← Praxis Core: Number & Quantity

Praxis Core Teaching Exam Study Guide

Key concepts, definitions, and exam tips organized by topic.

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Praxis Core: Number & Quantity — Study Guide


Overview

This study guide covers the core Number & Quantity concepts tested on the Praxis Core Mathematics exam. Topics include number properties, fraction and decimal operations, ratios and proportions, percentages, and integer arithmetic. Mastery of these foundational skills is essential, as they appear both directly and embedded within more complex problem types.


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Number Properties


Summary

Number properties describe how numbers behave under operations and how they are classified. Understanding classifications (rational, prime, composite) and operational properties (distributive, associative) allows you to simplify problems efficiently and recognize shortcuts on the exam.


Number Classifications

  • • Natural Numbers: Counting numbers — 1, 2, 3, ...
  • • Whole Numbers: Natural numbers plus zero — 0, 1, 2, 3, ...
  • • Integers: Whole numbers and their negatives — ..., −3, −2, −1, 0, 1, 2, 3, ...
  • • Rational Numbers: Any number expressible as p/q where p and q are integers and q ≠ 0

    • - Includes: integers, terminating decimals (0.625), repeating decimals (0.333...)

  • • Irrational Numbers: Cannot be expressed as p/q — non-terminating, non-repeating decimals (e.g., π, √2)

    • Prime vs. Composite Numbers

    | Type | Definition | Example |

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

    | Prime | Exactly 2 factors: 1 and itself | 2, 3, 5, 7, 11 |

    | Composite | More than 2 factors | 4, 6, 9, 12 |

    | Neither | The number 1 | 1 |


    > ⚠️ Watch Out For: The number 1 is neither prime nor composite — this is a frequent trap on the exam. Also, 2 is the only even prime number.


    Key Properties of Operations


    | Property | Addition | Multiplication |

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

    | Commutative | a + b = b + a | a × b = b × a |

    | Associative | (a + b) + c = a + (b + c) | (a × b) × c = a × (b × c) |

    | Identity | a + 0 = a | a × 1 = a |

    | Distributive | a(b + c) = ab + ac | (applies across addition/subtraction) |


    GCF and LCM

  • • Greatest Common Factor (GCF): The largest integer that divides evenly into two or more numbers

    • - GCF(36, 48) = 12

    - Method: List all factors of each number; identify the largest shared factor

  • • Least Common Multiple (LCM): The smallest positive integer divisible by two or more numbers

    • - LCM(8, 12) = 24

    - Method: List multiples of each number; identify the smallest shared multiple


    Absolute Value

  • • Absolute value = distance from zero on the number line; always non-negative
  • • |−7| = 7 | |7| = 7 | |0| = 0

    • Key Terms

  • • Rational number | Irrational number | Prime | Composite | GCF | LCM | Absolute value | Distributive Property | Associative Property

    • ---


    Fractions & Decimals


    Summary

    Fraction operations require finding common denominators for addition/subtraction and using reciprocals for division. Converting between fractions, decimals, and mixed numbers is a tested skill. Always simplify your final answer.


    Fraction Operations


    #### Addition & Subtraction — Unlike Denominators

    1. Find the LCM of the denominators (the LCD)

    2. Convert each fraction to an equivalent fraction using the LCD

    3. Add or subtract the numerators; keep the denominator


    > Example: 1/3 + 1/4 → LCD = 12 → 4/12 + 3/12 = 7/12


    #### Multiplication

  • • Multiply numerators together; multiply denominators together
  • • Simplify (cancel common factors before or after multiplying)

    • #### Division — "Keep, Change, Flip"

  • • Keep the first fraction
  • • Change ÷ to ×
  • • Flip (take the reciprocal of) the second fraction

    • > Example: (3/4) ÷ (2/5) = (3/4) × (5/2) = 15/8


    Mixed Numbers

  • • Convert to improper fraction before performing operations:

