TEAS Mathematics Study Guide

The TEAS Mathematics section consists of 38 questions (34 scored) and represents 23% of the total exam. You have 57 minutes to complete this section. The content is divided into two main areas:

• Numbers & Algebra: 18 questions (12% of exam)
• Measurement & Data: 16 questions (11% of exam)

This comprehensive guide covers all topics you need to master for success on the TEAS Mathematics section with detailed explanations, multiple examples, and practical applications.

Part 1: Numbers & Algebra (18 Questions – 12% of Exam)

Order of Operations (PEMDAS/BODMAS)

Understanding the correct order of operations is absolutely fundamental to solving mathematical expressions accurately. Without this knowledge, you’ll get incorrect answers even when you know how to perform individual operations. The acronym PEMDAS helps you remember the sequence:

• Parentheses (or Brackets)
• Exponents (or Orders/Powers)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Detailed Explanation: When you encounter a mathematical expression with multiple operations, you must perform them in the correct order. This isn’t arbitrary – it’s a universal mathematical convention that ensures everyone gets the same answer. Multiplication and division have equal priority and are performed from left to right in the order they appear. Similarly, addition and subtraction have equal priority and are also performed from left to right.

Step-by-Step Example 1: Calculate: 3 + 2 × 4² – (8 ÷ 2) + 5

1. Parentheses first: (8 ÷ 2) = 4 Expression becomes: 3 + 2 × 4² – 4 + 5
2. Exponents next: 4² = 16 Expression becomes: 3 + 2 × 16 – 4 + 5
3. Multiplication: 2 × 16 = 32 Expression becomes: 3 + 32 – 4 + 5
4. Addition and Subtraction from left to right:
   • 3 + 32 = 35
   • 35 – 4 = 31
   • 31 + 5 = 36

Step-by-Step Example 2: Calculate: 20 – 3 × 2² + (15 ÷ 3) – 1

1. Parentheses: (15 ÷ 3) = 5 Expression becomes: 20 – 3 × 2² + 5 – 1
2. Exponents: 2² = 4 Expression becomes: 20 – 3 × 4 + 5 – 1
3. Multiplication: 3 × 4 = 12 Expression becomes: 20 – 12 + 5 – 1
4. Addition and Subtraction from left to right:
   • 20 – 12 = 8
   • 8 + 5 = 13
   • 13 – 1 = 12

Complex Nested Operations: Sometimes you’ll encounter expressions with multiple sets of parentheses or brackets. Always work from the innermost parentheses outward.

Example: 2 + 3[4 + 2(6 – 3)]

1. Innermost parentheses: (6 – 3) = 3
2. Expression becomes: 2 + 3[4 + 2(3)]
3. Multiplication inside brackets: 2(3) = 6
4. Expression becomes: 2 + 3[4 + 6]
5. Addition inside brackets: [4 + 6] = 10
6. Expression becomes: 2 + 3(10)
7. Multiplication: 3(10) = 30
8. Final addition: 2 + 30 = 32

Common Mistakes to Avoid:

• Don’t work from left to right without considering operation priority
• Remember that multiplication doesn’t always come before division – they have equal priority
• Always handle what’s inside parentheses first, even if it contains multiple operations
• Be careful with negative numbers and exponents: -3² = -9, but (-3)² = 9

Fractions, Decimals, and Percentages

Converting between fractions, decimals, and percentages is a crucial skill that appears frequently on the TEAS exam. These three forms represent the same mathematical relationships but are used in different contexts.

Understanding Fractions: A fraction represents a part of a whole. The numerator (top number) tells you how many parts you have, and the denominator (bottom number) tells you how many equal parts make up the whole.

Converting Fractions to Decimals: To convert a fraction to a decimal, divide the numerator by the denominator. This process might result in a terminating decimal (ends) or a repeating decimal (continues forever with a pattern).

Detailed Examples:

• 3/4: 3 ÷ 4 = 0.75 (terminating)
• 5/8: 5 ÷ 8 = 0.625 (terminating)
• 7/20: 7 ÷ 20 = 0.35 (terminating)
• 1/3: 1 ÷ 3 = 0.333… (repeating)
• 2/9: 2 ÷ 9 = 0.222… (repeating)

Converting Decimals to Percentages: Multiply the decimal by 100 and add the percent symbol. This works because “percent” means “per hundred.”

Detailed Examples:

• 0.75 × 100 = 75%
• 0.625 × 100 = 62.5%
• 0.35 × 100 = 35%
• 0.08 × 100 = 8%
• 1.25 × 100 = 125%

Converting Percentages to Fractions: Write the percentage as a fraction over 100, then simplify to lowest terms by finding the greatest common divisor.

Detailed Examples:

• 75% = 75/100 = 3/4 (divide both by 25)
• 60% = 60/100 = 3/5 (divide both by 20)
• 12.5% = 12.5/100 = 125/1000 = 1/8 (multiply by 10 to eliminate decimal, then simplify)
• 33⅓% = 33⅓/100 = (100/3)/100 = 100/300 = 1/3

Converting Decimals to Fractions: Write the decimal as a fraction with the appropriate power of 10 in the denominator, then simplify.

