Overview
This study guide covers the core mathematical concepts tested on the GED Mathematical Reasoning exam, including number operations, algebra, geometry, data analysis, and measurement. The exam tests both procedural skills (knowing formulas and methods) and conceptual understanding (knowing why and when to apply them). Mastering these topics requires consistent practice with both computation and word problems.
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Number Operations & Number Sense
Key Concepts
Number sense forms the foundation of all GED math. You must be comfortable working with integers, fractions, decimals, percents, and understanding how numbers relate to one another.
Order of Operations (PEMDAS)
When multiple operations appear in one expression, always follow this sequence:
1. Parentheses — solve groupings first
2. Exponents — evaluate powers
3. Multiplication / Division — left to right
4. Addition / Subtraction — left to right
> Example: 3 + 2 × (4²) → 3 + 2 × 16 → 3 + 32 = 35
Working with Signed Numbers
• Negative × Negative = Positive
• Negative × Positive = Negative
• Adding a negative = subtracting that value
Fractions
• To add/subtract fractions: denominators must match (find a common denominator first)
• To convert a fraction to a percent: divide numerator ÷ denominator × 100
• Example: 3/4 → 0.75 × 100 = 75%
Number Classifications
| Type | Definition | Example |
|------|-----------|---------|
| Prime | Exactly 2 factors: 1 and itself | 2, 3, 5, 7, 11 |
| Composite | More than 2 factors | 4, 6, 8, 9 |
| Rational | Expressible as p/q (q ≠ 0) | ½, 0.75, -3 |
| Absolute Value | Distance from zero; always ≥ 0 | \|−7\| = 7 |
Key Terms
• PEMDAS — acronym for order of operations
• Absolute value — distance from zero on the number line
• Rational number — any number expressible as a fraction of two integers
• Common denominator — a shared multiple of two denominators
• Prime number — a number with exactly two distinct factors
Watch Out For ⚠️
• Do NOT simply go left to right — always follow PEMDAS order
• Multiplication and division are equal priority — work left to right between them; same rule applies to addition and subtraction
• Negative signs before parentheses: −(3 + 4) = −7, not −3 + 4
• When adding fractions, never add the denominators — only the numerators (after finding a common denominator)
• Remember: 0 is neither prime nor composite
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Algebra & Functions
Key Concepts
Solving for x
"Solve for x" means performing inverse operations to isolate x on one side. Whatever you do to one side, you must do to the other.
> Example: 2x + 5 = 13 → 2x = 8 → x = 4
Linear Equations & Slope
A linear equation has variables raised only to the first power and graphs as a straight line.
Slope-Intercept Form: y = mx + b
• m = slope (rise over run) = (y₂ − y₁) / (x₂ − x₁)
• b = y-intercept (where the line crosses the y-axis)
| Slope Type | Meaning | Appearance |
|-----------|---------|-----------|
| Positive | Line rises left → right | ↗ |
| Negative | Line falls left → right | ↘ |
| Zero | Horizontal line | → |
| Undefined | Vertical line | ↑ |
Systems of Equations (Substitution Method)
1. Solve one equation for one variable
2. Substitute that expression into the second equation
3. Solve the resulting single-variable equation
4. Back-substitute to find the other variable
Quadratic Formula
Used to solve ax² + bx + c = 0:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Functions
A function maps every input (x) to exactly one output (y). Use the vertical line test on a graph — if any vertical line crosses the graph more than once, it is NOT a function.
