3 Ratios, Proportions, and Dimensional Analysis

Learn how ratios, proportions, medication rates, and dimensional analysis connect ordered amounts with product strengths and quantities while keeping units consistent.

Ratios and equivalent expressions

A compares quantities in a specified order. For example, the product strength $250\,\text{mg}:5\,\text{mL}$ can also be written as $\frac{250\,\text{mg}}{5\,\text{mL}}$ . The order matters: milligrams per milliliter is not the same expression as milliliters per milligram.

An equivalent preserves the same relationship when both terms are multiplied or divided by the same nonzero number:

$\frac{250\,\text{mg}}{5\,\text{mL}}=\frac{50\,\text{mg}}{1\,\text{mL}}$
Since $1\,\text{g}=1{,}000\,\text{mg}$ , $\frac{1\,\text{g}}{1{,}000\,\text{mg}}=1$ .

A fraction equal to $1$ , such as $\frac{1\,\text{g}}{1{,}000\,\text{mg}}$ , can convert units without changing the quantity. Orient the fraction so that the unwanted unit cancels.

Solving proportions

A states that two ratios are equal. If a liquid contains $250\,\text{mg}$ per $5\,\text{mL}$ , the amount in $7.5\,\text{mL}$ can be found by setting up the ratios with corresponding units in the same positions:

\frac{250\,\text{mg}}{5\,\text{mL}}=\frac{x\,\text{mg}}{7.5\,\text{mL}}

Cross-multiply and solve:

5x=250\times 7.5

x=375\,\text{mg}

A can also find an unknown volume when the ordered amount is known:

\frac{250\,\text{mg}}{5\,\text{mL}}=\frac{375\,\text{mg}}{x\,\text{mL}}

Then $250x=5\times 375$ , so $x=7.5\,\text{mL}$ .

Rates and medication concentrations

A compares quantities, often with different units. Examples include $\frac{\text{mg}}{\text{mL}}$ and $\frac{\text{mL}}{\text{hr}}$ . A is a that relates the amount of drug to the volume or mass of the preparation.

For example, $\frac{250\,\text{mg}}{5\,\text{mL}}$ is equivalent to $\frac{50\,\text{mg}}{1\,\text{mL}}$ . To find the volume containing $375\,\text{mg}$ , multiply the desired amount by the reciprocal concentration:

375\,\text{mg}\times\frac{1\,\text{mL}}{50\,\text{mg}}=7.5\,\text{mL}

The $\text{mg}$ units cancel, leaving milliliters.

, also called the factor-label method, expresses a calculation as a product of labeled fractions. Arrange each fraction so units that are not wanted in the answer cancel. The unit remaining at the end should match the quantity requested.

A reliable setup process is:

Identify the quantity and unit required in the answer.
Write the given amount with its unit.
Add the product strength or conversion factors as fractions, arranging them so matching units cancel.
Check that only the desired unit remains, then calculate.
Check that the numerical result is reasonable and label it clearly.

Tablets from a dose and strength

An order is for $0.75\,\text{g}$ , and tablets contain $250\,\text{mg}$ each. First express the order in milligrams, then cancel grams and milligrams:

0.75\,\text{g}\times\frac{1{,}000\,\text{mg}}{1\,\text{g}}\times\frac{1\,\text{tablet}}{250\,\text{mg}}=3\,\text{tablets}

Liquid volume from a concentration

A product contains $125\,\text{mg}$ per $5\,\text{mL}$ , and the calculated amount needed is $250\,\text{mg}$ :

250\,\text{mg}\times\frac{5\,\text{mL}}{125\,\text{mg}}=10\,\text{mL}

The $\text{mg}$ units cancel, leaving $\text{mL}$ .

Choosing and checking a method

and describe the same underlying relationships but display them differently. A emphasizes that two ratios are equal; emphasizes cancellation of units. Either method can work when it is set up consistently.

Before accepting a result, check that:

Like units have been matched or converted before they are compared.
Ratios are oriented consistently in a .
Unwanted units cancel in .
The final unit answers the question.
The result's size makes sense; for example, at the same concentration, a larger ordered amount should require a larger volume.

Ratios compare quantities in a stated order, while equivalent expressions preserve the same relationship. Proportions equate two ratios and can be solved for an unknown. Rates, including medication concentrations, relate quantities with units such as $\frac{\text{mg}}{\text{mL}}$ . multiplies labeled fractions so unwanted units cancel and the requested unit remains. Correct unit setup is as important as the arithmetic.