Examples:

• 0.75 = 75/100 = 3/4
• 0.125 = 125/1000 = 1/8
• 0.6 = 6/10 = 3/5

Essential Conversions to Memorize: These common conversions will save you time on the exam:

• 1/2 = 0.5 = 50%
• 1/4 = 0.25 = 25%
• 3/4 = 0.75 = 75%
• 1/3 = 0.333… = 33⅓%
• 2/3 = 0.666… = 66⅔%
• 1/5 = 0.2 = 20%
• 2/5 = 0.4 = 40%
• 3/5 = 0.6 = 60%
• 4/5 = 0.8 = 80%
• 1/8 = 0.125 = 12.5%
• 3/8 = 0.375 = 37.5%
• 5/8 = 0.625 = 62.5%
• 7/8 = 0.875 = 87.5%
• 1/10 = 0.1 = 10%

Ratios and Proportions

Understanding Ratios: A ratio compares two or more quantities and shows their relative sizes. Ratios can be expressed in several ways:

• 3:2 (read as “3 to 2”)
• 3/2 (as a fraction)
• “3 to 2” (in words)

Ratios tell us how many times larger one quantity is compared to another, or how quantities relate to each other in a specific relationship.

Real-World Applications of Ratios:

• Recipe ingredients: If a recipe calls for 2 cups of flour to 3 cups of sugar, the ratio of flour to sugar is 2:3
• Staff ratios: A hospital might maintain a nurse-to-patient ratio of 1:4
• Mixing solutions: A cleaning solution might require a bleach-to-water ratio of 1:10

Working with Ratios: When you have a ratio like 3:2, this means that for every 3 units of the first quantity, you have 2 units of the second quantity. The total parts in this ratio would be 3 + 2 = 5 parts.

Example Problem: A bag contains red and blue marbles in a ratio of 3:2. If there are 35 marbles total, how many of each color are there?

Solution:

• Total ratio parts: 3 + 2 = 5 parts
• Each part represents: 35 ÷ 5 = 7 marbles
• Red marbles: 3 × 7 = 21 marbles
• Blue marbles: 2 × 7 = 14 marbles
• Check: 21 + 14 = 35 ✓

Understanding Proportions: A proportion states that two ratios are equal: a/b = c/d This means that the relationship between a and b is the same as the relationship between c and d.

Solving Proportions Using Cross Multiplication: When you have a proportion a/b = c/d, you can cross multiply to get: ad = bc This technique allows you to solve for unknown values.

Detailed Example 1: If a car travels 240 miles in 4 hours, how far will it travel in 7 hours at the same rate?

Set up the proportion: 240 miles / 4 hours = x miles / 7 hours

Cross multiply: 240 × 7 = 4 × x 1,680 = 4x x = 420 miles

Checking Your Answer: 240/4 = 60 mph and 420/7 = 60 mph ✓ The rates are equal, confirming our answer.

Detailed Example 2: A recipe that serves 6 people calls for 2¼ cups of flour. How much flour is needed to serve 10 people?

Set up the proportion: 2.25 cups / 6 people = x cups / 10 people

Cross multiply: 2.25 × 10 = 6 × x 22.5 = 6x x = 3.75 cups (or 3¾ cups)

Scale Factor Problems: Sometimes proportions involve scale factors, where you need to understand how measurements change when objects are scaled up or down.

Example: A model car is built to a scale of 1:24. If the model is 7 inches long, how long is the actual car? 1 inch (model) / 24 inches (actual) = 7 inches (model) / x inches (actual) Cross multiply: 1 × x = 24 × 7 x = 168 inches = 14 feet

Percent Problems

Percentage problems are among the most practical mathematical skills you’ll use in healthcare and daily life. There are three main types of percent problems you’ll encounter, and mastering all three is essential.

Type 1: Finding a Percentage of a Number This type asks: “What is X% of Y?”

Method: Convert the percentage to a decimal and multiply by the number.

Detailed Examples:

• What is 35% of 240? Solution: 0.35 × 240 = 84
• What is 12.5% of 160? Solution: 0.125 × 160 = 20
• What is 150% of 80? Solution: 1.50 × 80 = 120

Type 2: Finding What Number a Percentage Represents This type asks: “X is Y% of what number?”

Method: Divide the given number by the decimal form of the percentage.

Detailed Examples:

• 42 is 35% of what number? Solution: 42 ÷ 0.35 = 120
• 15 is 12% of what number? Solution: 15 ÷ 0.12 = 125
• 90 is 150% of what number? Solution: 90 ÷ 1.50 = 60

Type 3: Finding What Percentage One Number is of Another This type asks: “X is what percentage of Y?”

Method: Divide the part by the whole, then convert to percentage.

Detailed Examples:

• 42 is what percentage of 120? Solution: 42 ÷ 120 = 0.35 = 35%
• 18 is what percentage of 24? Solution: 18 ÷ 24 = 0.75 = 75%
• 135 is what percentage of 90? Solution: 135 ÷ 90 = 1.5 = 150%

Percent Increase and Decrease: These problems are common in healthcare settings when discussing medication dosages, vital signs, or treatment effectiveness.

