Mathematics Foundations | Free Pre-Nursing

Mathematics Foundations for Healthcare

Build the clinical numeracy confidence you need for safe medication calculation, unit conversion, and data interpretation. This module starts with arithmetic review and progressively applies each concept to real healthcare scenarios.

Fractions, Decimals, and Percentages

The arithmetic of parts and wholes

Fractions, decimals, and percentages are three ways to express the same concept: a part of a whole. Fluency in converting between them and performing arithmetic with them is the foundation of every drug calculation you will ever do.

Fractions

Numerator / Denominator.A proper fraction has numerator < denominator (3/4). An improper fraction has numerator > denominator (7/4). A mixed number combines a whole number and a fraction (1¾). To convert improper to mixed: divide and write the remainder as the new numerator (7÷4 = 1 remainder 3 → 1¾).

Decimals

Convert a fraction to a decimal by dividing numerator by denominator: 3/4 = 3 ÷ 4 = 0.75. In healthcare, always lead with a zero before the decimal (0.5 mg, never .5 mg) to prevent ten-fold dosing errors.

Percentages

Percent means "per hundred." Convert decimal to percent: multiply by 100 (0.75 × 100 = 75%). Convert percent to decimal: divide by 100 (75% ÷ 100 = 0.75). Convert fraction to percent: first to decimal, then × 100.

Fraction Arithmetic — Four Rules

Add/Subtract: Find LCD, convert both fractions, add/subtract numerators, keep denominator. Example: 1/4 + 1/3 → LCD 12 → 3/12 + 4/12 = 7/12.
Multiply: Straight across — numerator × numerator, denominator × denominator. 2/3 × 3/4 = 6/12 = 1/2. Simplify before multiplying when possible.
Divide: Multiply by the reciprocal (flip the second fraction). 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 = 1½.
Simplify: Divide numerator and denominator by their greatest common factor. 6/12 → GCF = 6 → 1/2.

LCD Method for Adding Fractions

When adding fractions with unlike denominators, find the Least Common Denominator (LCD) — the smallest number both denominators divide into evenly. Multiply each fraction's numerator and denominator to convert, then add. Example: 1/4 + 1/3 → LCD = 12 → 3/12 + 4/12 = 7/12.

Clinical Worked Examples — Fractions & Percentages

Common Error: Trailing Zeros and Missing Leading Zeros

Never write 1.0 mg (the trailing zero can look like 10 mg). Always write 0.5 mg, never .5 mg (the missing leading zero can look like 5 mg). These are the two most common decimal documentation errors in clinical practice.

Fractions, Decimals & Percentages Quiz

1/4

A patient ate 2/3 of a meal tray. The tray contained 600 kcal. How many kcal did the patient consume?

Ratios and Proportions

The language of medication dosing

A ratio expresses a relationship between two quantities. A proportion is a statement that two ratios are equal. Almost every drug dosing problem is, at its core, a proportion problem — you know three values and solve for the fourth.

Ratios

Written as 2:5 or 2/5, both mean "2 parts to 5 parts." Drug ratios express concentration: 1:1000 adrenaline = 1 g per 1000 mL = 1 mg/mL. Ratios can be converted to fractions and used in proportion equations.

Proportions — Cross Multiplication

If a/b = c/d, then a × d = b × c. Set up known concentration on the left, unknown on the right with x. Cross-multiply, then divide to solve for x. Always label units to confirm they match.

The D/H × Q Formula

D= Desired dose (what is ordered) | H= Have (available concentration) | Q = Quantity of the available form (mL, tablet, etc.)

Oral tablets: Have 250 mg/tablet, ordered 500 mg. D/H × Q = 500/250 × 1 tab = 2 tablets.

Liquid oral: Have 125 mg/5 mL, ordered 200 mg. D/H × Q = 200/125 × 5 mL = 8 mL.

Injection: Have 10 mg/mL, ordered 7.5 mg. D/H × Q = 7.5/10 × 1 mL = 0.75 mL.

Always Check Units First

The D/H × Q formula is the cornerstone of oral medication calculation: D = Desired (ordered) dose, H = Have (available concentration), Q = Quantity of the available form. Always write the formula first, substitute values second, cancel units third, calculate last. This sequence prevents transposition errors.