Exponent Rules
| Rule | Formula | Example |
|------|---------|---------|
| Zero exponent | x⁰ = 1 | 7⁰ = 1 |
| Multiply same base | xᵃ × xᵇ = xᵃ⁺ᵇ | x³ × x⁴ = x⁷ |
| Divide same base | xᵃ ÷ xᵇ = xᵃ⁻ᵇ | x⁵ ÷ x² = x³ |
Key Terms
• Variable — a letter representing an unknown value
• Linear equation — equation whose graph is a straight line
• Slope — measure of a line's steepness and direction
• Y-intercept — the point (0, b) where a line crosses the y-axis
• Quadratic equation — an equation with a variable squared (ax² + bx + c = 0)
• Function — a relation where each input has exactly one output
• System of equations — two or more equations solved simultaneously
Watch Out For ⚠️
• The slope formula is (y₂ − y₁) / (x₂ − x₁), not x over y
• In y = mx + b, m is slope and b is the y-intercept — don't reverse them
• 0⁰ is undefined — the zero exponent rule only applies to nonzero bases
• When using the quadratic formula, the ± means there are two possible solutions
• A vertical line (x = 3) is not a function — each x must map to exactly one y
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Geometry
Key Concepts
Triangles
• Interior angles always sum to 180°
• Area: A = ½ × base × height (height must be perpendicular to base)
• Pythagorean Theorem (right triangles only): a² + b² = c², where c = hypotenuse (longest side)
Circles
| Measurement | Formula |
|------------|---------|
| Circumference | C = 2πr or C = πd |
| Area | A = πr² |
> Remember: radius (r) = half the diameter. If given a diameter, divide by 2 first.
Rectangles & Prisms
• Perimeter of rectangle: P = 2l + 2w
• Area of rectangle: A = l × w
• Volume of rectangular prism: V = l × w × h
Coordinate Geometry
• Distance Formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
- This is derived directly from the Pythagorean Theorem
Angle Relationships
| Angle Type | Property |
|-----------|---------|
| Complementary | Two angles that sum to 90° |
| Supplementary | Two angles that sum to 180° |
| Right angle | Exactly 90° |
| Straight angle | Exactly 180° |
Key Terms
• Hypotenuse — the longest side of a right triangle, opposite the right angle
• Radius — distance from center to edge of a circle
• Diameter — distance across a circle through its center (d = 2r)
• Perimeter — total distance around the outside of a shape
• Area — amount of space inside a 2D shape (measured in square units)
• Volume — amount of space inside a 3D shape (measured in cubic units)
• Complementary angles — two angles summing to 90°
• Perpendicular — meeting at a right angle (90°)
Watch Out For ⚠️
• The height of a triangle must be perpendicular to the base — it is NOT always a side of the triangle
• The Pythagorean Theorem only works for right triangles
• c is always the hypotenuse (the side across from the 90° angle) in a² + b² = c²
• Area uses r² but circumference uses only r (or diameter) — don't mix them up
• Complementary = 90°, Supplementary = 180° — use "C comes before S, 90 comes before 180" to remember
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Data Analysis, Statistics & Probability
Key Concepts
Measures of Central Tendency
| Measure | How to Find | Best Used When |
|---------|------------|---------------|
| Mean | Sum all values ÷ number of values | Data has no extreme outliers |
| Median | Middle value when ordered (or average of two middle values) | Data has outliers |
| Mode | Most frequently occurring value | Looking for most common item |
| Range | Maximum − Minimum | Measuring spread of data |
Probability
$$P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$
> Example: 3 red marbles, 7 blue marbles → P(red) = 3/10 = 30%
Independent Events
Two events are independent if the outcome of one does NOT affect the other.