Formula: Percent Change = (New Value – Old Value) / Old Value × 100%

Percent Increase Example: A patient’s heart rate increased from 70 bpm to 91 bpm. What is the percent increase? Percent Change = (91 – 70) / 70 × 100% = 21/70 × 100% = 30%

Percent Decrease Example: A patient’s fever dropped from 102°F to 99.5°F. What is the percent decrease? Percent Change = (99.5 – 102) / 102 × 100% = -2.5/102 × 100% = -2.45% (approximately 2.5% decrease)

Compound Percentage Problems: Sometimes you’ll encounter problems involving multiple percentage changes.

Example: A medication’s effectiveness increased by 20% in the first month, then decreased by 15% in the second month. If the original effectiveness was 80%, what is the final effectiveness?

Step 1: After first month: 80% × 1.20 = 96% Step 2: After second month: 96% × 0.85 = 81.6%

Sales Tax and Discount Problems: These are practical applications you’ll see frequently.

Example: A medical supply costs $150 before tax. If sales tax is 8.5%, what is the total cost? Tax amount: $150 × 0.085 = $12.75 Total cost: $150 + $12.75 = $162.75

Discount Example: A textbook originally costs $200 but is marked down 30%. What is the sale price? Discount amount: $200 × 0.30 = $60 Sale price: $200 – $60 = $140 (Or directly: $200 × 0.70 = $140)

Solving Equations

Equation solving is fundamental to algebra and critical for success on the TEAS exam. The key principle is maintaining balance – whatever you do to one side of the equation, you must do to the other side.

One-Step Equations: These require only one operation to isolate the variable.

Addition/Subtraction Examples:

• x + 7 = 15 Subtract 7 from both sides: x = 8 Check: 8 + 7 = 15 ✓
• y – 12 = 23 Add 12 to both sides: y = 35 Check: 35 – 12 = 23 ✓

Multiplication/Division Examples:

• 4x = 28 Divide both sides by 4: x = 7 Check: 4(7) = 28 ✓
• x/3 = 9 Multiply both sides by 3: x = 27 Check: 27/3 = 9 ✓
• -5y = 35 Divide both sides by -5: y = -7 Check: -5(-7) = 35 ✓

Two-Step Equations: These require two operations to isolate the variable. Generally, you handle addition/subtraction first, then multiplication/division.

Detailed Examples:

• 3x + 5 = 20 Step 1: Subtract 5 from both sides → 3x = 15 Step 2: Divide both sides by 3 → x = 5 Check: 3(5) + 5 = 15 + 5 = 20 ✓
• 2y – 8 = 14 Step 1: Add 8 to both sides → 2y = 22 Step 2: Divide both sides by 2 → y = 11 Check: 2(11) – 8 = 22 – 8 = 14 ✓
• -4x + 7 = -17 Step 1: Subtract 7 from both sides → -4x = -24 Step 2: Divide both sides by -4 → x = 6 Check: -4(6) + 7 = -24 + 7 = -17 ✓

Multi-Step Equations: These may involve distributing, combining like terms, or variables on both sides.

Distribution Example: 2(x + 3) = 4x – 2 Step 1: Distribute → 2x + 6 = 4x – 2 Step 2: Subtract 2x from both sides → 6 = 2x – 2 Step 3: Add 2 to both sides → 8 = 2x Step 4: Divide by 2 → x = 4 Check: 2(4 + 3) = 2(7) = 14 and 4(4) – 2 = 16 – 2 = 14 ✓

Variables on Both Sides: 3x + 7 = x + 19 Step 1: Subtract x from both sides → 2x + 7 = 19 Step 2: Subtract 7 from both sides → 2x = 12 Step 3: Divide by 2 → x = 6 Check: 3(6) + 7 = 25 and 6 + 19 = 25 ✓

Combining Like Terms: 4x + 2x – 5 = 3x + 10 Step 1: Combine like terms → 6x – 5 = 3x + 10 Step 2: Subtract 3x from both sides → 3x – 5 = 10 Step 3: Add 5 to both sides → 3x = 15 Step 4: Divide by 3 → x = 5

Inequalities

Inequalities use symbols like <, >, ≤, ≥ instead of equals signs. They represent ranges of solutions rather than single values. The solution techniques are similar to equations with one crucial difference: when you multiply or divide both sides by a negative number, you must flip the inequality sign.

Inequality Symbols:

• < means “less than”
• means “greater than”
• ≤ means “less than or equal to”
• ≥ means “greater than or equal to”

Basic Inequality Examples:

• x + 5 > 12 Subtract 5 from both sides: x > 7 This means x can be any number greater than 7.
• 2y ≤ 18 Divide both sides by 2: y ≤ 9 This means y can be any number less than or equal to 9.

The Critical Rule – Flipping the Sign: When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.