    • - 2⅓ → (2 × 3 + 1)/3 = 7/3

  • • Convert back to mixed number if required in the answer

    • > Example: (7/3) × (3/7) = 21/21 = 1


    Fraction ↔ Decimal Conversions

    | To Convert | Method |

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

    | Fraction → Decimal | Divide numerator ÷ denominator |

    | Decimal → Fraction | Write as fraction over power of 10; simplify |


  • • Terminating decimal: Division ends — e.g., 5/8 = 0.625
  • • Repeating decimal: One or more digits repeat infinitely — e.g., 1/3 = 0.3̄ = 0.333...

    • > ⚠️ Watch Out For: Forgetting to convert mixed numbers to improper fractions before multiplying or dividing. Operating on the whole number and fraction parts separately is a common error.


    Key Terms

  • • Numerator | Denominator | Reciprocal | LCD (Least Common Denominator) | Improper fraction | Mixed number | Terminating decimal | Repeating decimal

    • ---


    Ratios, Rates & Proportions


    Summary

    Ratios compare two quantities; rates compare quantities with different units; proportions state that two ratios are equal. Cross-multiplication is the primary tool for solving proportions. Unit rates simplify comparisons.


    Core Concepts


  • • Ratio: A comparison of two quantities — written as a:b, a/b, or "a to b"
  • • Rate: A ratio comparing quantities with different units (e.g., miles per hour)
  • • Unit Rate: A rate with a denominator of 1 — found by dividing

    • - 150 miles in 3 hours → 150 ÷ 3 = 50 miles per hour

  • • Proportion: Two equal ratios → a/b = c/d, which means a × d = b × c (cross-products are equal)

    • Solving Proportions — Step-by-Step

    1. Set up the equation: a/b = c/d

    2. Cross-multiply: a × d = b × c

    3. Solve for the unknown variable


    > Example: Recipe calls for 2 cups flour per 3 cups sugar. How much flour for 9 cups sugar?

    > 2/3 = x/9 → 3x = 18 → x = 6 cups of flour


    Ratio Problems with Parts

  • • Identify the total number of parts in the ratio
  • • Determine the value of one part
  • • Multiply to find the target quantity

    • > Example: Ratio of boys to girls is 3:5. If there are 25 girls → 5 parts = 25 → 1 part = 5 → boys = 3 × 5 = 15 boys


    > ⚠️ Watch Out For: In ratio problems, the numbers given must align with the ratio parts. If your answer is not a whole number (e.g., 14.4 boys), re-read the problem — the given values may not perfectly match the ratio, or the question may ask for an approximation. Look for consistent setups.


    Key Terms

  • • Ratio | Rate | Unit rate | Proportion | Cross-multiplication | Equivalent ratios

    • ---


    Percentages


    Summary

    Percentages express a part out of 100. Three core problem types appear on the exam: finding a percent of a number, finding what percent one number is of another, and calculating percent change. Know the formula for each.


    Essential Conversions

    | Convert | Method | Example |

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

    | Percent → Decimal | Divide by 100 | 35% = 0.35 |

    | Decimal → Percent | Multiply by 100 | 0.35 = 35% |

    | Percent → Fraction | Write over 100; simplify | 35% = 35/100 = 7/20 |


    The Three Core Percent Problems


    #### 1. Finding a Percent of a Number

    > Formula: Part = Percent (decimal) × Whole

    > Example: 40% of 85 → 0.40 × 85 = 34


    #### 2. Finding What Percent One Number Is of Another

    > Formula: (Part ÷ Whole) × 100

    > Example: 18 is what percent of 72? → (18 ÷ 72) × 100 = 25%


    #### 3. Percent Change (Increase or Decrease)

    > Formula: [(New Value − Old Value) / Old Value] × 100

    > - Positive result = percent increase

    > - Negative result = percent decrease

    > Example: Salary from $40,000 → $46,000 → [(46,000 − 40,000) / 40,000] × 100 = 15% increase


