Mathematics is a key component of the ASVAB (Armed Services Vocational Aptitude Battery). Mastering fundamental math concepts will help improve your score, particularly in Arithmetic Reasoning and Mathematics Knowledge. This guide covers essential topics with detailed explanations and examples to help you build a strong foundation.
Order of Operations (PEMDAS)
Definition: The order of operations determines the correct sequence to evaluate mathematical expressions. The acronym PEMDAS helps you remember the order:
- Parentheses ( )
- Exponents (²,³, etc.)
- Multiplication & Division (from left to right)
- Addition & Subtraction (from left to right)
Example 1:
Solve 6 + 4 × 2
- Incorrect way: (Adding first) (6 + 4) × 2 = 10 × 2 = 20
- Correct way: (Multiplication first) 6 + (4 × 2) = 6 + 8 = 14
Example 2:
Solve (5 + 3)² ÷ 4
- Parentheses first: (5 + 3) = 8
- Exponents next: 8² = 64
- Division last: 64 ÷ 4 = 16
- Final Answer: 16
Absolute Value
Definition: The absolute value of a number is its distance from zero on a number line. It is always positive.
Examples:
- |7| = 7 (7 is 7 units from 0)
- |-9| = 9 (-9 is 9 units from 0)
Absolute value removes the negative sign but does not make positive numbers negative.
Example Problem:
Solve |5 – 12| + |3 – 8|
- First, simplify inside the absolute values:
- 5 – 12 = -7 → | -7 | = 7
- 3 – 8 = -5 → | -5 | = 5
- Now, add the results: 7 + 5 = 12
Factorials (!)
Definition: A factorial (n!) is the product of a number and all positive whole numbers below it.
Exponents
Definition: An exponent tells you how many times a number is multiplied by itself.
- 2³ = 2 x 2 x 2 = 8
- 5² = 5 x 5 = 25
Exponent Rules:
- Multiplication Rule: a^m x a^n = a^(m+n)
- Example: 2³ x 2² = 2^(3+2) = 2⁵ = 32
- Division Rule: a^m ÷ a^n = a^(m-n)
- Example: 3⁵ ÷ 3² = 3^(5-2) = 3³ = 27
- Power of a Power Rule: (a^m)^n = a^(mxn)
- Example: (2²)³ = 2^(2×3) = 2⁶ = 64
Square Roots & Cube Roots
Definition:
- The square root of a number is the value that, when squared, gives the original number.
- The cube root of a number is the value that, when cubed, gives the original number.
Examples:
- √25 = 5 because 5 x 5 = 25
- ∛27 = 3 because 3 x 3 x 3 = 27
Inverse Operations
Definition: Inverse operations undo each other.
- Addition & Subtraction are inverses:
- If x + 7 = 10, then x = 10 – 7 = 3
- Multiplication & Division are inverses:
- If 3x = 12, then x = 12 ÷ 3 = 4
Example Problem:
Solve for x in 4x – 5 = 15
- Add 5 to both sides: 4x = 20
- Divide by 4: x = 5
Naming Mathematical Results
- Sum → Addition (8 + 4 = 12)
- Difference → Subtraction (9 – 3 = 6)
- Product → Multiplication (5 x 2 = 10)
- Quotient → Division (20 ÷ 4 = 5)
Number Sequences & Patterns
Definition: Identifying patterns helps predict the next number in a sequence.
- Arithmetic Sequence (constant difference):
- 2, 5, 8, 11, … (Next: 14 because +3)
- Geometric Sequence (constant ratio):
- 3, 6, 12, 24, … (Next: 48 because x2)
Example Problem:
Find the next number: 10, 17, 24, 31, …
- Difference = +7
- Next number: 31 + 7 = 38
Averages (Mean, Median, Mode)
- Mean (Average):
- Formula: (Sum of numbers) ÷ (Total numbers)
- Example: (8 + 10 + 12) ÷ 3 = 10
- Median (Middle Value):
- Odd number of values: The middle value
- Even number of values: The average of the two middle values
- Example: Numbers: 4, 6, 7, 8, 10 → Median = 7
- Mode (Most Frequent Number):
- Example: 2, 5, 5, 6, 8, 5, 9 → Mode = 5 (appears most often)
Final Review
- Follow PEMDAS for Order of Operations
- Absolute Value is always positive
- Factorials involve multiplying all positive integers below a number
- Exponents & roots are inverse operations
- Inverse operations help solve equations
- Know the differences between mean, median, and mode