4.1 Ratios & Rates

4.1 Ratios & Rates

Mathematical comparisons form the foundation of nursing practice and clinical dosage calculations. On the ATI TEAS 7 Mathematics section, you will frequently encounter problems requiring you to set up, simplify, and interpret ratios, rates, and unit rates. Mastering these concepts ensures accuracy when calculating medication dosages, IV infusion speeds, staffing ratios, and patient vital statistic trends.

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Understanding Ratios: Definitions and Formats

A ratio is a mathematical comparison of two numbers or quantities by division. It expresses how much of one quantity exists relative to another quantity. Ratios can be expressed in three distinct mathematical formats, all of which are read identically as "$a$ to $b$":

1. Colon Format: $a : b$ (e.g., $3 : 5$)
2. Fraction Format: $\frac{a}{b}$ (e.g., $\frac{3}{5}$)
3. Word Format: "$a$ to $b$" (e.g., "3 to 5")

When writing a ratio, order matters. The first quantity mentioned must always correspond to the first number (or numerator), and the second quantity corresponds to the second number (or denominator). For example, if a medical surgical unit has 15 patients and 3 nurses, the ratio of patients to nurses is $15 : 3$, whereas the ratio of nurses to patients is $3 : 15$.

Part-to-Part vs. Part-to-Whole Ratios

A critical distinction tested on the TEAS exam is the difference between part-to-part comparisons and part-to-whole comparisons:

• Part-to-Part Ratio: Compares one subgroup to another subgroup within the same collection.
• Part-to-Whole Ratio: Compares one subgroup to the total combined quantity of all groups.

Clinical Scenario: An outpatient clinic sees 12 pediatric patients and 18 adult patients during a morning shift.

• Total Patients = $12 + 18 = 30$ patients.
• Part-to-Part Ratio (Pediatric to Adult) = $12 : 18 = 2 : 3$.
• Part-to-Whole Ratio (Pediatric to Total) = $12 : 30 = 2 : 5$.

Common Student Trap: Confusing part-to-part with part-to-whole when converting ratios to fractions or percentages. If a question asks for the fraction of pediatric patients in the clinic above, you must use the total count ($30$) as the denominator, resulting in $\frac{12}{30} = \frac{2}{5} = 40\%$, rather than $\frac{12}{18} = \frac{2}{3} \approx 66.7\%$.

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Simplifying Ratios to Lowest Terms

Like fractions, ratios should always be simplified to their lowest terms (simplest form). A ratio $a : b$ is in lowest terms when $a$ and $b$ are whole numbers that share no common factors other than $1$.

Step-by-Step Procedure for Ratio Simplification

1. Ensure Identical Units: If the two quantities have different units of measurement within the same category (e.g., grams and milligrams, or hours and minutes), convert both quantities to the smaller unit first.
2. Find the Greatest Common Divisor (GCD): Identify the largest whole number that divides evenly into both numbers.
3. Divide Both Terms: Divide each term by the GCD to produce the simplified ratio.

Worked Example 1: Simplifying Ratios with Mixed Units

Problem: A nurse compares a dose of $500\text{ mg}$ of medication to $2\text{ g}$ of the same medication. Express the ratio of milligrams to grams in lowest terms.

• Step 1 (Convert Units): Recall that $1\text{ g} = 1,000\text{ mg}$. Convert $2\text{ g}$ into milligrams: $$2\text{ g} \times 1,000\text{ mg/g} = 2,000\text{ mg}$$
• Step 2 (Set up Ratio): Compare $500\text{ mg}$ to $2,000\text{ mg}$: $$\text{Ratio} = \frac{500}{2,000}$$
• Step 3 (Divide by GCD): Both numbers can be divided by their GCD, $500$: $$\frac{500 \div 500}{2,000 \div 500} = \frac{1}{4}$$
• Final Answer: The simplified ratio is $1 : 4$ (or $\frac{1}{4}$, or "1 to 4").

Worked Example 2: Simplifying Decimal Ratios

Problem: Simplify the decimal ratio $1.5 : 4.5$.

• Step 1 (Eliminate Decimals): Multiply both terms by $10$ to obtain whole numbers: $$1.5 \times 10 = 15, \quad 4.5 \times 10 = 45 \implies 15 : 45$$
• Step 2 (Divide by GCD): The GCD of $15$ and $45$ is $15$: $$\frac{15 \div 15}{45 \div 15} = \frac{1}{3}$$
• Final Answer: The simplified ratio is $1 : 3$.

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Rates vs. Ratios: What Changes?

While a ratio compares two quantities measured in the same units (making the ratio dimensionless), a rate compares two quantities measured in different units (such as miles per hour, dollars per gallon, or milligrams per milliliter).

$$\text{Rate} = \frac{\text{Quantity in Unit A}}{\text{Quantity in Unit B}}$$

What is a Unit Rate?

A unit rate is a specialized rate in which the denominator is reduced to exactly $1$ unit. To find a unit rate, divide the numerator by the denominator using basic division.

$$\text{Unit Rate} = \frac{\text{Numerator}}{\text{Denominator}} \text{ per } 1 \text{ unit of Denominator}$$

Worked Example 3: Calculating IV Infusion Unit Rate

Problem: An IV infusion pump is programmed to deliver $1,200\text{ mL}$ of $0.9\%$ Normal Saline over an $8\text{-hour}$ surgical shift. What is the unit rate of infusion in $\text{mL/hr}$?

• Setup: Write the rate as a fraction of milliliters over hours: $$\text{Rate} = \frac{1,200\text{ mL}}{8\text{ hr}}$$
• Calculation: Divide $1,200$ by $8$ on your calculator: $$1,200 \div 8 = 150$$
• Final Answer: The unit rate is $150\text{ mL/hr}$ (150 milliliters per 1 hour).

Worked Example 4: Vital Signs Heart Rate Calculation

Problem: A nurse counts $28$ heartbeats in $15\text{ seconds}$. What is the patient's heart rate expressed in the standard unit rate of beats per minute (bpm)?

• Step 1: Express the measurement as a rate: $$\text{Rate} = \frac{28\text{ beats}}{15\text{ seconds}}$$
• Step 2: Convert seconds to minutes ($60\text{ seconds} = 1\text{ minute}$): $$\text{Heart Rate (bpm)} = \frac{28\text{ beats}}{15\text{ seconds}} \times \frac{60\text{ seconds}}{1\text{ minute}}$$
• Step 3: Simplify using multiplication and division on your 4-function calculator: $$28 \times 60 = 1,680 \implies 1,680 \div 15 = 112$$
• Final Answer: The patient's heart rate is $112\text{ bpm}$.

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Healthcare Applications & Common Nursing Rates

The following table summarizes common ratios and rates frequently encountered on the TEAS 7 exam and in clinical nursing practice:

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TEAS 4-Function Calculator Tips

1. No Fraction Key: The onscreen TEAS calculator does not have a dedicated fraction input key ($a\,b/c$). To simplify a ratio or find a unit rate decimal, divide the top number by the bottom number directly ($a \div b$).
2. Order of Keying: For unit rates, enter the numerator first, press the division button ($\div$), enter the denominator, and press equals ($=$).
3. Unit Checking: Never enter units into your calculator! Perform unit conversions (e.g., hours to minutes or grams to milligrams) on paper before typing numbers into the calculator.