GED Mathematical Reasoning: Comprehensive Study Guide

Overview

The GED Mathematical Reasoning test covers four core domains: Number Sense & Operations, Algebra & Functions, Geometry, and Data Analysis & Statistics. This guide organizes all key concepts, formulas, and strategies you need to succeed. Mastery of these topics requires both conceptual understanding and the ability to apply skills to real-world problem-solving scenarios.

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Number Sense & Operations

Key Concepts

Order of Operations (PEMDAS)

Always solve expressions in this sequence:

1. Parentheses — solve anything inside grouping symbols first

2. Exponents — evaluate powers and roots

3. Multiplication / Division — left to right (equal priority)

4. Addition / Subtraction — left to right (equal priority)

> Example: 3 + 4 × (2²) → 3 + 4 × 4 → 3 + 16 = 19

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Factors and Multiples

| Concept | Definition | Example (36 & 48) |

|---|---|---|

| GCF (Greatest Common Factor) | Largest number that divides evenly into both numbers | GCF = 12 |

| LCM (Least Common Multiple) | Smallest number both numbers divide into evenly | LCM of 4 & 6 = 12 |

How to find GCF: List all factors of each number, identify the largest one they share.

How to find LCM: List multiples of each number until you find the first one in common.

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Fractions, Decimals, and Percents

Converting between forms:

• Fraction → Decimal: Divide numerator ÷ denominator

• Decimal → Percent: Multiply by 100, add the % sign

• Fraction → Percent: Combine both steps above

> Example: 3/4 → 3 ÷ 4 = 0.75 → 0.75 × 100 = 75%

Percent Discount Formula:

• Discount Amount = Original Price × Discount Rate

• Sale Price = Original Price − Discount Amount

> Example: $80 jacket at 25% off → $80 × 0.25 = $20 off → Sale price = $60

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Absolute Value

• Absolute value is a number's distance from zero on the number line — always non-negative

• Written as |−15| = 15

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Signed Number Rules

| Operation | Same Signs | Different Signs |

|---|---|---|

| Multiply / Divide | Result is positive | Result is negative |

| Add | Add values, keep the sign | Subtract values, keep sign of larger |

> Key rule: (−4) × (−3) = +12 ✓

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Rational Numbers

A rational number is any number expressible as a fraction p/q where:

• p and q are integers

• q ≠ 0

Includes: whole numbers, integers, fractions, terminating decimals, repeating decimals

Key Terms — Number Sense

• Factor — a number that divides evenly into another

• Multiple — the product of a number and any whole number

• Absolute Value — distance from zero; always ≥ 0

• Rational Number — expressible as a fraction p/q (q ≠ 0)

• Percent — a ratio out of 100

⚠️ Watch Out For

• PEMDAS mistake: Multiplication does NOT always come before division — they are solved left to right together

• Percent problems: Always convert percent to a decimal (divide by 100) before multiplying

• Negative × Negative: Students often forget this yields a positive result

• GCF vs. LCM confusion: GCF is for simplifying; LCM is for finding common denominators

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Algebra & Functions

Key Concepts

Solving Equations

To solve for a variable, isolate it using inverse operations applied equally to both sides.

> Example: 3x + 7 = 22

> Step 1: Subtract 7 → 3x = 15

> Step 2: Divide by 3 → x = 5

Inverse Operations:

• Addition ↔ Subtraction

• Multiplication ↔ Division

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Slope

Slope (m) measures the steepness and direction of a line.

$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$$

| Slope Type | Meaning |

|---|---|

| Positive | Line rises left to right |

| Negative | Line falls left to right |

| Zero | Horizontal line |

| Undefined | Vertical line |

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Slope-Intercept Form

$$y = mx + b$$

• m = slope

• b = y-intercept (value of y when x = 0; where line crosses the y-axis)

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Linear Inequalities

| Symbol | Meaning | Graph |

|---|---|---|

| < | Less than | Open circle, shade left |

| > | Greater than | Open circle, shade right |

| ≤ | Less than or equal to | Closed circle, shade left |

| ≥ | Greater than or equal to | Closed circle, shade right |

A linear inequality has a range of solutions, not just one value.

> Critical Rule: When multiplying or dividing both sides by a negative number, FLIP the inequality sign.