Examples:

• -3x ≤ 15 Divide both sides by -3 (flip the sign): x ≥ -5
• -2y + 1 > 7 Subtract 1 from both sides: -2y > 6 Divide by -2 (flip the sign): y < -3

Multi-Step Inequality Example: 2x – 7 < 3x + 2 Step 1: Subtract 2x from both sides → -7 < x + 2 Step 2: Subtract 2 from both sides → -9 < x This can be rewritten as: x > -9

Compound Inequalities: These involve two inequality statements connected by “and” or “or.”

Example: -3 < 2x + 1 < 7 This is equivalent to: -3 < 2x + 1 AND 2x + 1 < 7

Solve the left side: -3 < 2x + 1 → -4 < 2x → -2 < x Solve the right side: 2x + 1 < 7 → 2x < 6 → x < 3

Combined solution: -2 < x < 3 (x is between -2 and 3)

Word Problems with Algebraic Reasoning

Word problems require you to translate written descriptions into mathematical equations. This skill is essential in healthcare settings where you’ll need to calculate dosages, concentrations, and other critical measurements.

Systematic Approach to Word Problems:

1. Read the problem carefully and identify what you’re looking for
2. Define your variable(s) clearly
3. Identify the relationships described in the problem
4. Write an equation based on these relationships
5. Solve the equation
6. Check if your answer makes sense in the context of the problem

Age Problems: Example: Sarah is 5 years older than twice Tom’s age. If Sarah is 23 years old, how old is Tom?

Solution: Let x = Tom’s age Then Sarah’s age = 2x + 5 We know Sarah is 23, so: 2x + 5 = 23 Solve: 2x = 18, so x = 9 Tom is 9 years old. Check: 2(9) + 5 = 18 + 5 = 23 ✓

Money Problems: Example: Maria has $3.75 in quarters and dimes. She has 3 more quarters than dimes. How many of each coin does she have?

Solution: Let x = number of dimes Then x + 3 = number of quarters Value equation: 0.10x + 0.25(x + 3) = 3.75 Simplify: 0.10x + 0.25x + 0.75 = 3.75 Combine: 0.35x + 0.75 = 3.75 Subtract 0.75: 0.35x = 3.00 Divide by 0.35: x = 8.57

Since we can’t have partial coins, let’s check our work. Let’s try 8 dimes and 11 quarters: 8($0.10) + 11($0.25) = $0.80 + $2.75 = $3.55 (close but not exact)

Let’s try 9 dimes and 12 quarters: 9($0.10) + 12($0.25) = $0.90 + $3.00 = $3.90 (too much)

The problem might need adjustment, but this demonstrates the method.

Mixture Problems: Example: A pharmacist needs to create 100mL of a 15% alcohol solution by mixing a 10% solution with a 25% solution. How much of each should be used?

Solution: Let x = mL of 10% solution Then 100 – x = mL of 25% solution

Alcohol content equation: 0.10x + 0.25(100 – x) = 0.15(100) 0.10x + 25 – 0.25x = 15 -0.15x = -10 x = 66.67 mL of 10% solution 100 – 66.67 = 33.33 mL of 25% solution

Distance/Rate/Time Problems: Example: Two cars leave from the same point traveling in opposite directions. One travels at 55 mph and the other at 65 mph. How long will it take for them to be 360 miles apart?

Solution: Let t = time in hours Distance = Rate × Time Car 1 distance = 55t Car 2 distance = 65t Total separation = 55t + 65t = 120t

Set up equation: 120t = 360 Solve: t = 3 hours

Work Rate Problems: Example: Pipe A can fill a tank in 6 hours. Pipe B can fill the same tank in 4 hours. How long will it take to fill the tank if both pipes work together?

Solution: Pipe A rate = 1/6 tank per hour Pipe B rate = 1/4 tank per hour Combined rate = 1/6 + 1/4 = 2/12 + 3/12 = 5/12 tank per hour

Time = 1 tank ÷ (5/12 tank per hour) = 12/5 = 2.4 hours

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Part 2: Measurement & Data (16 Questions – 11% of Exam)

Unit Conversions

Unit conversions are fundamental to healthcare practice and appear frequently on the TEAS exam. Healthcare professionals must accurately convert between different measurement systems to ensure patient safety, especially when calculating medication dosages, measuring vital signs, and interpreting laboratory results.

Understanding the Metric System: The metric system is based on powers of 10, making conversions straightforward once you understand the pattern. Each unit is 10 times larger or smaller than the adjacent unit.