Common Error: Mismatched Units

If the order is in mg and the available drug is in mcg, you MUST convert to the same unit before applying D/H × Q. Failure to match units is the leading cause of ten-fold dosing errors. Write the unit next to every number before calculating.

Ratios and Proportions Quiz

1/5

Ordered: amoxicillin 500 mg. Available: 250 mg/capsule. How many capsules?

Metric System and Unit Conversions

The universal language of clinical measurement

Healthcare uses the metric system universally because its base-10 structure minimises calculation errors. Understanding SI prefixes, mass, volume, temperature, and weight conversions is non-negotiable for safe clinical practice.

Why the Metric System Reduces Errors

The metric system is built on powers of 10, which makes conversions straightforward — you only multiply or divide by 10, 100, or 1000. In healthcare, the most used conversions are: kg ↔ g (×1000), g ↔ mg (×1000), mg ↔ mcg (×1000), L ↔ mL (×1000). Memorise these four and you can handle 95% of clinical calculations.

SI Prefix Table

Prefix	Symbol	Power of 10	Clinical Use
kilo	k	10³ (×1,000)	kilogram (kg)
— base —	—	10⁰ (×1)	gram (g), litre (L), metre (m)
milli	m	10⁻³ (÷1,000)	milligram (mg), millilitre (mL)
micro	mc / μ	10⁻⁶ (÷1,000,000)	microgram (mcg)
nano	n	10⁻⁹ (÷1,000,000,000)	nanogram (ng) — lab assays

Critical Mass Conversions

1 kg = 1,000 g = 1,000,000 mg = 1,000,000,000 mcg

1 g = 1,000 mg | 1 mg = 1,000 mcg

To go smaller → multiply; To go larger → divide.

Critical Volume Conversions

1 L = 1,000 mL

1 mL = 1 cc (cubic centimetre)

1 tbsp = 15 mL | 1 tsp = 5 mL

Weight: Pounds ↔ Kilograms

1 kg = 2.2 lb

lb → kg: divide by 2.2

kg → lb: multiply by 2.2

Always convert to kg for dosing — orders are weight-based in kg.

Temperature Conversions

°C → °F: (°C × 9/5) + 32

°F → °C: (°F − 32) × 5/9

Normal body temp: 37°C = 98.6°F

Fever threshold: 38°C = 100.4°F

Calculation Tip: Military Time in Healthcare

Healthcare uses 24-hour (military) time to eliminate AM/PM ambiguity. 1:00 AM = 0100 | 12:00 PM (noon) = 1200 | 1:00 PM = 1300 | 11:00 PM = 2300. To convert 12-hour PM → 24-hour: add 12 to the hour. Midnight = 0000 or 2400.

Step-by-Step Unit Conversion Examples

Metric System and Conversions Quiz

1/5

Convert 2.5 g to milligrams.

Dimensional Analysis (Factor-Label Method)

Systematic unit cancellation — the gold standard for drug calculations

Dimensional analysis (DA) is a problem-solving method that converts one unit to another by multiplying by conversion factors expressed as fractions. Units cancel like algebra, leaving only the desired unit. It works for any calculation — simple or complex — and automatically tells you whether to multiply or divide.

Why Dimensional Analysis Prevents Errors

Dimensional analysis (DA) is a systematic method that eliminates guesswork about whether to multiply or divide. By writing every quantity as a fraction with units and cancelling units like algebra, you guarantee the correct arithmetic path. Nurses who use DA make significantly fewer medication calculation errors than those who rely on memory or mental math shortcuts.

The DA Template

[Known quantity] × [Conversion 1] × [Conversion 2] = Answer in desired units

Step 1:Identify what you HAVE (given quantity + unit). Step 2:Identify what you WANT (target unit). Step 3:Write conversion factors as fractions so unwanted units cancel. Step 4:Check unit cancellation before calculating. Step 5: Multiply all numerators, divide by all denominators.

Step-by-Step DA Healthcare Examples

Calculation Tip: Unit Cancellation Check

Before computing any number, verify that only your desired unit survives after cancellation. If a wrong unit remains, you have a setup error — find it before calculating. This five-second check prevents the most common dimensional analysis mistakes.

Dimensional Analysis Quiz

1/5

Infuse 500 mL over 4 hours. What is the rate in mL/hr?

