4.3 Practical Ratio & Percentage Applications

4.3 Practical Ratio & Percentage Applications

In healthcare and daily life, ratios, proportions, and percentages are deeply intertwined. A percentage is simply a ratio that compares a quantity to $100$ (from the Latin per centum, meaning "out of one hundred"). On the ATI TEAS 7 exam, practical word problems will challenge your ability to translate between ratios, decimals, and percentages, and apply these conversions to clinical scenarios such as weight-based drug dosages, IV solution concentrations, percent change in lab values, and scale drawing dimensions.

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Inter-Converting Ratios, Fractions, Decimals, and Percentages

To navigate TEAS math questions efficiently, you must be comfortable converting between representations:

• Ratio to Percentage: Convert the ratio $a : b$ to a fraction $\frac{a}{b}$, divide $a \div b$ to obtain a decimal, and multiply by $100\%$.
• Percentage to Ratio: Express $p\%$ as $\frac{p}{100}$ and simplify the fraction to lowest terms.

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Weight-Based Clinical Dosage Calculations

In pediatric and critical care nursing, drug dosages are frequently prescribed based on patient body weight (e.g., $\text{mg/kg}$). Because patient weight in the United States is often measured in pounds (lbs), you must convert weight to kilograms (kg) using the standard conversion ratio:

$$1\text{ kg} = 2.2\text{ lbs} \implies \text{Weight in kg} = \frac{\text{Weight in lbs}}{2.2}$$

3-Step Weight-Based Dosage Protocol

1. Convert Body Weight: Divide patient weight in pounds by $2.2$ to obtain weight in kilograms.
2. Calculate Total Required Dose: Multiply weight in kilograms by the prescribed dosage rate ($\text{mg/kg}$).
3. Calculate Administered Volume: Use ratio-proportion cross-multiplication to determine the liquid volume ($\text{mL}$) or tablet count needed.

Worked Example 1: Pediatric Weight-Based Dose

Problem: A pediatric patient weighs $44\text{ lbs}$. The provider prescribes an oral antibiotic at a rate of $15\text{ mg/kg/dose}$. The medication concentration on hand is $100\text{ mg / 5 mL}$. How many milliliters should be administered per dose?

• Step 1 (Convert Weight): $$\text{Weight in kg} = \frac{44\text{ lbs}}{2.2\text{ lbs/kg}} = 20\text{ kg}$$
• Step 2 (Calculate Required Dose): $$\text{Total Dose} = 20\text{ kg} \times 15\text{ mg/kg} = 300\text{ mg}$$
• Step 3 (Solve Proportion for Volume): $$\frac{100\text{ mg}}{5\text{ mL}} = \frac{300\text{ mg}}{x\text{ mL}}$$ $$100 \cdot x = 5 \cdot 300 \implies 100x = 1,500 \implies x = \frac{1,500}{100} = 15\text{ mL}$$
• Final Answer: The nurse should administer $15\text{ mL}$.

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Percentage Concentrations of IV Solutions

In medical therapies, liquid solution concentrations expressed as percentages represent weight-to-volume percentage (% w/v). By definition:

$$\text{An } X\% \text{ solution contains } X \text{ grams of solute per } 100\text{ mL of total solution.}$$

Common intravenous fluids include:

• $0.9\%$ Normal Saline ($0.9\%\text{ NaCl}$) = $0.9\text{ g of NaCl per } 100\text{ mL}$.
• $5\%$ Dextrose in Water ($D_5W$) = $5\text{ g of Dextrose per } 100\text{ mL}$.

Worked Example 2: Grams of Solute in an IV Bag

Problem: How many total grams of dextrose are present in a $500\text{ mL}$ IV bag of $D_5W$ ($5\%$ Dextrose in Water)?

• Step 1 (Define Percentage Ratio): $5\%$ means $\frac{5\text{ g}}{100\text{ mL}}$.
• Step 2 (Set Up Proportion): $$\frac{5\text{ g}}{100\text{ mL}} = \frac{x\text{ g}}{500\text{ mL}}$$
• Step 3 (Cross-Multiply): $$100 \cdot x = 5 \cdot 500 \implies 100x = 2,500 \implies x = 25\text{ g}$$
• Final Answer: The bag contains $25\text{ grams}$ of dextrose.

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Percent Change: Increase and Decrease

Percent change measures how much a value has grown or shrunk relative to its original starting value. The TEAS 7 frequently tests percent change through patient clinical trends (e.g., weight loss, blood pressure changes, or blood cell counts).

$$\text{Percent Change} = \frac{| \text{New Value} - \text{Original Value} |}{\text{Original Value}} \times 100\%$$

Exam Trap Alert: The denominator MUST always be the original starting value, never the new value!

Worked Example 3: Calculating Percent Decrease in Lab Values

Problem: A patient presenting with an infection has an initial white blood cell (WBC) count of $12,000 / \mu\text{L}$. Following antibiotic therapy, the WBC count drops to $8,400 / \mu\text{L}$. What is the percentage decrease in the patient's WBC count?

• Step 1 (Find Absolute Difference): $$\text{Difference} = 12,000 - 8,400 = 3,600 / \mu\text{L}$$
• Step 2 (Divide by Original Starting Value): $$\text{Decimal Change} = \frac{3,600}{12,000} = 0.30$$
• Step 3 (Multiply by 100%): $$0.30 \times 100\% = 30\%$$
• Final Answer: The WBC count decreased by $30\%$.

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Scale Ratios and Practical Measurement

Scale ratios compare a representative drawing, blueprint, or map distance to actual physical distances:

$$\frac{\text{Scale Measurement}}{\text{Actual Measurement}} = \frac{\text{Drawing Unit}}{\text{Real-World Unit}}$$

Worked Example 4: Microscope Specimen Enlargement Scale

Problem: A cell micrograph uses a scale ratio of $1\text{ mm} : 5\text{ micrometers } (\mu\text{m})$. If a cell measures $14\text{ mm}$ on the micrograph image, what is its actual physical size in micrometers?

• Set Up Proportion: $$\frac{1\text{ mm}}{5\ \mu\text{m}} = \frac{14\text{ mm}}{x\ \mu\text{m}}$$
• Cross-Multiply: $$1 \cdot x = 5 \cdot 14 \implies x = 70\ \mu\text{m}$$
• Final Answer: The cell's actual physical size is $70\ \mu\text{m}$.