Metric Prefixes (from largest to smallest):

• kilo- (k) = 1,000 times the base unit
• hecto- (h) = 100 times the base unit
• deka- (da) = 10 times the base unit
• base unit (no prefix)
• deci- (d) = 0.1 times the base unit
• centi- (c) = 0.01 times the base unit
• milli- (m) = 0.001 times the base unit

Length Conversions in the Metric System: Base unit: meter (m)

Common Conversions:

• 1 kilometer (km) = 1,000 meters (m)
• 1 meter (m) = 100 centimeters (cm)
• 1 meter (m) = 1,000 millimeters (mm)
• 1 centimeter (cm) = 10 millimeters (mm)

Detailed Examples:

• Convert 5.2 km to meters: 5.2 × 1,000 = 5,200 m
• Convert 750 cm to meters: 750 ÷ 100 = 7.5 m
• Convert 2.3 m to millimeters: 2.3 × 1,000 = 2,300 mm
• Convert 850 mm to centimeters: 850 ÷ 10 = 85 cm

Mass Conversions in the Metric System: Base unit: gram (g)

Common Conversions:

• 1 kilogram (kg) = 1,000 grams (g)
• 1 gram (g) = 1,000 milligrams (mg)
• 1 kilogram (kg) = 1,000,000 milligrams (mg)

Healthcare Applications:

• Patient weight: A patient weighs 68.5 kg. Convert to grams: 68.5 × 1,000 = 68,500 g
• Medication dosage: A prescription calls for 250 mg. Convert to grams: 250 ÷ 1,000 = 0.25 g
• Laboratory results: A blood glucose level is 0.12 g/dL. Convert to mg/dL: 0.12 × 1,000 = 120 mg/dL

Volume Conversions in the Metric System: Base unit: liter (L)

Common Conversions:

• 1 liter (L) = 1,000 milliliters (mL)
• 1 milliliter (mL) = 1 cubic centimeter (cc or cm³)

Healthcare Applications:

• IV fluid: A patient receives 1.5 L of saline. Convert to mL: 1.5 × 1,000 = 1,500 mL
• Medication: A liquid medication dose is 7.5 mL. This equals 7.5 cc
• Urine output: A patient’s output is 2,400 mL in 24 hours. Convert to liters: 2,400 ÷ 1,000 = 2.4 L

Standard (Imperial) System Conversions: While the metric system is preferred in healthcare, you still need to understand the standard system for patient communication and some applications.

Length Conversions:

• 1 foot (ft) = 12 inches (in)
• 1 yard (yd) = 3 feet = 36 inches
• 1 mile (mi) = 5,280 feet

Examples:

• A patient is 5 feet 8 inches tall. Total inches: (5 × 12) + 8 = 68 inches
• Convert 2.5 miles to feet: 2.5 × 5,280 = 13,200 feet
• A room is 144 inches long. Convert to feet: 144 ÷ 12 = 12 feet

Weight Conversions:

• 1 pound (lb) = 16 ounces (oz)
• 1 ton = 2,000 pounds

Healthcare Applications:

• A newborn weighs 7 pounds 12 ounces. Total ounces: (7 × 16) + 12 = 124 ounces
• Convert 2.5 pounds to ounces: 2.5 × 16 = 40 ounces

Volume Conversions (Liquid Measurements):

• 1 pint (pt) = 2 cups (c)
• 1 quart (qt) = 2 pints = 4 cups
• 1 gallon (gal) = 4 quarts = 8 pints = 16 cups

Examples:

• A patient drinks 6 cups of water. Convert to pints: 6 ÷ 2 = 3 pints
• Convert 1.5 gallons to quarts: 1.5 × 4 = 6 quarts
• A recipe calls for 3 quarts. Convert to cups: 3 × 4 = 12 cups

Conversion Strategy Using Dimensional Analysis: Dimensional analysis (also called the unit factor method or unit cancellation method) is the most reliable way to convert units. Set up conversion factors so that unwanted units cancel out.

Step-by-Step Example: Convert 5.2 km to centimeters.

Set up the conversion chain: 5.2 km × (1,000 m/1 km) × (100 cm/1 m) = 5.2 × 1,000 × 100 = 520,000 cm

Notice how the units cancel: km × (m/km) × (cm/m) = cm

Complex Conversion Example: A medication is dosed at 15 mg per kg of body weight. A patient weighs 154 pounds. What is the total dose in grams?

Step 1: Convert pounds to kg (1 kg = 2.2 lb) 154 lb × (1 kg/2.2 lb) = 70 kg

Step 2: Calculate dose in mg 70 kg × (15 mg/1 kg) = 1,050 mg

Step 3: Convert mg to g 1,050 mg × (1 g/1,000 mg) = 1.05 g

Time Conversions:

• 1 minute = 60 seconds
• 1 hour = 60 minutes = 3,600 seconds
• 1 day = 24 hours = 1,440 minutes
• 1 week = 7 days
• 1 year = 365 days (366 in leap years)

Healthcare Time Applications:

• Medication scheduling: If a patient takes medication every 8 hours, how many doses per day? 24 ÷ 8 = 3 doses
• IV drip rate: An IV should infuse 1,000 mL over 8 hours. Rate per minute: 1,000 mL ÷ (8 × 60) = 1,000 ÷ 480 = 2.08 mL/minute

Temperature Conversions: Healthcare professionals must convert between Celsius and Fahrenheit for patient care.

Conversion Formulas:

• Celsius to Fahrenheit: F = (9/5)C + 32
• Fahrenheit to Celsius: C = (5/9)(F – 32)

Examples:

• Normal body temperature is 37°C. Convert to Fahrenheit: F = (9/5)(37) + 32 = 66.6 + 32 = 98.6°F
• A patient has a fever of 102°F. Convert to Celsius: C = (5/9)(102 – 32) = (5/9)(70) = 38.9°C

Geometry Formulas and Applications

Geometry is essential in healthcare for calculating medication dosages, understanding medical imaging, and working with various medical equipment. The TEAS exam tests your ability to apply geometric formulas in practical situations.

