Study Guide

Mathematics Knowledge Overview

  • The Mathematics Knowledge section of the ASVAB tests knowledge of key concepts in high school math, emphasizing procedures and ideas over computation.

  • There are 25 Mathematics Knowledge questions on the test with a time limit of 24 minutes.

Number Theory

Factors

  • Whole Numbers: The set includes 0, 1, 2, 3, 4, 5, etc.

  • Definition of a Factor: A number that divides another number evenly (resulting in a whole number).     - Example: For 24, both 6 and 4 are factors: 6 × 4 = 24.     - Checking Factors: Divide the given number (e.g., 72) by the candidate factor (e.g., 8). If no remainder, it's a factor.

Common Factors

  • Definition: A factor that divides more than one number.

  • Greatest Common Factor (GCF): The largest common factor of two or more numbers.     - Example:         - For 36: Factors are 1, 2, 3, 4, 6, 9, 12, 18, 36.         - For 48: Factors are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.         - Common factors of 36 and 48: 1, 2, 3, 4, 6, 12; GCF = 12.

Prime Numbers

  • A Prime Number has no divisors other than 1 and itself.     - The smallest prime number is 2.     - First 10 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

  • Identifying Primes:     - Example (Checking if a number is prime):         - 33 = 3 × 11 (not prime).         - 37 (prime).         - 39 = 3 × 13 (not prime).         - 42 = 2 × 21 (not prime).         - 43 (prime).

  • A Composite Number is not prime and can be expressed as a product of primes.     - Example: 35 is composite; prime factorization is 5 × 7.

Prime Factorization

  • To find prime factors:     - Example: Prime factor the number 150:         - 150 = 15 × 10; 15 = 3 × 5; 10 = 2 × 5.         - Prime factorization: 2 × 3 × 5 × 5.         - In exponential notation: 21imes31imes522^1 imes 3^1 imes 5^2.

Multiples

  • A Multiple is the product of a number and an integer.     - Example for multiples of 7: 7, 14, 21, 28…

  • Common Multiple: A number that is a multiple of two or more numbers.

  • Least Common Multiple (LCM): The smallest common multiple.     - Finding LCM: Write multiples of each number and find the smallest one.     - Finding LCM of 3 and 8: First few multiples of 3: 3, 6, 9, 12, 15; for 8: 8, 16, 24, 32. LCM = 24.

Exponents

  • Exponential Notation: Simplifies repeated multiplication.     - Example: 72=7imes77^2 = 7 imes 7; 54=5imes5imes5imes55^4 = 5 imes 5 imes 5 imes 5.     - Prime factorization of 150 can be expressed as: 21imes31imes522^1 imes 3^1 imes 5^2.

  • Calculating Values:     - Example: What is the value of 353^5?         - 35=3imes3imes3imes3imes3=2433^5 = 3 imes 3 imes 3 imes 3 imes 3 = 243.

Square Roots

  • The Square Root of a number is the number that, when multiplied by itself, yields that number.     - Example: extSquarerootof25=5ext{Square root of } 25 = 5 because 5imes5=255 imes 5 = 25.     - Square roots can be expressed in radical form (e.g., extSquarerootof2extisapproximately1.414213562…ext{Square root of } 2 ext{ is approximately } 1.414213562…).

  • Perfect Squares: Have integer square roots (e.g., 1, 4, 9, 16, 25…).

Cube Roots

  • The Cube Root of a number is the number that, when multiplied by itself three times, gives that number.     - Example: extCuberootof8=2ext{Cube root of } 8 = 2 since 2imes2imes2=82 imes 2 imes 2 = 8.

Order of Operations

  • The standard order of operations is:     1. Parentheses     2. Exponents     3. Multiplication and Division (left to right)     4. Addition and Subtraction (left to right)

  • Mnemonic: PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).     - Example: Evaluate 18−6÷3×7+418 - 6 ÷ 3 × 7 + 4:         - Perform division: 6÷3=26 ÷ 3 = 2         - Replace and evaluate multiplications then add/subtract left to right.         - Final answer is 0.

