ASVAB Study Guide

Comprehensive preparation for the Armed Services Vocational Aptitude Battery

Mathematics Knowledge

Understanding mathematical concepts and principles for the ASVAB

Core Concepts

The Mathematics Knowledge (MK) section tests your understanding of mathematical concepts and principles. You'll need to demonstrate knowledge in arithmetic, algebra, geometry, and other mathematical areas.

Basic Arithmetic Concepts

Number Systems

  • Natural Numbers: Counting numbers (1, 2, 3, ...)
  • Whole Numbers: Natural numbers and zero (0, 1, 2, ...)
  • Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...)
  • Rational Numbers: Numbers expressible as fractions (p/q where q ≠ 0)
  • Irrational Numbers: Numbers with non-terminating, non-repeating decimals (√2, π)
  • Real Numbers: All rational and irrational numbers

Properties of Operations

  • Commutative: a + b = b + a, a × b = b × a
  • Associative: (a + b) + c = a + (b + c), (a × b) × c = a × (b × c)
  • Distributive: a × (b + c) = a × b + a × c
  • Identity: a + 0 = a, a × 1 = a
  • Inverse: a + (-a) = 0, a × (1/a) = 1 (a ≠ 0)

Order of Operations (PEMDAS)

  1. Parentheses
  2. Exponents
  3. Multiplication and Division (left to right)
  4. Addition and Subtraction (left to right)

Factors and Multiples

  • Factors: Numbers that divide evenly into another number
  • Multiples: Products of a number and an integer
  • GCF: Greatest Common Factor - largest number that divides two numbers
  • LCM: Least Common Multiple - smallest number that is a multiple of two numbers
  • Prime Numbers: Numbers greater than 1 with only two factors: 1 and themselves

Fractions, Decimals, and Percentages

Fractions

  • Types: Proper (numerator < denominator), Improper (numerator ≥ denominator), Mixed (whole number + fraction)
  • Operations:
    • Addition/Subtraction: Common denominator required
    • Multiplication: Multiply numerators and denominators
    • Division: Multiply by reciprocal

Decimals

  • Types: Terminating (finite digits), Repeating (digit pattern repeats)
  • Operations: Align decimals for addition/subtraction, count decimal places for multiplication/division

Percentages

  • Percentage = Fraction × 100%
  • Decimal = Percentage ÷ 100
  • Percentage of a number: Number × (Percentage ÷ 100)

Algebra

Variables and Expressions

Algebraic expressions combine variables, numbers, and operations (e.g., 3x + 2, 5y² - 7y + 1).

Equations and Inequalities

  • Linear Equations: ax + b = c
  • Quadratic Equations: ax² + bx + c = 0 (solved by factoring, quadratic formula, or completing the square)
  • Inequalities: Use <, >, ≤, ≥; reverse inequality when multiplying/dividing by negative

Functions

A function assigns exactly one output to each input. Notation: f(x)

Systems of Equations

Solve using substitution, elimination, or graphing.

Geometry

Basic Concepts

  • Angles: Acute (<90°), Right (90°), Obtuse (>90°), Straight (180°)
  • Complementary: Sum to 90°, Supplementary: Sum to 180°
  • Parallel Lines: Never intersect, Perpendicular: Intersect at 90°

Polygons

  • Triangles: Sum of angles = 180°, Area = (1/2) × base × height, Pythagorean theorem: a² + b² = c²
  • Quadrilaterals:
    • Rectangle: Area = length × width
    • Square: Area = side²
    • Parallelogram: Area = base × height
    • Trapezoid: Area = (1/2) × (sum of parallel sides) × height
  • Circles: Circumference = 2πr, Area = πr²

Solid Geometry

  • Rectangular Prism: Volume = length × width × height
  • Cube: Volume = side³
  • Cylinder: Volume = πr²h
  • Sphere: Volume = (4/3)πr³
  • Cone: Volume = (1/3)πr²h

Coordinate Geometry

  • Distance Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
  • Midpoint Formula: ((x₁ + x₂)/2, (y₁ + y₂)/2)
  • Slope Formula: m = (y₂ - y₁)/(x₂ - x₁)
  • Equation of a Line: Slope-intercept form: y = mx + b