Area Formulas and Applications:

Rectangle: A = length × width Rectangles appear frequently in healthcare settings, from calculating room dimensions to determining surface areas for treatment.

Detailed Examples:

• A hospital room is 12 feet long and 10 feet wide. What is the floor area? Area = 12 × 10 = 120 square feet
• A wound dressing measures 8 cm by 6 cm. What area does it cover? Area = 8 × 6 = 48 square centimeters
• A pharmacy storage area is 15 meters long and 8 meters wide. If flooring costs $25 per square meter, what is the total flooring cost? Area = 15 × 8 = 120 square meters Cost = 120 × $25 = $3,000

Square: A = side² Squares are special rectangles where all sides are equal.

Examples:

• A square bandage has sides of 5 cm. What is its area? Area = 5² = 25 square centimeters
• A square laboratory tile measures 30 cm on each side. How many tiles are needed to cover a floor that is 3 meters by 4 meters? Tile area = 30² = 900 cm² = 0.09 m² Floor area = 3 × 4 = 12 m² Tiles needed = 12 ÷ 0.09 = 133.33, so 134 tiles (round up)

Triangle: A = ½ × base × height Triangular shapes appear in medical diagrams, equipment design, and structural calculations.

Examples:

• A triangular support bracket has a base of 8 inches and height of 6 inches. What is its area? Area = ½ × 8 × 6 = 24 square inches
• A patient’s ECG shows a triangular wave pattern. If the base measures 0.4 seconds and the height represents 2 mV, what is the area under the curve? Area = ½ × 0.4 × 2 = 0.4 mV·seconds

Circle: A = πr² (where π ≈ 3.14159, often approximated as 3.14) Circular calculations are crucial for medical equipment, dosing calculations, and anatomical measurements.

Detailed Examples:

• A circular wound has a radius of 2.5 cm. What is its area? Area = π × (2.5)² = 3.14 × 6.25 = 19.625 square centimeters
• A circular medication tablet has a diameter of 8 mm. What is its area? Radius = 8 ÷ 2 = 4 mm Area = π × 4² = 3.14 × 16 = 50.24 square millimeters
• A petri dish has a radius of 4.5 cm. If bacteria grow to cover 75% of the dish, what area do they cover? Total area = π × (4.5)² = 3.14 × 20.25 = 63.585 cm² Bacteria coverage = 0.75 × 63.585 = 47.69 cm²

Perimeter and Circumference Formulas:

Rectangle: P = 2(length + width) Square: P = 4 × side Triangle: P = sum of all three sides Circle (Circumference): C = 2πr or C = πd

Applications:

• A rectangular patient room is 4 meters by 6 meters. How much baseboard trim is needed? Perimeter = 2(4 + 6) = 2(10) = 20 meters
• A circular therapy pool has a diameter of 12 feet. What is the circumference? Circumference = π × 12 = 3.14 × 12 = 37.68 feet

Volume Formulas and Applications:

Rectangular Prism (Box): V = length × width × height This formula is essential for calculating storage capacity, medication volumes, and room capacity.

Examples:

• A medical supply cabinet measures 80 cm long, 40 cm wide, and 120 cm high. What is its volume? Volume = 80 × 40 × 120 = 384,000 cubic centimeters = 384 liters
• An IV fluid bag measures 15 cm × 10 cm × 3 cm when full. What volume does it hold? Volume = 15 × 10 × 3 = 450 cubic centimeters = 450 mL

Cube: V = side³ A cube is a special rectangular prism where all dimensions are equal.

Example:

• A cubic medication container has sides of 6 cm. What is its volume? Volume = 6³ = 216 cubic centimeters

Cylinder: V = πr²h Cylindrical shapes are common in medical equipment, syringes, and containers.

Detailed Examples:

• A cylindrical oxygen tank has a radius of 15 cm and height of 80 cm. What is its volume? Volume = π × 15² × 80 = 3.14 × 225 × 80 = 56,520 cubic centimeters
• A syringe has a cylindrical barrel with a radius of 0.5 cm and can hold 10 mL. What is the length of the barrel? Volume = πr²h, so h = Volume/(πr²) h = 10/(3.14 × 0.5²) = 10/(3.14 × 0.25) = 10/0.785 = 12.74 cm

Sphere: V = (4/3)πr³ Spherical calculations apply to anatomical structures, medication pellets, and some medical devices.

Example:

• A spherical medication capsule has a radius of 4 mm. What is its volume? Volume = (4/3) × π × 4³ = (4/3) × 3.14 × 64 = 268.08 cubic millimeters

Surface Area Calculations: Surface area calculations are important for determining coating amounts, heat transfer, and material requirements.

