2.2 Percentages & Percent Conversions

2.2 Percentages & Percent Conversions

Quick Summary: Percentages represent proportions out of 100. Master three-way conversions between percents, decimals, and fractions. Understand how to solve for the Part, Percent, or Whole using the core percent equation Part = Percent × Whole, and learn to calculate Percent Change using the baseline (original) value as the denominator.

Percentage calculations appear throughout healthcare—from determining IV fluid concentration strengths and enteral feeding dilutions to calculating patient weight loss percentages and dosage adjustments. The ATI TEAS 7 tests both direct mathematical conversions and multi-step word problems involving percentages.

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1. The Percent Concept & Three-Way Conversions

The word percent comes from the Latin per centum, meaning "per hundred." A percentage is simply a ratio or fraction whose denominator is always 100.

$75\% = \frac{75}{100} = 0.75$

The Three-Way Conversion System

          ┌───────────────────────────┐
          │         PERCENT           │
          └─────┬───────────────▲─────┘
                │ ÷ 100         │ × 100
                │ (Shift 2 L)   │ (Shift 2 R)
                ▼               │
          ┌─────────────────────┴─────┐
          │         DECIMAL           │
          └─────┬───────────────▲─────┘
                │ Digits over   │ Divide
                │ Place Value   │ Numerator by
                ▼               │ Denominator
          ┌─────────────────────┴─────┐
          │        FRACTION           │
          └───────────────────────────┘

Rules for Conversions:

1. Percent to Decimal: Remove the % sign and divide by 100 (move decimal point 2 places to the left).
   Example: 8.5% → 8.5 ÷ 100 = 0.085.
2. Decimal to Percent: Multiply by 100 (move decimal point 2 places to the right) and attach the % sign.
   Example: 0.425 → 0.425 × 100 = 42.5%.
3. Fraction to Percent: Divide numerator by denominator to get decimal form, then multiply by 100.
   Example: 3/5 = 3 ÷ 5 = 0.60 → 0.60 × 100 = 60%.
4. Percent to Fraction: Write percent over 100 and simplify to lowest terms.
   Example: 15% = 15/100 = 3/20.

Conversion Reference Grid

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2. Solving Basic Percent Problems: Part, Percent, & Whole

All basic percent problems relate three quantities: Part, Percent (Rate), and Whole (Base). These are unified by the fundamental percent equation:

$\text{Part} = \text{Percent (as decimal)} \times \text{Whole}$

Depending on which variable is unknown, rearrange the formula as shown in the table below:

Worked Example 1 (Finding Part - Solute Mass): An IV bag contains 500 mL of 5% Dextrose in Water (D5W). How many grams of pure dextrose solute are dissolved in the entire bag?

1. Identify given values: Whole = 500 mL, Percent = 5% = 0.05.
2. Apply formula: Part = Percent × Whole.
3. Calculate: Part = 0.05 × 500 = 25 grams.

Result: There are 25 grams of pure dextrose in the IV bag.

Worked Example 2 (Finding Whole - Total Patient Population): In a clinical trial, 42 patients experienced mild nausea, representing 15% of the total enrolled study group. How many total patients were enrolled in the trial?

1. Identify given values: Part = 42, Percent = 15% = 0.15.
2. Apply formula: Whole = Part / Percent (decimal).
3. Calculate: Whole = 42 ÷ 0.15 = 280 patients.

Result: A total of 280 patients were enrolled.

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3. Percent Increase, Percent Decrease, & Percent Change

Percent change measures the relative change between an original baseline value and a new value. It is expressed as a percentage of the original value.

$\text{Percent Change} = \left( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \right) \times 100\%$

• If New Value > Original Value, the result is positive → Percent Increase.
• If New Value < Original Value, the result is negative → Percent Decrease.

$\text{Percent Decrease} = \left( \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \right) \times 100\%$

CRITICAL RULE: The denominator MUST ALWAYS be the Original (Baseline) Value, never the new value!

Worked Example 3 (Patient Weight Loss Percentage): A patient admitted with congestive heart failure weighs 88 kg on Day 1. Following intensive IV diuretic therapy, the patient's weight drops to 81.4 kg on Day 4. Calculate the percentage decrease in body weight.

1. Identify values: Original Value = 88 kg, New Value = 81.4 kg.
2. Find absolute change (weight loss): 88 - 81.4 = 6.6 kg.
3. Divide by original baseline weight: 6.6 ÷ 88 = 0.075.
4. Multiply by 100%: 0.075 × 100 = 7.5%.

Result: The patient experienced a 7.5% weight decrease.

Worked Example 4 (Percent Increase in Medication Cost): The wholesale cost of an antibiotic vial increases from $40.00 to $46.00. What is the percent increase?

1. Absolute increase: $46.00 - $40.00 = $6.00.
2. Divide by original cost: 6.00 ÷ 40.00 = 0.15.
3. Convert to percent: 0.15 × 100 = 15%.

Result: The cost increased by 15%.

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4. TEAS Calculator Hacks for Percent Problems

On the computer-based TEAS 7 calculator:

• There is no % button on the standard screen interface.
• Hack 1 (Direct Decimal Input): Always divide the percentage by 100 in your head or on screen before multiplying. For 18% of 450, enter 0.18 * 450 = 81.
• Hack 2 (Percent Increase Shortcut): To calculate a 15% increase on $250, multiply directly by 1.15 (1 + 0.15): 250 * 1.15 = 287.50.
• Hack 3 (Percent Decrease Shortcut): To calculate a 20% discount or reduction on 150 mg, multiply directly by 0.80 (1 - 0.20): 150 * 0.80 = 120.

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5. Common Student Traps & Healthcare Applications

Common Student Traps on the TEAS

• Trap 1: Using New Value in Denominator. In weight loss or pressure change problems, students often divide by the new value instead of the original value. Always ask: "What was the starting point?"
• Trap 2: Forgetting to shift decimal point twice. Converting 0.4% to a decimal gives 0.004, NOT 0.04. Moving only 1 place is a frequent distractor on TEAS multiple-choice options!
• Trap 3: Confusing Percentage Points with Percent Change. An increase from 10% to 15% is a 5 percentage point increase, but a 50% relative increase! ((15 - 10) / 10 = 5/10 = 50%).

Healthcare Application: Enteral Tube Feeding Dilutions

Clinical Scenario: A physician orders 400 mL of a 70% strength enteral formula for a tube feeding patient. How much full-strength formula and how much water are needed?

Calculation:

• Full-Strength Formula needed: 70% × 400 mL = 0.70 × 400 = 280 mL.
• Water needed: Total Volume - Formula Volume = 400 mL - 280 mL = 120 mL.

Nurse Action: Mix 280 mL of formula with 120 mL of sterile water.