Exponents and Radicals

Exponent Properties

aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐ*ⁿ
(a × b)ⁿ = aⁿ × bⁿ
a⁻ⁿ = 1/aⁿ
a⁰ = 1 (a ≠ 0)

Radical Properties

√(a × b) = √a × √b
√(a ÷ b) = √a ÷ √b
√aⁿ = a^(n/2)

Probability and Statistics

Probability

P(event) = (number of favorable outcomes) / (total possible outcomes)

  • Independent Events: P(A and B) = P(A) × P(B)
  • Mutually Exclusive: P(A or B) = P(A) + P(B)

Statistics

  • Mean: Average (sum of values / number of values)
  • Median: Middle value in ordered list
  • Mode: Most frequent value
  • Range: Difference between highest and lowest values

Logarithms

If aˣ = b, then logₐ(b) = x

logₐ(xy) = logₐ(x) + logₐ(y)
logₐ(x/y) = logₐ(x) - logₐ(y)
logₐ(xⁿ) = n × logₐ(x)

Trigonometry

sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent

Key Identity: sin²(θ) + cos²(θ) = 1

Examples and Solved Problems

Step-by-step solutions to various mathematics problems you might encounter on the ASVAB.

Example 1: Order of Operations

Problem: Calculate the value of 3 + 4 × 2² - 6 ÷ 2.

Solution:

  1. Exponent: 2² = 4
  2. Multiplication and division: 4 × 4 = 16, 6 ÷ 2 = 3
  3. Addition and subtraction: 3 + 16 - 3 = 16

Answer: 16

Example 2: Properties of Operations

Problem: Simplify: 5(x + 2) - 3(x - 4)

Solution:

  1. Distribute: 5(x + 2) = 5x + 10, 3(x - 4) = 3x - 12
  2. Substitute: 5x + 10 - (3x - 12) = 5x + 10 - 3x + 12
  3. Combine like terms: 5x - 3x + 10 + 12 = 2x + 22

Answer: 2x + 22

Example 3: Solving Linear Equations

Problem: Solve for x: 3(x - 2) = 2x + 7

Solution:

  1. Expand: 3(x - 2) = 3x - 6
  2. Equation: 3x - 6 = 2x + 7
  3. Subtract 2x: 3x - 2x - 6 = 7 → x - 6 = 7
  4. Add 6: x = 13

Answer: x = 13

Example 4: Solving Quadratic Equations

Problem: Solve for x: x² - 5x + 6 = 0

Solution:

  1. Factor: (x - 2)(x - 3) = 0
  2. Set each factor to zero: x - 2 = 0 or x - 3 = 0
  3. Solve: x = 2 or x = 3

Answer: x = 2 or x = 3

Example 5: Geometry Application

Problem: Find the hypotenuse of a right triangle with legs measuring 5 cm and 12 cm.

Solution:

  1. Pythagorean theorem: c² = a² + b²
  2. c² = 5² + 12² = 25 + 144 = 169
  3. c = √169 = 13

Answer: 13 cm

Example 6: Probability

Problem: A bag contains 3 red, 4 blue, and 5 green marbles. What is the probability of drawing a blue marble?

Solution:

  1. Total marbles: 3 + 4 + 5 = 12
  2. Number of blue marbles: 4
  3. P(blue) = 4/12 = 1/3

Answer: 1/3

Review Section

Key concepts summary, common pitfalls, and memory aids for the Mathematics Knowledge section.