Percentages in Healthcare

From IV solutions to burn assessment

Percentages appear throughout clinical practice in ways beyond simple arithmetic — solution concentrations, burn surface area, lab value trending, and research statistics all rely on percentage reasoning. This lesson builds percent fluency in each of those clinical contexts.

Percent Solutions — w/v Convention

Percent solutions in healthcare always mean weight-in-volume (w/v): grams of solute per 100 mL of solution. This convention is universal for IV fluids, antiseptics, and topical preparations. A 1% lidocaine solution = 1 g per 100 mL = 10 mg per mL. Knowing this conversion instantly tells you the dose per milliliter without any formula.

Common IV Solution Concentrations

0.9% NaCl (Normal Saline) = 0.9 g/100 mL = 9 g/L

0.45% NaCl (Half Normal Saline) = 0.45 g/100 mL = 4.5 g/L

5% Dextrose (D5W) = 5 g/100 mL = 50 g/L

50% Dextrose (D50) = 50 g/100 mL = 500 mg/mL

Rule of Nines — Body Surface Area Burns

Head and neck: 9% | Each arm: 9%

Anterior trunk: 18% | Posterior trunk: 18%

Each leg: 18% | Perineum: 1%

Total = 100% BSA. Guides fluid resuscitation.

Percent Change — Lab Value Trending

Formula: Percent Change = (New value − Old value) / Old value × 100

Example: Haemoglobin was 12 g/dL; now 10 g/dL. Percent change = (10 − 12) / 12 × 100 = −2/12 × 100 = −16.7%. A 16.7% decrease. Clinical significance threshold varies by parameter.

Check Your Answer: Does the Percent Make Sense?

When calculating percent of a group (e.g., 30% of 200 patients), the answer must be less than the total. When calculating percent concentration, the gram amount should be less than the total volume in mL. If your answer violates these sanity checks, re-examine your setup.

Percentages in Healthcare Quiz

1/4

How many grams of NaCl are in 1 L of 0.45% normal saline?

Basic Algebra for Healthcare

Using formulas to solve for clinical unknowns

Algebra provides the framework for working with clinical formulas. When you know most variables in a formula but need to find one — solving for stroke volume from cardiac output, calculating creatinine clearance, or determining BSA — algebra gives you the tools to isolate and solve for the unknown.

Algebra Rule: Balance the Equation

Golden rule of algebra: whatever you do to one side of the equation, you must do to the other. To isolate a variable: (1) Identify the variable. (2) Identify what operation surrounds it. (3) Apply the inverse operation to both sides. Multiplication → divide both sides. Division → multiply both sides. Addition → subtract both sides. Subtraction → add both sides.

Key Healthcare Formulas

Cardiac Output: CO = HR × SV. (CO in L/min, HR in beats/min, SV in mL/beat). To find SV: SV = CO / HR.

BMI: BMI = weight (kg) / height² (m). Height must be in metres. 70 kg ÷ (1.75 m)² = 70 ÷ 3.0625 = 22.9 kg/m².

Anion Gap: AG = Na⁺ − (Cl⁻ + HCO₃⁻). Normal range ≈ 8–12 mEq/L. Elevated AG suggests metabolic acidosis with unmeasured anions (e.g., lactate, ketones).

Cockcroft-Gault CrCl: CrCl = [(140 − age) × weight (kg)] / (72 × serum Cr). Multiply by 0.85 for females. Used to estimate kidney function and adjust drug dosing.

BSA — Mosteller formula: BSA (m²) = √[(height cm × weight kg) / 3600]. Important for chemotherapy dosing.

Step-by-Step Formula Applications

Calculation Tip: Plug Before You Simplify

Always substitute your known values into the formula before simplifying. Do not try to simplify the formula algebraically in your head first — substituting concrete numbers reduces the chance of sign or order-of-operations errors.

Basic Algebra for Healthcare Quiz

1/4

Cardiac output is 6 L/min and heart rate is 75 bpm. What is stroke volume?

Statistics Basics for Healthcare

Understanding data, lab ranges, and research results

Healthcare workers encounter statistics daily — interpreting lab reference ranges, reading research studies, understanding screening test performance, and evaluating treatment outcomes. You do not need to calculate statistics; you need to interpret them correctly.

Mean (Average)

Sum of all values ÷ number of values. Sensitive to extreme values (outliers). Best when data is symmetrically distributed. Example: average pain score across 10 patients.

