Proportions¶

What Is a Proportion?¶

A proportion is a statement that two ratios are equal:

\[\frac{a}{b} = \frac{c}{d}\]

In nursing, proportions let us solve for an unknown quantity when we know three of the four values. This is the core of most dosage calculations.

Setting Up a Proportion¶

The key is keeping units consistent — the same unit must appear in the same position on both sides.

\[\frac{\text{dose on hand}}{\text{quantity on hand}} = \frac{\text{desired dose}}{\text{desired quantity}}\]

Example: Stock: 250 mg per tablet Order: 500 mg How many tablets?

\[\frac{250 \text{ mg}}{1 \text{ tablet}} = \frac{500 \text{ mg}}{x \text{ tablets}}\]

Solving for the Unknown¶

Once the proportion is set up, solve using cross-multiplication (covered in detail in the next section):

[250x = 500 \times 1] [x = \frac{500}{250} = 2 \text{ tablets}]

The Means and Extremes¶

In a proportion written in colon format:

\[a : b = c : d\]

Extremes are the outer values: a and d
Means are the inner values: b and c
The product of the means always equals the product of the extremes

\[a \times d = b \times c\]

This is the mathematical basis for cross-multiplication.

Checking Your Answer¶

Always verify by substituting your answer back into the original proportion:

\[\frac{250 \text{ mg}}{1 \text{ tablet}} = \frac{500 \text{ mg}}{2 \text{ tablets}}\]

Simplify the right side: [\frac{500}{2} = 250 ✅]

Both sides are equal — the answer is correct.

Recognizing Unreasonable Answers¶

Reasonableness Check

Before accepting any answer ask yourself:

Would a nurse realistically give this many tablets?
Is this volume reasonable to administer?
Does the answer make sense given the order and stock?

General guidelines: - Oral tablets: rarely more than 3 tablets per dose - Oral liquid: typically between 5 mL and 30 mL per dose - If your answer falls outside these ranges, recheck your work

Clinical Application¶

Example 1: Stock: 125 mg/5 mL oral suspension Order: 250 mg How many mL?

\[\frac{125 \text{ mg}}{5 \text{ mL}} = \frac{250 \text{ mg}}{x \text{ mL}}\]

[125x = 250 \times 5] [125x = 1250] [x = \frac{1250}{125} = 10 \text{ mL}]

Example 2: Stock: 40 mg/mL injectable solution Order: 100 mg How many mL?

\[\frac{40 \text{ mg}}{1 \text{ mL}} = \frac{100 \text{ mg}}{x \text{ mL}}\]

[40x = 100] [x = \frac{100}{40} = 2.5 \text{ mL}]

Practice Problems¶

Problem 1

Stock: 500 mg per tablet Order: 1000 mg How many tablets?

Answer

[\frac{500 \text{ mg}}{1 \text{ tablet}} = \frac{1000 \text{ mg}}{x \text{ tablets}}] [500x = 1000] [x = 2 \text{ tablets}]

Problem 2

Stock: 250 mg/5 mL Order: 375 mg How many mL?

Answer

[\frac{250 \text{ mg}}{5 \text{ mL}} = \frac{375 \text{ mg}}{x \text{ mL}}] [250x = 375 \times 5] [250x = 1875] [x = \frac{1875}{250} = 7.5 \text{ mL}]

Problem 3

Stock: 0.25 mg per tablet Order: 0.5 mg How many tablets?

Answer

[\frac{0.25 \text{ mg}}{1 \text{ tablet}} = \frac{0.5 \text{ mg}}{x \text{ tablets}}] [0.25x = 0.5] [x = \frac{0.5}{0.25} = 2 \text{ tablets}]

Problem 4

Stock: 10 mg/mL Order: 25 mg How many mL?

Answer

[\frac{10 \text{ mg}}{1 \text{ mL}} = \frac{25 \text{ mg}}{x \text{ mL}}] [10x = 25] [x = 2.5 \text{ mL}]

Problem 5

Stock: 80 mg/2 mL Order: 60 mg How many mL?

Answer

[\frac{80 \text{ mg}}{2 \text{ mL}} = \frac{60 \text{ mg}}{x \text{ mL}}] [80x = 60 \times 2] [80x = 120] [x = \frac{120}{80} = 1.5 \text{ mL}]

Double Check Method

After solving, always verify: [\frac{80 \text{ mg}}{2 \text{ mL}} = \frac{60 \text{ mg}}{1.5 \text{ mL}}] [80 \times 1.5 = 120 = 60 \times 2 ✅]