• P(A and B) = P(A) × P(B)
Data Displays
| Display | What It Shows |
|---------|--------------|
| Scatter plot | Relationship/correlation between two variables |
| Box plot (box-and-whisker) | Five-number summary: Min, Q1, Median, Q3, Max |
| Bar graph | Comparison of categories |
| Line graph | Change over time |
Correlation in Scatter Plots
• Positive correlation — points trend upward (↗); both variables increase together
• Negative correlation — points trend downward (↘); one increases as other decreases
• No correlation — points are scattered with no pattern
Key Terms
• Mean — arithmetic average
• Median — middle value in an ordered data set
• Mode — most frequently occurring value
• Range — difference between maximum and minimum values
• Probability — likelihood of an event, expressed as a fraction, decimal, or percent
• Independent events — events where one outcome does not influence the other
• Box plot — visual display of the five-number summary
• Correlation — the relationship/trend between two variables in a scatter plot
• Quartile — values that divide data into four equal parts
Watch Out For ⚠️
• To find the median with an even number of values, average the two middle numbers
• A data set can have more than one mode or even no mode at all
• Probability values must always be between 0 and 1 (or 0% and 100%) — never negative or greater than 1
• Correlation does not mean causation — two things trending together doesn't mean one causes the other
• Don't confuse mean and median — outliers heavily affect the mean but not the median
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Measurement & Problem Solving
Key Concepts
Percent Change
$$\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100$$
• Positive result = percent increase
• Negative result = percent decrease
Speed, Distance, and Time
$$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$
The formula triangle:
• D = S × T
• S = D ÷ T
• T = D ÷ S
> Example: 240 miles ÷ 4 hours = 60 mph
Proportions
A proportion states that two ratios are equal: a/b = c/d
Cross multiplication: a × d = b × c, then solve for the unknown variable.
> Example: 3/4 = x/20 → 4x = 60 → x = 15
Simple Interest
$$I = P \times r \times t$$
• P = Principal (initial amount)
• r = Annual interest rate (as a decimal)
• t = Time in years
> Example: $1,000 at 5% for 3 years → I = 1000 × 0.05 × 3 = $150
Key Terms
• Percent change — the relative change from an original value, expressed as a percentage
• Rate — a ratio comparing two different units (e.g., miles per hour)
• Proportion — an equation stating two ratios are equal
• Cross multiplication — method for solving proportions: a/b = c/d → ad = bc
• Simple interest — interest calculated only on the principal amount
• Principal — the original amount of money invested or borrowed
Watch Out For ⚠️
• For percent change, always divide by the original value — never the new value
• For simple interest, convert the rate to a decimal before calculating (5% → 0.05)
• In proportion problems, set up ratios consistently (same units in same positions)
• For time/distance/speed problems, make sure units match (don't mix hours and minutes without converting)
• Percent increase vs. decrease — read the problem carefully to know which direction the change goes
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Quick Review Checklist ✅
Before your GED Math exam, confirm you can do each of the following:
Number Operations
• [ ] Apply PEMDAS correctly to multi-step expressions
• [ ] Add, subtract, multiply, and divide fractions
• [ ] Convert between fractions, decimals, and percents
• [ ] Identify prime vs. composite numbers
• [ ] Evaluate absolute value expressions
Algebra & Functions
• [ ] Solve one- and two-step equations for x
• [ ] Calculate slope from two points or from an equation
• [ ] Identify slope (m) and y-intercept (b) in y = mx + b
• [ ] Solve a system of two equations using substitution
• [ ] Apply the quadratic formula to find roots
• [ ] Apply exponent rules (zero power, multiply/divide same base)
• [ ] Determine whether a relation is a function
Geometry
• [ ] Calculate area and perimeter of rectangles and triangles
• [ ] Find the circumference and area of a circle
• [ ] Apply the Pythagorean Theorem to right triangles
• [ ] Use the distance formula between two coordinate points
• [ ] Distinguish complementary (90°) from supplementary (180°) angles
• [ ] Calculate volume of a rectangular prism
Data Analysis & Probability
• [ ] Calculate mean, median, mode, and range
• [ ] Compute basic probability as a fraction/percent
• [ ] Interpret scatter plots (positive, negative, no correlation)
• [ ] Read and interpret a box plot's five-number summary
• [ ] Identify independent events and calculate combined probability
Measurement & Problem Solving
• [ ] Calculate percent increase and percent decrease
• [ ] Solve distance = rate × time problems
• [ ] Set up and solve proportions using cross multiplication
• [ ] Apply the simple interest formula I = Prt
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> 💡 Final Exam Tip: The GED provides a formula sheet during the exam — but knowing the formulas by heart saves time and reduces errors. Focus your practice on setting up problems correctly and checking your work before moving on.