    Discounts & Sale Prices

  • • Method 1: Calculate discount amount → subtract from original

    • - 25% off $120 → 0.25 × 120 = $30 → $120 − $30 = $90

  • • Method 2: Multiply original by (1 − discount rate)

    • - $120 × 0.75 = $90 (faster on exam)


    > ⚠️ Watch Out For:

    > - In percent change problems, always divide by the original (old) value, not the new value

    > - Don't confuse "percent of" with "percent more than" — they require different setups

    > - Moving the decimal: 35% → 0.35 (move left 2 places); not 3.5 or 0.035*


    Key Terms

  • • Percent | Decimal equivalent | Percent increase | Percent decrease | Discount | Part/Whole/Percent relationship

    • ---


    Integers & Order of Operations


    Summary

    Integers include all positive and negative whole numbers and zero. Rules for signs under multiplication and division, combined with strict adherence to order of operations (PEMDAS), are foundational skills that appear in nearly every math problem on the exam.


    Integer Sign Rules


    | Operation | Rule | Example |

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

    | (+) × (+) | Positive | 4 × 3 = 12 |

    | (−) × (−) | Positive | −5 × −3 = 15 |

    | (+) × (−) | Negative | 4 × −3 = −12 |

    | (−) × (+) | Negative | −4 × 3 = −12 |

    | Same rule applies to division | | |


    > Memory tip: Same signs → Positive; Different signs → Negative


    Order of Operations — PEMDAS


    | Step | Letter | Operation |

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

    | 1 | P | Parentheses (innermost first) |

    | 2 | E | Exponents |

    | 3 | M/D | Multiplication & Division (left to right) |

    | 4 | A/S | Addition & Subtraction (left to right) |


    > ⚠️ Watch Out For: Multiplication and Division are performed left to right — not multiplication always before division. Same rule applies to Addition and Subtraction. This is a very common source of errors.


    Worked Examples


    Example 1: −3 × (−4) + 2² − 7

    1. Exponent: 2² = 4

    2. Multiply: −3 × (−4) = 12

    3. Combine: 12 + 4 − 7 = 9 ✓


    Example 2: 3 + 6 × 2 − (4 + 1)

    1. Parentheses: (4 + 1) = 5

    2. Multiply: 6 × 2 = 12

    3. Left to right: 3 + 12 − 5 = 10 ✓


    Key Terms

  • • Integer | PEMDAS | Exponent | Order of operations | Negative product | Absolute value

    • ---


    Quick Review Checklist


    Use this checklist in the final days before your exam:


  • • [ ] I can identify rational vs. irrational numbers and give examples of each
  • • [ ] I know that 1 is neither prime nor composite and 2 is the only even prime
  • • [ ] I can state the Distributive, Associative, and Commutative Properties and identify them in equations
  • • [ ] I can find the GCF and LCM of two numbers using factor lists or prime factorization
  • • [ ] I can add, subtract, multiply, and divide fractions, including with unlike denominators
  • • [ ] I can convert mixed numbers to improper fractions before multiplying or dividing
  • • [ ] I can convert between fractions, decimals, and percents
  • • [ ] I can identify terminating vs. repeating decimals
  • • [ ] I can set up and solve proportions using cross-multiplication
  • • [ ] I can calculate unit rates by dividing total quantity by total units
  • • [ ] I can solve all three percent problem types: percent of a number, what percent, and percent change
  • • [ ] I use the original value as the denominator in percent change problems
  • • [ ] I follow PEMDAS correctly, treating M/D and A/S as left-to-right pairs
  • • [ ] I know that negative × negative = positive and negative × positive = negative
  • • [ ] I can evaluate multi-step integer expressions with exponents and parentheses accurately

    • ---


    Focus extra attention on percent change, proportion setup, and PEMDAS — these are among the most frequently tested and most frequently missed topics on the Praxis Core.

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