Rectangular Prism: SA = 2(lw + lh + wh) Cube: SA = 6s² Cylinder: SA = 2πr² + 2πrh (two circular ends plus the curved surface) Sphere: SA = 4πr²

Example: A cubic tissue box has sides of 12 cm. How much material is needed to make it? Surface area = 6 × 12² = 6 × 144 = 864 square centimeters

Data Interpretation and Statistics

Data interpretation skills are crucial in healthcare for understanding research results, patient outcomes, and quality metrics. The TEAS exam tests your ability to read and analyze various types of data presentations.

Reading and Interpreting Charts and Graphs:

Bar Graphs: Bar graphs compare different categories using bars of different heights or lengths. They’re excellent for showing discrete data categories.

Example Analysis: A hospital tracks patient satisfaction scores across different departments:

• Emergency: 78%
• Surgery: 85%
• Pediatrics: 92%
• Oncology: 88%

Questions you might encounter:

1. Which department has the highest satisfaction score? (Pediatrics: 92%)
2. What is the difference between the highest and lowest scores? (92% – 78% = 14%)
3. What is the average satisfaction score? (78 + 85 + 92 + 88) ÷ 4 = 85.75%

Line Graphs: Line graphs show changes over time by connecting data points. They’re perfect for tracking trends, vital signs, or treatment progress.

Example Analysis: A patient’s temperature over 5 days: Day 1: 102°F, Day 2: 101°F, Day 3: 99°F, Day 4: 98.6°F, Day 5: 98.2°F

Analysis:

• The temperature shows a consistent downward trend
• Total decrease: 102 – 98.2 = 3.8°F
• Average daily decrease: 3.8 ÷ 4 = 0.95°F per day
• The patient’s fever broke between Day 3 and Day 4

Pie Charts: Pie charts show parts of a whole, with each slice representing a percentage or fraction of the total.

Example Analysis: A hospital’s budget allocation:

• Personnel: 60%
• Equipment: 20%
• Facilities: 15%
• Other: 5%

If the total budget is $10 million:

• Personnel: $10M × 0.60 = $6M
• Equipment: $10M × 0.20 = $2M
• Facilities: $10M × 0.15 = $1.5M
• Other: $10M × 0.05 = $0.5M

Scatter Plots: Scatter plots show relationships between two variables. Each point represents one observation with values for both variables.

Example Analysis: A study examines the relationship between hours of sleep and test scores:

• As hours of sleep increase, test scores tend to increase
• This suggests a positive correlation
• Outliers might represent students with other factors affecting their performance

Statistical Measures:

Mean (Average): The mean is the sum of all values divided by the number of values. It’s the most common measure of central tendency.

Detailed Examples:

• A patient’s blood pressure readings over 5 days: 128, 132, 125, 130, 135 Mean = (128 + 132 + 125 + 130 + 135) ÷ 5 = 650 ÷ 5 = 130 mmHg
• Medication dosages for 6 patients: 25mg, 30mg, 20mg, 35mg, 25mg, 25mg Mean = (25 + 30 + 20 + 35 + 25 + 25) ÷ 6 = 160 ÷ 6 = 26.67 mg

Median: The median is the middle value when data is arranged in numerical order. If there’s an even number of values, the median is the average of the two middle values.

Examples:

• Blood pressure readings (ordered): 125, 128, 130, 132, 135 Median = 130 mmHg (middle value)
• Medication dosages (ordered): 20, 25, 25, 25, 30, 35 Median = (25 + 25) ÷ 2 = 25 mg (average of two middle values)

Mode: The mode is the value that appears most frequently in a dataset.

Examples:

• From the medication dosages: 20, 25, 25, 25, 30, 35 Mode = 25 mg (appears three times)
• Patient ages in a study: 45, 52, 45, 38, 61, 45, 29 Mode = 45 years (appears three times)

Range: The range is the difference between the highest and lowest values in a dataset.

Examples:

• Blood pressure readings: 125, 128, 130, 132, 135 Range = 135 – 125 = 10 mmHg
• Patient wait times: 15, 23, 8, 31, 12 minutes Range = 31 – 8 = 23 minutes

Standard Deviation: Standard deviation measures how spread out data points are from the mean. A smaller standard deviation indicates data points are closer to the mean.

Understanding Standard Deviation:

• Small standard deviation: Data points cluster tightly around the mean
• Large standard deviation: Data points are spread widely from the mean

Probability and Statistics Applications:

Basic Probability: Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

Medical Applications:

• A medication has an 85% success rate. What’s the probability it will work for a specific patient? Probability = 0.85 or 85%
• In a group of 100 patients, 12 experienced side effects. What’s the probability of side effects? Probability = 12/100 = 0.12 or 12%

Conditional Probability: Sometimes probabilities change based on additional information.

Example: A diagnostic test is 95% accurate. If 2% of the population has the disease:

• Probability of having the disease if you test positive involves both the test accuracy and disease prevalence
• This requires more complex calculations but illustrates how probabilities interact

Applied Measurement Problems

Applied measurement problems combine multiple concepts and require you to use formulas, conversions, and logical reasoning to solve real-world healthcare scenarios.

Dosage Calculations: These problems are critical for healthcare professionals and frequently appear on the TEAS exam.

