Overview
This study guide covers the core mathematical concepts tested on the TEAS exam, including arithmetic operations, fractions, algebra, and proportional reasoning. Mastery of these topics requires understanding both procedural rules (how to solve) and conceptual reasoning (why the method works). Use this guide alongside active practice problems to build speed and accuracy for test day.
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Order of Operations & Basic Arithmetic
Summary
Every multi-step math problem requires a strict sequence of operations to arrive at the correct answer. Without a consistent order, the same expression could yield multiple different results. Equally important are LCM and GCF, which appear in fraction simplification, scheduling problems, and ratio questions.
Key Concepts
• PEMDAS — the universal order of operations:
1. Parentheses (innermost first)
2. Exponents
3. Multiplication / Division (left to right, same priority)
4. Addition / Subtraction (left to right, same priority)
• Least Common Multiple (LCM): The smallest number that is divisible by two or more given numbers
- Example: LCM(4, 6) = 12
- Use for: adding/subtracting fractions, finding common denominators
• Greatest Common Factor (GCF): The largest number that divides evenly into two or more given numbers
- Example: GCF(36, 48) = 12
- Use for: simplifying fractions, reducing ratios
Worked Example
> Evaluate: 3 + 6 × (5 + 4) ÷ 3 − 7
> 1. Parentheses: (5 + 4) = 9
> 2. Multiply: 6 × 9 = 54
> 3. Divide: 54 ÷ 3 = 18
> 4. Add/Subtract left to right: 3 + 18 − 7 = 14
Key Terms
• PEMDAS — mnemonic for order of operations
• LCM — Least Common Multiple
• GCF — Greatest Common Factor
• Factor — a number that divides evenly into another
• Multiple — the result of multiplying a number by any integer
⚠️ Watch Out For
• Multiplication and Division have equal priority — always work left to right, do NOT always multiply before dividing
• Same rule applies to Addition and Subtraction — left to right, not always add first
• Forgetting to simplify inside parentheses before moving outward
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Fractions, Decimals & Percentages
Summary
Fractions, decimals, and percentages are three representations of the same value and must be converted fluidly on the TEAS. Percent problems are among the most frequently tested real-world applications, especially in healthcare contexts (dosage calculations, discounts, increases/decreases).
Key Concepts
Converting Between Forms:
| Fraction | Decimal | Percentage |
|----------|---------|------------|
| 3/8 | 0.375 | 37.5% |
| 1/4 | 0.25 | 25% |
| 1/2 | 0.5 | 50% |
• Fraction → Decimal: Divide numerator ÷ denominator
• Decimal → Percentage: Multiply by 100 (move decimal right 2 places)
• Percentage → Decimal: Divide by 100 (move decimal left 2 places)
Fraction Operations:
• Multiplying fractions: Multiply numerators × numerators, denominators × denominators, then simplify
- Example: 2/3 × 3/4 = 6/12 = 1/2
• Dividing fractions — "Keep, Change, Flip":
1. Keep the first fraction
2. Change ÷ to ×
3. Flip the second fraction (take reciprocal)
- Example: 3/4 ÷ 1/2 = 3/4 × 2/1 = 3/2
Percent Calculations:
• Finding a percent of a number: Convert percent to decimal, then multiply
- 15% of 200 = 200 × 0.15 = 30
• Finding a discount: Calculate the discount amount, then subtract from original
- 25% off $40 → $40 × 0.25 = $10 off → Sale price = $30
Key Terms
• Numerator — the top number of a fraction
• Denominator — the bottom number of a fraction
• Reciprocal — the flipped version of a fraction (e.g., reciprocal of 3/4 is 4/3)
• Percent — "per hundred"; a ratio out of 100
⚠️ Watch Out For
• When adding or subtracting fractions, you must have a common denominator first — you cannot simply add numerators and denominators separately
• Confusing percent off (subtraction) with percent of (multiplication only)
• Moving the decimal the wrong direction when converting between decimals and percentages
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Solving Equations & Inequalities
Summary
Algebra on the TEAS tests your ability to isolate variables using inverse operations and to understand how solution sets change under different conditions. Inequalities follow the same rules as equations with one critical exception involving negative numbers.