> Example: −2x > 6 → x < −3

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Functions & The Vertical Line Test

• A function assigns exactly one output (y-value) for each input (x-value)

• Vertical Line Test: If any vertical line crosses the graph more than once → NOT a function

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Simplifying Expressions

Steps:

1. Distribute (apply the distributive property)

2. Combine like terms (same variable and exponent)

> Example: 4(2x − 3) + 5x

> = 8x − 12 + 5x

> = 13x − 12

Key Terms — Algebra

• Variable — a letter representing an unknown value

• Coefficient — the number multiplied by a variable

• Like Terms — terms with the same variable and exponent

• Slope — rise over run; rate of change

• Y-Intercept — where the line crosses the y-axis (x = 0)

• Function — each input has exactly one output

• Inequality — a mathematical statement using <, >, ≤, or ≥

⚠️ Watch Out For

• Forgetting to flip the inequality sign when multiplying or dividing by a negative number — this is one of the most common algebra errors

• Distributing negatives: 4(2x − 3) — don't forget to distribute the 4 to both terms

• Confusing slope and y-intercept in y = mx + b (m = slope, b = y-intercept)

• Functions: A vertical line, like x = 3, is not a function

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Geometry

Key Concepts

Triangle Formulas

$$\text{Area} = \frac{1}{2} \times base \times height$$

> Note: The height must be perpendicular to the base — it is not always a side of the triangle.

Sum of Interior Angles of a Triangle = 180°

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Pythagorean Theorem

Used only with right triangles:

$$a^2 + b^2 = c^2$$

• a and b are the two legs (shorter sides)

• c is the hypotenuse (longest side, opposite the right angle)

> Use when: You know two sides and need to find the third.

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Circle Formulas

| Measurement | Formula |

|---|---|

| Circumference | C = 2πr or C = πd |

| Area | A = πr² |

• r = radius (center to edge)

• d = diameter (edge to edge through center) = 2r

• π ≈ 3.14

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Volume of a Rectangular Prism

$$V = l \times w \times h$$

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Perimeter vs. Area

| Concept | Measures | Units |

|---|---|---|

| Perimeter | Distance around a shape | Units (cm, ft, m) |

| Area | Surface inside a shape | Square units (cm², ft²) |

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Angles

| Angle Pair | Sum | Example |

|---|---|---|

| Supplementary | 180° | 65° + 115° = 180° |

| Complementary | 90° | 30° + 60° = 90° |

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Congruent Figures

Two figures are congruent if they have:

• The same shape and same size

• All corresponding sides and angles are equal

• One may be a flip (reflection), slide (translation), or rotation of the other

Key Terms — Geometry

• Hypotenuse — 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 the center (= 2r)

• Perimeter — distance around a shape

• Area — surface space inside a shape

• Volume — space inside a 3D figure

• Supplementary Angles — two angles summing to 180°

• Complementary Angles — two angles summing to 90°

• Congruent — same shape and size

⚠️ Watch Out For

• Pythagorean theorem applies ONLY to right triangles — don't use it on other triangles

• Radius vs. Diameter: Many circle problems give diameter — remember r = d ÷ 2

• Triangle height: The height is always perpendicular to the base, not necessarily a visible side

• Perimeter uses regular units; Area uses SQUARE units — don't mix them up

• Supplementary (180°) vs. Complementary (90°): Use the memory trick — "S" comes after "C" in the alphabet, and 180 > 90

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Data Analysis & Statistics

Key Concepts

Measures of Central Tendency

| Measure | How to Find It | Best Used When... |

|---|---|---|

| Mean | Sum of all values ÷ count | Data has no extreme outliers |

| Median | Middle value (data in order) | Outliers are present |

| Mode | Most frequently occurring value | Finding the most common value |

Finding the Median:

• Odd number of values → middle value

• Even number of values → average the two middle values

> Example (even): {2, 4, 6, 8} → (4 + 6) ÷ 2 = 5

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Range

$$\text{Range} = \text{Highest Value} - \text{Lowest Value}$$

Measures the spread of data.

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Probability

$$P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$$

> Example: 5 blue marbles out of 10 total → P(blue) = 5/10 = 1/2 = 50%

Probability is always between 0 (impossible) and 1 (certain).