Integer Operations

Adding Signed Numbers

  1. Same Sign: Add and keep the sign.    - Example: (+4) + (+7) = +11; (−5) + (−9) = −14.

  2. Different Sign: Subtract the smaller absolute value from the larger absolute value; sign of answer matches the number with the larger size.    - Example: (+9) + (−5) yields +4.    - Example: (−9) + (+5) yields −4.

Subtracting Signed Numbers

  • Change subtraction to addition by changing the sign of the second number.     - Example: (−7) − (+2) = (−7) + (−2) = −9.     - Example solving for: (−7) − (−3) = (−7) + (+3) = −4.

Multiplication and Division of Signed Numbers

  • Ignore signs and operate normally; determine the sign based on the number of negatives:    - Even number of negatives yields a positive result.    - Odd number of negatives yields a negative result.    - Example: (−2) × (−3) = +6; (−2) × (+3) = −6.    

Operations with Fractions

Equivalent Fractions

  • Simplifying: Rewrite fractions in a simpler form by dividing numerator and denominator by a common factor.     - Example: 10/5 simplifies to 2/1.

Adding and Subtracting Fractions

  • Common denominators: Add/Subtract numerators and keep the denominator.

  • Different denominators: Convert to equivalent fractions with a common denominator.     - Example: To add 43+31\frac{4}{3} + \frac{3}{1} find a common denominator like 12.

Multiplying and Dividing Fractions

  • Multiply: Numerators multiplied together, denominators multiplied together.

  • Division: Change to multiplication and take reciprocal of the second fraction.     - Example: 23÷14=23×41=83\frac{2}{3} \div \frac{1}{4} = \frac{2}{3} \times \frac{4}{1} = \frac{8}{3}.

Algebraic Operations

Numerical Evaluation

  • Constants: Fixed values (e.g., 3, −4).

  • Variables: Symbols representing unknown values (e.g., x, y).

  • Evaluate numerical expressions with given variable values using order of operations.

Solving Equations

  • Rearranging equations involves adding, subtracting, multiplying or dividing both sides by the same quantity.     - Example: Solve x+7=20x + 7 = 20 leads to x=13x = 13 after subtracting 7 from both sides.

Solving Word Problems

  • Converting words into equations and then solving them.     - Recognizing key terms and building the correct equations to find answers.

Geometry and Measurement

Angle Measurement

  • Angles measured in degrees, total of one complete rotation = 360°.

  • Types of angles: acute (<90°), obtuse (>90°), straight (=180°).     - Complementary angles: Sum to 90°.     - Supplementary angles: Sum to 180°.     - Vertical angles: Opposite angles formed when two lines intersect, are equal.

Properties of Triangles

  • Sum of internal angles in any triangle = 180°.

  • Isosceles triangle: Two equal sides and angles.

  • Equilateral triangle: All sides and angles equal (60° each).

Pythagorean Theorem

  • In right triangles, a2+b2=c2a^2 + b^2 = c^2 where c is the hypotenuse.

  • Can be used to find missing side lengths.     - Example: Given sides of 6 and hypotenuse of 10, calculate missing side a: a2+62=102a^2 + 6^2 = 10^2 leads to a=8a = 8.

Properties of Circles

Circle Definitions

  • A circle is the set of points equidistant from a center point.     - Radius: Distance from center to circle.     - Diameter: Twice the radius.

Circle Formulas

  • Circumference: C=2extπrC = 2 ext{π}r;

  • Area: A=extπr2A = ext{π}r^2.     - Finding dimensions by reversing the formulas if given area or circumference.

Coordinates and Slope

  • Coordinates: Points represented as (x, y) in a Cartesian plane.

  • Slope of a Line: Change in y over change in x between two points.     - Slope Formula: m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.     - Example: Calculate slope between points (9, 5) and (3, −2).

Practice Questions

  • Review practice questions to reinforce knowledge on factors, prime numbers, solving equations, and solving real-world problems with mathematical operations.

Answers and Explanations

  • Answers provided to practice questions can clarify understanding and reinforce concepts.