Key Concepts
Solving Linear Equations — Golden Rule:
> Whatever you do to one side of the equation, you must do to the other side.
Step-by-step approach:
1. Distribute if needed (remove parentheses)
2. Combine like terms on each side
3. Move variable terms to one side using inverse operations
4. Isolate the variable
5. Check your answer by substituting back in
Worked Examples:
> 3x + 7 = 22
> Subtract 7: 3x = 15 → Divide by 3: x = 5
> 2(x − 4) = 10
> Distribute: 2x − 8 = 10 → Add 8: 2x = 18 → Divide by 2: x = 9
Inequalities — The Critical Rule:
> Flip the inequality sign when multiplying or dividing by a negative number.
> −4x > 20
> Divide both sides by −4 → flip sign: x < −5
Systems of Equations:
• One solution: Lines intersect at exactly one point (different slopes)
• No solution: Lines are parallel (same slope, different y-intercepts) — never intersect
• Infinite solutions: Lines are identical (same slope, same y-intercept)
Solving a System by Addition/Elimination:
> x + y = 10 and x − y = 4
> Add equations: 2x = 14 → x = 7
> Substitute: 7 + y = 10 → y = 3
Key Terms
• Equation — a mathematical statement that two expressions are equal
• Inequality — a statement using <, >, ≤, or ≥
• Variable — a letter representing an unknown value
• System of equations — two or more equations with the same variables
• Parallel lines — lines with equal slopes that never intersect
⚠️ Watch Out For
• Forgetting to flip the inequality sign when dividing or multiplying by a negative number — this is the #1 inequality mistake
• Failing to distribute the negative sign when subtracting a parenthetical expression (e.g., −(x − 3) = −x + 3, not −x − 3)
• Not checking your solution by plugging it back into the original equation
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Ratios, Proportions & Rates
Summary
Proportional reasoning is essential in healthcare math and appears frequently on the TEAS in contexts like medication dosing, staffing ratios, recipe scaling, and rate problems. The key skill is correctly setting up equivalent ratios before solving.
Key Concepts
Proportion: A statement that two ratios are equal
> a/b = c/d → Cross-multiply: ad = bc
Setting Up Proportions — Always keep units consistent:
> 150 miles / 3 hours = x miles / 5 hours
> Cross-multiply: 3x = 750 → x = 250 miles
Unit Rate: The amount per single unit (denominator = 1)
> $180 ÷ 8 hours = $22.50 per hour
Ratio Applications:
• Nurse-to-patient ratio 1:4 with 28 patients → 28 ÷ 4 = 7 nurses
• Recipe scaling: 2 cups flour : 3 cups sugar → for 9 cups sugar: 2/3 = x/9 → x = 6 cups flour
Quick Method for Unit Rates:
> Divide the total by the number of units to find the per-unit value
Key Terms
• Ratio — a comparison of two quantities (written as a:b or a/b)
• Proportion — an equation stating two ratios are equal
• Unit rate — a ratio with a denominator of 1
• Cross-multiplication — multiplying diagonally across a proportion to solve for an unknown
• Scale factor — the multiplier used to scale a ratio up or down
⚠️ Watch Out For
• Setting up the proportion with inconsistent units (e.g., putting miles in one ratio and hours in the other numerator)
• Forgetting to simplify the ratio before scaling — work with the reduced form first
• Confusing ratio (comparison) with rate (comparison of two different units like miles per hour)
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Algebra & Functions
Summary
This section covers linear equations, graphing, function notation, and quadratic equations — all of which test your ability to connect algebraic expressions to real-world meaning and visual representations. Understanding slope deeply (not just as a formula) is especially critical.
