text{ mg}$. These conventions help reduce misreading.

Percentages and quantities

Percent means per hundred, so the percent symbol can be replaced by a fraction with denominator $100$.

• Percent to decimal: divide by $100$. For example, $7.5\%=0.075$.

• Decimal to percent: multiply by $100$. For example, $0.32=32\%$.

• Percent to fraction: write the number over $100$ and simplify. For example, $25\%=\frac{25}{100}=\frac{1}{4}$.

• Fraction to percent: convert the fraction to a decimal, then multiply by $100$. For example, $\frac{3}{5}=0.6=60\%$.

To find a percentage of a quantity, convert the percentage to a decimal and multiply. For example, $15\%$ of $80$ units is $0.15\times80=12$ units.

Ratios and proportions

A ratio compares quantities, and a proportion states that two ratios are equal. For example, if $\frac{2}{5}=\frac{x}{20}$, cross-multiplication gives $5x=40$, so $x=8$.

Ratios can include units, as in $\frac{250\text{ mg}}{5\text{ mL}}$. Keep units with the numbers so the meaning of each quantity remains clear.

Unit labeling and conversions

Write the unit beside every number, including in intermediate steps. In SI writing, separate a number from its unit symbol with a space, as in $5\text{ mg}$ or $12\text{ mL}$.

Dimensional analysis organizes a conversion by identifying the starting and desired units and arranging a conversion factor so the unwanted unit cancels. A conversion factor is a ratio of equivalent quantities, so it equals one. For example, given $1\text{ g}=1000\text{ mg}$:

The grams cancel, leaving milligrams. Units can also reveal errors: adding $2\text{ mg}$ to $3\text{ mL}$ is not meaningful because the units measure different quantities. When adding measured quantities, identify both units and express them in compatible units.

Rounding and final checks

Round only when the problem, measurement method, or applicable procedure specifies how. Keep extra digits during intermediate steps because rounding early can change the final result. Retain unrounded values during a conversion, then round the final calculated quantity to an appropriate precision. Preserve the required unit in the rounded answer.

For ordinary rounding to a specified decimal place, inspect the digit immediately to the right of the last place to keep:

• If that digit is from $0$ through $4$, leave the retained digit unchanged.

• If that digit is from $5$ through $9$, increase the retained digit by one.

To the nearest hundredth, $3.146$ rounds to $3.15$, while $3.142$ rounds to $3.14$. Do not assume one rounding rule applies to every medication, device, or setting; follow the stated instructions and established workplace procedure.

Practice with arithmetic and units

These exercises practice arithmetic and unit notation; they are not directions for preparing or administering medication. Try each problem before checking the answers.

Problems

1. Simplify $\frac{12}{18}$.

2. Calculate $\frac{1}{3}+\frac{1}{6}$.

3. Calculate $0.48+1.7$.

4. Calculate $0.25\times0.8$.

5. Convert $0.035$ to a percentage.

6. Find $12\%$ of $250$.

7. Solve the proportion $\frac{3}{4}=\frac{x}{20}$.

8. Convert $0.65\text{ g}$ to milligrams, given $1\text{ g}=1000\text{ mg}$. Show the unit cancellation.

9. Round $6.287$ to the nearest hundredth.

10. What is missing from $5+2\text{ mL}$ if it is meant to represent adding two measured quantities?

Answers

1. $\frac{2}{3}$.

2. $\frac{1}{2}$.

3. $2.18$.

4. $0.20=0.2$.

5. $3.5\%$.

6. $0.12\times250=30$.

7. $4x=60$, so $x=15$.

8. $0.65\text{ g}\times\frac{1000\text{ mg}}{1\text{ g}}=650\text{ mg}$; grams cancel.

9. $6.29$.

10. The first quantity has no unit. Both quantities must be identified and expressed in compatible units before they can be added.