Median

Middle value when data is ranked in order. Not affected by outliers. Preferred when data is skewed. Example: median length of hospital stay (a few very long stays skew the mean).

Mode

Most frequently occurring value. Useful for categorical data. Example: the most common adverse reaction reported in a drug trial.

Lab Reference Ranges and the Normal Distribution

Laboratory reference ranges are typically the middle 95% of values from a healthy population — approximately ±2 standard deviations from the mean. This means about 5% of perfectly healthy individuals will have a result outside the normal range by chance alone. Before concluding a result is pathological, context matters: is it a borderline deviation or far outside normal? Trend over time often matters more than a single value.

Screening vs Diagnostic Tests — Sensitivity and Specificity

Sensitivity: Ability to correctly identify true positives. A highly sensitive test rarely misses a disease. High sensitivity → few false negatives. Use for screening (rule OUT disease with a negative result). Mnemonic: SnNout — Sensitive, Negative rules OUT.

Specificity: Ability to correctly identify true negatives. A highly specific test rarely misidentifies healthy people as diseased. High specificity → few false positives. Use to confirm diagnosis (rule IN disease with a positive result). Mnemonic: SpPin — Specific, Positive rules IN.

Reading Research: Key Concepts

p < 0.05: If the null hypothesis were true, there is less than a 5% chance of seeing results this extreme by chance. Conventionally "statistically significant" — but statistical significance does not equal clinical significance.
Relative Risk (RR): RR = risk in treatment group / risk in control group. RR < 1 means the treatment reduces risk.
Number Needed to Treat (NNT): How many patients must be treated to prevent one additional bad outcome. NNT = 1 / absolute risk reduction. Lower NNT = more effective treatment.
Number Needed to Harm (NNH): How many treated patients for one additional harmful outcome to occur. Higher NNH = safer treatment.

Statistics for Healthcare Quiz

1/4

A dataset of hospital stays (in days): 2, 3, 3, 4, 5, 30. Which measure best represents the typical stay?

Scientific Notation and Orders of Magnitude

Working with very large and very small numbers in science

Healthcare involves numbers that span many orders of magnitude — from nanograms of hormones to billions of bacteria to the logarithmic pH scale. Scientific notation is the compact way to express these extremes and reason about their relative sizes.

Scientific Notation Format

a × 10ⁿ where 1 ≤ a < 10 and n is any integer.
Large numbers: n is positive. 6,000,000 = 6 × 10⁶
Small numbers: n is negative. 0.000003 = 3 × 10⁻⁶
To convert: count decimal places moved. Moving left → positive exponent. Moving right → negative exponent.

Clinical Examples of Scale

Bacteria (E. coli): ~2 × 10⁻⁶ m in length
Normal WBC count: ~7 × 10³ cells/μL
Serum digoxin level: ~1–2 × 10⁻⁹ g/mL (nanograms!)
Testosterone (male): ~3–10 × 10⁻⁹ g/mL
E. coli doubling time ~20 min; after 7 hours (21 doublings): 2²¹ = ~2 million bacteria from one cell.

Arithmetic in Scientific Notation

Multiply: Multiply the coefficients, add the exponents. (3 × 10⁴) × (2 × 10³) = 6 × 10⁷.

Divide: Divide the coefficients, subtract the exponents. (8 × 10⁶) ÷ (4 × 10²) = 2 × 10⁴.

Add/Subtract: First convert to the same exponent, then add/subtract coefficients. (3 × 10⁴) + (5 × 10³) = (3 × 10⁴) + (0.5 × 10⁴) = 3.5 × 10⁴.

pH is Logarithmic — Small Changes Mean Big Differences

pH is defined as −log[H⁺]. At pH 7, [H⁺] = 10⁻⁷ mol/L. At pH 6, [H⁺] = 10⁻⁶ mol/L — 10 times more acidic. Moving from pH 7.4 (normal blood) to pH 7.1 (severe acidosis) triples the H⁺ concentration. This is why seemingly small pH changes on the logarithmic scale represent massive changes in actual acid load and enzyme function.

pH and Hydrogen Ion Concentration Table

pH	[H⁺] mol/L	Clinical Context
6.8	10⁻⁶·⁸ (≈1.6 × 10⁻⁷)	Severe acidosis — life-threatening
7.1	7.9 × 10⁻⁸	Severe acidosis — urgent treatment
7.35	4.5 × 10⁻⁸	Lower limit of normal blood pH
7.4	4.0 × 10⁻⁸	Normal arterial blood pH
7.45	3.5 × 10⁻⁸	Upper limit of normal blood pH
7.6	2.5 × 10⁻⁸	Severe alkalosis

Calculation Tip: Bacterial Exponential Growth

If E. coli doubles every 20 minutes, after 1 hour (3 doublings) you have 2³ = 8 bacteria per starting cell. After 4 hours (12 doublings): 2¹² = 4,096 per starting cell. This exponential growth pattern explains why early antibiotic treatment matters — the bacterial load doubles relentlessly.