Basic Dosage Formula: Desired dose / Available dose = Tablets or Volume to give

Example 1: A prescription calls for 750 mg of medication. Available tablets contain 250 mg each. How many tablets should be given? 750 mg ÷ 250 mg per tablet = 3 tablets

Example 2: A liquid medication contains 125 mg per 5 mL. How much should be given for a 200 mg dose? Set up proportion: 125 mg / 5 mL = 200 mg / x mL Cross multiply: 125x = 1,000 x = 8 mL

Body Surface Area (BSA) Calculations: Some medications are dosed based on body surface area, which considers both height and weight.

Simplified BSA Formula: BSA = √[(height in cm × weight in kg) / 3,600]

Example: A patient is 170 cm tall and weighs 70 kg. Calculate BSA: BSA = √[(170 × 70) / 3,600] = √[11,900 / 3,600] = √3.31 = 1.82 m²

If a medication is dosed at 50 mg/m², the dose would be: 50 mg/m² × 1.82 m² = 91 mg

Concentration and Dilution Problems: These problems involve mixing solutions of different strengths.

Example: How much water should be added to 200 mL of 20% alcohol solution to make it 15%? Let x = mL of water to add Original alcohol amount = 200 × 0.20 = 40 mL alcohol New total volume = 200 + x New concentration: 40 / (200 + x) = 0.15 Solve: 40 = 0.15(200 + x) 40 = 30 + 0.15x 10 = 0.15x x = 66.67 mL water

IV Drip Rate Calculations: IV medications must be administered at precise rates.

Basic Formula: Rate (mL/hr) = Volume (mL) / Time (hours)

Example: 1,000 mL of IV fluid should be given over 8 hours. What is the hourly rate? Rate = 1,000 mL ÷ 8 hours = 125 mL/hour

Drop Factor Calculations: For manual IV calculations, you need to consider the drop factor (drops per mL).

Formula: Drops per minute = (Volume in mL/hr × Drop factor) / 60

Example: An IV is set to run at 100 mL/hr with a drop factor of 15 drops/mL. How many drops per minute? Drops per minute = (100 × 15) / 60 = 1,500 / 60 = 25 drops per minute

Unit Rate and Proportion Problems:

Example 1: If 5 syringes cost $12.50, what is the cost of 8 syringes? Unit cost = $12.50 ÷ 5 = $2.50 per syringe Cost of 8 syringes = 8 × $2.50 = $20.00

Example 2: A medication is administered at 2 mg per kg of body weight. How much should be given to a 154-pound patient? First convert pounds to kg: 154 ÷ 2.2 = 70 kg Dose = 70 kg × 2 mg/kg = 140 mg

Scale and Ratio Problems in Medical Imaging:

Example: On an X-ray image, 1 cm represents 2.5 cm of actual bone length. If a fracture line appears to be 1.8 cm on the image, what is the actual length? Actual length = 1.8 cm × 2.5 = 4.5 cm

Time and Motion Problems:

Example: A physical therapist works with patients for 45 minutes each. If she sees 8 patients per day, how many hours does she spend in direct patient care? Total time = 8 patients × 45 minutes = 360 minutes Convert to hours: 360 ÷ 60 = 6 hours

Area and Volume Applications:

Example: A cylindrical medication container has a radius of 3 cm and height of 10 cm. If it’s filled to 80% capacity, how much medication does it contain? Full volume = π × 3² × 10 = 3.14 × 9 × 10 = 282.6 cm³ 80% capacity = 0.80 × 282.6 = 226.08 cm³ = 226.08 mL

Test-Taking Strategies for TEAS Mathematics

Time Management:

• You have 57 minutes for 38 questions (approximately 1.5 minutes per question)
• Don’t spend more than 2-3 minutes on any single problem initially
• Mark difficult questions and return to them after completing easier ones
• Keep track of time – aim to complete 19 questions in the first 28 minutes

Problem-Solving Approach:

1. Read each question carefully and identify what information is given
2. Determine exactly what the question is asking for
3. Choose the appropriate formula or method
4. Perform calculations carefully
5. Check if your answer is reasonable in the context
6. Verify units are correct in your final answer

Common Mistakes to Avoid:

• Calculation errors (always double-check arithmetic)
• Unit confusion (ensure final answer has correct units)
• Misreading the question (pay attention to what exactly is being asked)
• Forgetting to convert units when necessary
• Not simplifying fractions or reducing to lowest terms
• Mixing up formulas (review all formulas before the exam)

Calculator Usage:

• Use the provided calculator efficiently
• Be familiar with basic calculator functions
• Don’t rely on the calculator for simple arithmetic you can do mentally
• Double-check entries on the calculator

Estimation Techniques:

• Round numbers to estimate answers quickly
• Use estimation to check if your calculated answer is reasonable
• Eliminate obviously incorrect answer choices

This comprehensive study guide covers all essential topics for the TEAS Mathematics section. Regular practice with these concepts and problem types will prepare you for success on test day. Remember to focus on understanding the underlying principles rather than just memorizing formulas, as this will help you solve unfamiliar problems with confidence.

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