Key Concepts Summary

Concept Key Points
Order of Operations PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction
Properties Commutative, Associative, Distributive, Identity, Inverse
Factors & Multiples GCF: Greatest Common Factor, LCM: Least Common Multiple
Fractions Addition/Subtraction: Common denominator, Multiplication: Multiply across, Division: Multiply by reciprocal
Percentages Percentage = Fraction × 100%, Decimal = Percentage ÷ 100
Linear Equations ax + b = c
Quadratic Equations ax² + bx + c = 0, Solved by factoring, quadratic formula, or completing the square
Geometry Formulas Triangle Area: (1/2)bh, Circle Area: πr², Pythagorean Theorem: a² + b² = c²

Common Pitfalls and Misconceptions

Memory Aids and Tricks

PEMDAS: "Please Excuse My Dear Aunt Sally" (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)

Fraction Division: "Keep, Change, Flip" (Keep first fraction, change division to multiplication, flip second fraction)

Percentage Calculations: "Percent means per hundred, so move the decimal point"

Quadratic Formula: "Negative b plus or minus the square root of b-squared minus 4ac, all over 2a"

Trigonometric Ratios: "SOH-CAH-TOA" (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent)

Exponent Rules: "Same base, adding exponents", "Same base, subtracting exponents", "Power to a power, multiply exponents"

Quick Review Questions

1. Calculate: 2³ × 2² ÷ 2⁴

Solution: 2³⁺²⁻⁴ = 2¹ = 2

2. Simplify: 3(x + 2) - 2(x - 1)

Solution: 3x + 6 - 2x + 2 = x + 8

3. Find the GCF and LCM of 18 and 24

Solution: GCF = 6, LCM = 72

4. Calculate: (2/3) + (1/4)

Solution: (8/12) + (3/12) = 11/12

5. Convert 0.625 to a fraction in lowest terms

Solution: 0.625 = 625/1000 = 5/8

6. 15% of what number is 45?

Solution: 0.15x = 45 → x = 45/0.15 = 300

7. Solve for x: 2(x - 3) = 3x + 1

Solution: 2x - 6 = 3x + 1 → -x = 7 → x = -7

8. Factor: x² - 7x + 12

Solution: (x - 3)(x - 4)

Final Tips for the Mathematics Knowledge Section

Practice Test

This practice test contains 15 multiple-choice questions covering various aspects of mathematics knowledge. Try to answer all questions within 12 minutes to simulate the time constraints of the actual ASVAB test.

Question 1

Calculate: 3 + 2 × 5 - 8 ÷ 4

11
12
13
14

Explanation: Following PEMDAS: 2 × 5 = 10, 8 ÷ 4 = 2, then 3 + 10 - 2 = 11

Question 2

Simplify: (2³ × 2²) ÷ 2³

2
4
8
16

Explanation: (8 × 4) ÷ 8 = 32 ÷ 8 = 4

Question 3

What is the greatest common factor (GCF) of 36 and 48?

6
12
18
24

Explanation: Factors of 36: 1,2,3,4,6,9,12,18,36; Factors of 48: 1,2,3,4,6,8,12,16,24,48; Largest common is 12

Question 4

What is the least common multiple (LCM) of 12 and 18?

6
36
72
216

Explanation: Multiples of 12: 12,24,36,48,...; Multiples of 18: 18,36,54,...; Smallest common is 36

Question 5

Which of the following is a prime number?

51
57
67
91

Explanation: 67 is only divisible by 1 and itself

Question 6

Calculate: (2/3) + (3/4) - (1/6)

5/12
7/6
5/4
15/12

Explanation: Common denominator is 12: 8/12 + 9/12 - 2/12 = 15/12 = 5/4

Question 7

What is 35% of 80?

24
28
35
45

Explanation: 0.35 × 80 = 28

Question 8

Solve for x: 3(x - 2) = 2x + 7

x = 13
x = 7
x = 5
x = -1

Explanation: 3x - 6 = 2x + 7 → 3x - 2x = 7 + 6 → x = 13

Question 9

Solve for x: x² - 5x + 6 = 0

x = 2 or x = 3
x = -2 or x = -3
x = 2 or x = -3
x = -2 or x = 3

Explanation: (x - 2)(x - 3) = 0 → x = 2 or x = 3

Question 10

What is the area of a triangle with base 6 cm and height 8 cm?

14 cm²
24 cm²
28 cm²
48 cm²

Explanation: Area = (1/2) × base × height = 0.5 × 6 × 8 = 24 cm²

Test Results

Your Score: 0/10

Review the explanations for each question to understand the correct answers.