Key Concepts
Slope-Intercept Form: y = mx + b
| Variable | Meaning | How to Find It |
|----------|---------|----------------|
| m | Slope (rate of change) | m = (y₂ − y₁)/(x₂ − x₁) |
| b | Y-intercept (starting value) | Where line crosses the y-axis |
> Line through (2, 5) and (6, 13): m = (13−5)/(6−2) = 8/4 = 2
Slope Relationships:
• Positive slope: Line rises left to right (x↑, y↑)
• Negative slope: Line falls left to right (x↑, y↓)
• Parallel lines: Same slope (m₁ = m₂), different y-intercepts
• Perpendicular lines: Negative reciprocal slopes (m₁ × m₂ = −1)
- Example: slopes of 2 and −1/2 are perpendicular
Absolute Value:
> |number| = distance from zero (always ≥ 0)
> |−7| = 7, |7| = 7
Combining Like Terms:
> 4x² + 3x − 2x² + 5x − 1
> = (4x² − 2x²) + (3x + 5x) − 1
> = 2x² + 8x − 1
> (Only combine terms with identical variables AND identical exponents)
Functions:
• Definition: Every input (x) has exactly one output (y)
• Vertical Line Test: If any vertical line crosses the graph more than once → NOT a function
• Function notation: f(x) means "the output when the input is x"
- f(x) = 2x² − 3x + 1, f(4) = 2(16) − 12 + 1 = 21
Quadratic Formula:
> For ax² + bx + c = 0:
> $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
> The ± means there are potentially two solutions (x-intercepts)
Key Terms
• Slope — steepness/rate of change of a line; rise over run
• Y-intercept — the point where a line crosses the y-axis (x = 0)
• Linear equation — an equation whose graph is a straight line
• Function — a relation with exactly one output per input
• Vertical Line Test — graphical method to check if a relation is a function
• Quadratic equation — an equation with a squared variable (ax² + bx + c = 0)
• Like terms — terms with the same variable(s) raised to the same power(s)
• Absolute value — a number's distance from zero; always non-negative
• Negative reciprocal — flip a fraction and change its sign (used for perpendicular slopes)
⚠️ Watch Out For
• When finding slope, always subtract coordinates in the same order: (y₂ − y₁)/(x₂ − x₁) — mixing up the order gives the wrong sign
• Like terms only: You can combine 3x and 5x, but NOT 3x and 5x²
• Confusing parallel (same slope) with perpendicular (negative reciprocal slopes)
• In the quadratic formula, the entire expression −b ± √(b² − 4ac) is divided by 2a, not just part of it
• A circle fails the vertical line test — it is not a function
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Quick Review Checklist
Use this checklist before your exam. Check off each item when you feel confident:
Order of Operations & Arithmetic
• [ ] I can apply PEMDAS correctly, especially handling multiplication/division left to right
• [ ] I can find the LCM and GCF of two numbers
• [ ] I can evaluate multi-step expressions without a calculator
Fractions, Decimals & Percentages
• [ ] I can convert fluently between fractions, decimals, and percentages
• [ ] I know "Keep, Change, Flip" for dividing fractions
• [ ] I can calculate percent discounts and percent of a number
• [ ] I remember to find a common denominator before adding/subtracting fractions
Equations & Inequalities
• [ ] I can solve multi-step linear equations by isolating the variable
• [ ] I remember to flip the inequality sign when multiplying/dividing by a negative
• [ ] I can solve a system of equations using addition/elimination
• [ ] I can identify when a system has no solution (parallel lines)
Ratios, Proportions & Rates
• [ ] I can set up and solve proportions using cross-multiplication
• [ ] I keep units consistent when setting up proportions
• [ ] I can calculate unit rates by dividing
• [ ] I can apply ratios to real-world healthcare staffing and recipe problems
Algebra & Functions
• [ ] I know the slope formula: m = (y₂ − y₁)/(x₂ − x₁)
• [ ] I understand slope-intercept form y = mx + b and what m and b represent
• [ ] I know that perpendicular lines have negative reciprocal slopes
• [ ] I can apply the vertical line test to determine if a graph is a function
• [ ] I can evaluate a function like f(4) by substituting x = 4
• [ ] I can simplify expressions by combining like terms only
• [ ] I know the quadratic formula and when to apply it
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> 💡 Final Exam Tip: On the TEAS, read each word problem carefully to identify what is being asked before setting up your equation or proportion. Label your variables and units, and always check that your answer makes logical sense in context.