1.3 Ratios, Proportions & Applied Math

Ratios and proportions are fundamental to nursing math. They appear in medication concentrations, IV mixtures, patient-to-nurse staffing, and many other clinical scenarios.

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Ratios

A ratio compares two quantities. It can be written as:

• 1:3 (read "1 to 3")
• 1/3 (fraction form)
• 1 to 3 (word form)

Example: A solution has a ratio of 1:4 (medication to saline). If you have 20 mL total, how much is medication?

• Total parts = 1 + 4 = 5
• Medication = 1/5 x 20 = 4 mL
• Saline = 4/5 x 20 = 16 mL

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Proportions

A proportion is an equation stating that two ratios are equal:

a/b = c/d

To solve a proportion with an unknown, use cross-multiplication:

If a/b = c/d, then a x d = b x c

Example: If 3 tablets contain 750 mg, how many tablets are needed for 1,250 mg?

3/750 = x/1250

Cross-multiply: 3 x 1250 = 750 x x

3750 = 750x

x = 3750 / 750 = 5 tablets

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Percentage Calculations

Finding a Percentage of a Number

Convert the percentage to a decimal and multiply.

Example: What is 35% of 200? 0.35 x 200 = 70

Finding What Percentage One Number Is of Another

Divide the part by the whole and multiply by 100.

Example: A student answered 42 out of 50 questions correctly. What percentage is that? 42/50 x 100 = 84%

Percent Increase and Decrease

Percent Change = (New Value - Original Value) / Original Value x 100

Example: A medication price increased from $40 to $52. What is the percent increase? (52 - 40) / 40 x 100 = 12/40 x 100 = 30% increase

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Applied Word Problems

Word problems on the HESI A2 test your ability to extract mathematical relationships from real-world scenarios.

Strategy for Word Problems:

1. Read carefully — identify what is being asked
2. Identify known values — what numbers are given?
3. Determine the operation — addition, subtraction, multiplication, division, or a combination?
4. Set up the equation — write it out before solving
5. Solve — perform the calculation
6. Check reasonableness — does your answer make sense?

Example Word Problem: A nurse works 12-hour shifts. She earns $32 per hour for the first 8 hours and $48 per hour (time and a half) for any hours over 8. How much does she earn in one shift?

• Regular pay: 8 hours x $32 = $256
• Overtime pay: 4 hours x $48 = $192
• Total: $256 + $192 = $448

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Estimation and Reasonableness

Always check your answer using estimation:

• If a medication dose calculates to 15 tablets, something is likely wrong
• If a temperature converts to 500°F, recheck the formula
• If a patient weight calculates to 5 kg for an adult, the conversion is backward

Common sense checks prevent dangerous medication errors in clinical practice.

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Greatest Common Factor (GCF) and Least Common Multiple (LCM)

Use of GCF: Reducing fractions to lowest terms (divide numerator and denominator by GCF) Use of LCM: Finding common denominators for adding/subtracting fractions

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Number Line and Inequalities

• A number line shows numbers in order from left (smallest) to right (largest)
• Inequality symbols:
  • < means "less than" (3 < 5)

  • means "greater than" (5 > 3)

  • ≤ means "less than or equal to"
  • ≥ means "greater than or equal to"

Nursing application: Lab values are often reported with reference ranges using inequalities. For example, potassium must be ≥ 3.5 and ≤ 5.0 mEq/L to be within normal limits.