3.4 Ratios and Proportions

Ratios and Proportions on the TEAS

Ratios and proportions are the single most practical math skill for nursing, and the ATI TEAS 7 tests them heavily within Numbers and Algebra. Almost every medication calculation — how many tablets, how many milliliters, how many drops per minute — is a proportion in disguise. Mastering the setup once lets you answer a whole family of questions with the same four steps.

Ratios

A ratio compares two quantities and can be written three equivalent ways.

Ratios should be reduced just like fractions: 6:8 simplifies to 3:4. They come in three flavors.

Unit Rates

A unit rate has a denominator of 1, which makes comparison easy. Divide to get "per one."

Worked Example: A nurse walks 12 miles in 3 hours. The unit rate is 12 ÷ 3 = 4 miles per hour. Unit rates also tell you the better buy or the faster infusion when two options are stated over different amounts.

Proportions and Cross-Multiplication

A proportion is an equation stating two ratios are equal, such as 1/2 = 2/4. The defining property is cross-multiplication: if a/b = c/d, then ad = bc. This turns a fraction equation into a simple linear equation you can solve for the unknown.

Worked Example: Solve 3/4 = x/12.

1. Cross-multiply: 3 × 12 = 4 × x.
2. Simplify: 36 = 4x.
3. Divide both sides by 4: x = 9.

Follow four steps every time: (1) identify the knowns and the unknown, (2) set up the proportion with units labeled, (3) cross-multiply and solve, and (4) check that the answer is reasonable.

Setting Up Proportions Correctly

The golden rule is keep like units on the same side (or in the same position top-and-bottom) of the proportion. Mixing milligrams with milliliters across the equals sign is the classic error.

Worked Example: If 500 mg is contained in 10 mL, how many mL contain 250 mg? Set up mg over mL on both sides: 500 mg / 10 mL = 250 mg / x mL. Cross-multiply: 500x = 250 × 10 = 2,500, so x = 5 mL.

Dosage Calculations Using Proportions

A reliable nursing formula is Have / Quantity = Desired / x, where "Have" is the strength on hand, "Quantity" is the form it comes in (one tablet or some mL), and "Desired" is the ordered dose.

Worked Example (tablets): Order: 750 mg. Available: 250 mg tablets. 250 mg / 1 tablet = 750 mg / x → 250x = 750 → x = 3 tablets.

Worked Example (liquid with unit conversion): Order: 0.5 g. Available: 250 mg per 5 mL. First convert grams to milligrams: 0.5 g = 500 mg. Then 250 mg / 5 mL = 500 mg / x mL → 250x = 2,500 → x = 10 mL. Converting units before you set up the proportion is essential — a 0.5 versus 500 mismatch is a tenfold error.

Scale, Maps, and Similar Figures

Proportions also handle scale drawings and similar figures, where corresponding sides keep the same ratio.

Worked Example: A map scale reads 1 inch = 50 miles. Two cities are 3.5 inches apart. Set up 1 inch / 50 miles = 3.5 inches / x miles → x = 50 × 3.5 = 175 miles.

Direct vs. Inverse Proportion

For a direct proportion you set the ratios equal and cross-multiply. For an inverse proportion you set the products equal.

Worked Example (inverse): If 3 nurses finish a task in 8 hours, how long would 6 nurses take? Inverse means workers × time is constant: 3 × 8 = 6 × x → 24 = 6x → x = 4 hours. Doubling the workers halves the time, which passes the reasonableness check.

Common Healthcare Proportion Setups

Whenever a TEAS question gives you a known relationship and asks for a fourth value, reach for a proportion, label the units, cross-multiply, and confirm the magnitude makes sense before answering.