Scientific Notation and Orders of Magnitude Quiz

1/4

Express 0.00045 in scientific notation.

Match the Mathematical Concept to Its Clinical Application

0/8 matched

Terms

Definitions

Comprehensive Mathematics Foundations Quiz

1/10

A patient is ordered acetaminophen 1,000 mg. Available: 500 mg/tablet. How many tablets?

Pre-nursing comprehensive review

1/20

Which organelle contains its own DNA and is inherited exclusively from the mother?

Prefix

Symbol

Power of 10

Clinical Use

kilo

10³ (×1,000)

kilogram (kg)

— base —

—

10⁰ (×1)

gram (g), litre (L), metre (m)

milli

10⁻³ (÷1,000)

milligram (mg), millilitre (mL)

micro

mc / μ

10⁻⁶ (÷1,000,000)

microgram (mcg)

nano

10⁻⁹ (÷1,000,000,000)

nanogram (ng) — lab assays

[H⁺] mol/L

Clinical Context

6.8

10⁻⁶·⁸ (≈1.6 × 10⁻⁷)

Severe acidosis — life-threatening

7.1

7.9 × 10⁻⁸

Severe acidosis — urgent treatment

7.35

4.5 × 10⁻⁸

Lower limit of normal blood pH

7.4

4.0 × 10⁻⁸

Normal arterial blood pH

7.45

3.5 × 10⁻⁸

Upper limit of normal blood pH

7.6

2.5 × 10⁻⁸

Severe alkalosis

Mathematics Foundations for Healthcare

Fractions, Decimals, and Percentages

Clinical Worked Examples — Fractions & Percentages

Fractions, Decimals & Percentages Quiz

Ratios and Proportions

Ratios and Proportions Quiz

Metric System and Unit Conversions

Step-by-Step Unit Conversion Examples

Metric System and Conversions Quiz

Dimensional Analysis (Factor-Label Method)

Step-by-Step DA Healthcare Examples

Dimensional Analysis Quiz

Percentages in Healthcare

Percentages in Healthcare Quiz

Basic Algebra for Healthcare

Step-by-Step Formula Applications

Basic Algebra for Healthcare Quiz

Statistics Basics for Healthcare

Statistics for Healthcare Quiz

Scientific Notation and Orders of Magnitude

Scientific Notation and Orders of Magnitude Quiz

Match the Mathematical Concept to Its Clinical Application

Comprehensive Mathematics Foundations Quiz

Pre-nursing comprehensive review

Save your progress across devices

Your progress · Mathematics Foundations

Stay in Pre-Nursing

Ready for exam-style prep

Mathematics Foundations for Healthcare

Fractions, Decimals, and Percentages

Clinical Worked Examples — Fractions & Percentages

Fractions, Decimals & Percentages Quiz

Ratios and Proportions

Ratios and Proportions Quiz

Metric System and Unit Conversions

Step-by-Step Unit Conversion Examples

Metric System and Conversions Quiz

Dimensional Analysis (Factor-Label Method)

Step-by-Step DA Healthcare Examples

Dimensional Analysis Quiz

Percentages in Healthcare

Percentages in Healthcare Quiz

Basic Algebra for Healthcare

Step-by-Step Formula Applications

Basic Algebra for Healthcare Quiz

Statistics Basics for Healthcare

Statistics for Healthcare Quiz

Scientific Notation and Orders of Magnitude

Scientific Notation and Orders of Magnitude Quiz

Match the Mathematical Concept to Its Clinical Application

Comprehensive Mathematics Foundations Quiz

Pre-nursing comprehensive review

Save your progress across devices

Your progress · Mathematics Foundations

Stay in Pre-Nursing

Ready for exam-style prep