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Scatterplots and Correlation

| Correlation | Pattern | Trend Direction |

|---|---|---|

| Positive | Both variables increase together | Points trend upward ↗ |

| Negative | One increases, other decreases | Points trend downward ↘ |

| No Correlation | No clear pattern | Points scattered randomly |

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Outliers and Their Effect

• An outlier is a value significantly higher or lower than the rest of the data

• Outliers pull the mean up or down, making it less representative

• The median is more resistant to outliers — often a better measure of center when outliers exist

Key Terms — Data Analysis

• Mean — arithmetic average

• Median — middle value in ordered data

• Mode — most frequent value

• Range — difference between highest and lowest values

• Outlier — extreme value far from the others

• Probability — likelihood of an event (0 to 1)

• Scatterplot — graph showing relationship between two variables

• Positive Correlation — both variables increase together

⚠️ Watch Out For

• Even vs. Odd median: When there's an even number of data points, you must average the two middle values

• Outliers affect the mean more than the median — if a problem has an extreme value, the median is likely more appropriate

• Probability can never be greater than 1 or less than 0

• Mode can have multiple values (bimodal/multimodal), or sometimes no mode at all

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Problem Solving & Reasoning

Key Concepts

Ratios

A ratio compares two quantities and can be written three ways:

• As a fraction: 3/4

• With a colon: 3:4

• In words: 3 to 4

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Proportions

A proportion states that two ratios are equal: a/b = c/d

Solve by cross-multiplication:

$$a \times d = b \times c$$

Then solve for the unknown variable.

> Example: Car travels 180 miles in 3 hours. How far in 5 hours?

> Rate = 180 ÷ 3 = 60 mph → 60 × 5 = 300 miles

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Scientific Notation

Expresses very large or very small numbers as:

$$a \times 10^n$$

where 1 ≤ a < 10 and n is an integer.

| Standard Form | Scientific Notation |

|---|---|

| 4,500,000 | 4.5 × 10⁶ |

| 0.00032 | 3.2 × 10⁻⁴ |

Moving the decimal right → negative exponent

Moving the decimal left → positive exponent

Key Terms — Problem Solving

• Ratio — comparison of two quantities

• Proportion — equation stating two ratios are equal

• Cross-multiplication — method for solving proportions

• Unit Rate — ratio with a denominator of 1 (e.g., miles per hour)

• Scientific Notation — a × 10ⁿ where 1 ≤ a < 10

⚠️ Watch Out For

• Scientific notation: The first number must be between 1 and 10 — don't write 45 × 10⁵ (should be 4.5 × 10⁶)

• Unit rates: Always find the rate per one unit before scaling up

• Proportion setup: Make sure the units match across the ratios (miles/hours = miles/hours)

• Ratios are not always whole numbers — simplify when possible

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Essential Formula Reference Card

| Category | Formula |

|---|---|

| Area of Triangle | A = ½bh |

| Area of Circle | A = πr² |

| Circumference | C = 2πr or πd |

| Pythagorean Theorem | a² + b² = c² |

| Volume of Rectangular Prism | V = lwh |

| Slope | m = (y₂ − y₁) / (x₂ − x₁) |

| Slope-Intercept Form | y = mx + b |

| Probability | P = favorable outcomes / total outcomes |

| Percent Discount | Sale Price = Original − (Original × Rate) |

| Mean | Sum of values ÷ number of values |

| Range | Highest − Lowest |

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Quick Review Checklist

Use this checklist to confirm your readiness before test day:

Number Sense & Operations

• [ ] I can apply PEMDAS

GED High School Equivalency Exam Study Guide

GED Mathematical Reasoning: Comprehensive Study Guide

Overview

Number Sense & Operations

Key Concepts

Key Terms — Number Sense

⚠️ Watch Out For

Algebra & Functions

Key Concepts

Key Terms — Algebra

⚠️ Watch Out For

Geometry

Key Concepts

Key Terms — Geometry

⚠️ Watch Out For

Data Analysis & Statistics

Key Concepts

Key Terms — Data Analysis

⚠️ Watch Out For

Problem Solving & Reasoning

Key Concepts

Key Terms — Problem Solving

⚠️ Watch Out For

Essential Formula Reference Card

Quick Review Checklist

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