GED Mathematical Reasoning Quantitative Reasoning Integers Positive and Negative Numbers Rules for Operations

What Is This?

Integers are whole numbers, either positive, negative, or zero, without any fractional part. This topic is crucial in Quantitative Reasoning, as it deals with the rules for operations involving integers.

You'll encounter questions that test your understanding of integer properties, arithmetic operations, and problem-solving strategies. Be prepared for a mix of straightforward calculations and more complex problems that require logical reasoning.

Why It Matters

This topic appears in various exams, including the Graduate Management Admission Test (GMAT), Graduate Record Examination (GRE), and Professional Accounting (CPA) exams. It typically carries 10-20% of the total marks and tests your ability to apply mathematical concepts to solve problems.

The examiner wants to assess your understanding of integer properties, your ability to follow rules and procedures, and your capacity to think logically and reason correctly.

Core Concepts

To tackle integer-related questions, you must understand the following foundational ideas:

• Integers are whole numbers, either positive, negative, or zero.
• Order of Operations (PEMDAS/BODMAS) applies when dealing with multiple arithmetic operations.
• Integer Properties:
  • Addition: The sum of two integers is an integer.
  • Subtraction: The difference between two integers is an integer.
  • Multiplication: The product of two integers is an integer.
  • Division: The quotient of two integers is an integer only when the divisor is a factor of the dividend.
• Absolute Value: The absolute value of an integer is its distance from zero on the number line.

Prerequisites

Before diving into this topic, ensure you have a solid grasp of:

• Basic arithmetic operations (addition, subtraction, multiplication, and division)
• Fractions and decimals
• Algebraic expressions and equations

If you're missing these concepts, revisit them before proceeding, as they form the foundation for integer-related operations.

The Rule-Book (How It Works)

The primary rule for integer operations is:

• When adding or subtracting integers, combine like terms and follow the order of operations.
• When multiplying or dividing integers, follow the order of operations and apply the rules for multiplication and division.

Sub-rules and exceptions:

• Multiplication:
  • The product of two integers is an integer.
  • The product of an integer and a fraction is a fraction.
• Division:
  • The quotient of two integers is an integer only when the divisor is a factor of the dividend.
  • The quotient of an integer and a fraction is a fraction.

Visual pattern: Imagine a number line with positive and negative integers. When adding or subtracting integers, move along the number line accordingly.

Exam / Job / Audit Weighting

Frequency: 15-20% Difficulty Rating: Intermediate Question Type or Real-World Task Type: Multiple-choice questions, problem-solving exercises, and case studies.

Difficulty Level

Intermediate

Must-Know Rules, Formulas, Standards, or Principles

1. Addition and Subtraction:
   • When adding or subtracting integers, combine like terms and follow the order of operations.
2. Multiplication:
   • The product of two integers is an integer.
   • The product of an integer and a fraction is a fraction.
3. Division:
   • The quotient of two integers is an integer only when the divisor is a factor of the dividend.
   • The quotient of an integer and a fraction is a fraction.

Worked Examples (Step-by-Step)

Easy

Question: 2 + (-3) = ? Reasoning: Combine like terms and follow the order of operations.
Answer: -1 Key rule applied: Addition of integers

Medium

Question: (-4) × 3 = ? Reasoning: Multiply the integers and follow the order of operations.
Answer: -12 Key rule applied: Multiplication of integers

Hard

Question: (-15) ÷ (-3) = ? Reasoning: Divide the integers and follow the order of operations.
Answer: 5 Key rule applied: Division of integers

Common Exam Traps & Mistakes

1. Mistaking addition and subtraction:
   • Wrong answer: 2 + (-3) = 5
   • Correct approach: Combine like terms and follow the order of operations.
2. Ignoring the order of operations:
   • Wrong answer: (-4) × 3 + 2 = -12 + 2 = -10
   • Correct approach: Follow the order of operations and evaluate the expression step-by-step.
3. Not considering absolute value:
   • Wrong answer: |-15| = 15, but (-15) ÷ (-3) ≠ 15
   • Correct approach: Evaluate the expression using the correct rules and formulas.
4. Not recognizing integer properties:
   • Wrong answer: (-4) × 3 = 12 (incorrect product)
   • Correct approach: Recognize that the product of two integers is an integer.
5. Not following the rules for division:
   • Wrong answer: (-15) ÷ (-3) = 5 (incorrect quotient)
   • Correct approach: Recognize that the quotient of two integers is an integer only when the divisor is a factor of the dividend.

Shortcut Strategies & Exam Hacks

1. Memory aid: Use the acronym PEMDAS/BODMAS to remember the order of operations.
2. Elimination strategy: Eliminate answer choices that violate the rules for integer operations.
3. Pattern recognition: Recognize patterns in integer operations, such as the product of two integers being an integer.

Question-Type Taxonomy

1. Multiple-choice questions: Choose the correct answer from a set of options.
2. Problem-solving exercises: Solve a problem using integer operations.
3. Case studies: Apply integer operations to real-world scenarios.

Practice Set (MCQs)

1. Question: 3 + (-2) = ?
   • Options: A) 1, B) 3, C) 5, D) 7
   • Correct Answer: B) 1
   • Explanation: Combine like terms and follow the order of operations.
   • Why the Distractors Are Tempting: Options A and D are plausible but incorrect, while option C is too large.
2. Question: (-6) × 2 = ?
   • Options: A) -12, B) -8, C) 12, D) 8
   • Correct Answer: A) -12
   • Explanation: Multiply the integers and follow the order of operations.
   • Why the Distractors Are Tempting: Options B and D are plausible but incorrect, while option C is too large.
3. Question: (-9) ÷ (-3) = ?
   • Options: A) 3, B) 9, C) 1, D) -1
   • Correct Answer: A) 3
   • Explanation: Divide the integers and follow the order of operations.
   • Why the Distractors Are Tempting: Options B and D are plausible but incorrect, while option C is too small.
4. Question: 4 + (-5) = ?
   • Options: A) -1, B) 1, C) 3, D) 5
   • Correct Answer: A) -1
   • Explanation: Combine like terms and follow the order of operations.
   • Why the Distractors Are Tempting: Options B and D are plausible but incorrect, while option C is too large.
5. Question: (-8) × (-2) = ?
   • Options: A) 16, B) 8, C) -8, D) -16
   • Correct Answer: A) 16
   • Explanation: Multiply the integers and follow the order of operations.
   • Why the Distractors Are Tempting: Options B and C are plausible but incorrect, while option D is too small.

30-Second Cheat Sheet

• Addition and subtraction: Combine like terms and follow the order of operations.
• Multiplication: The product of two integers is an integer.
• Division: The quotient of two integers is an integer only when the divisor is a factor of the dividend.
• Absolute value: The absolute value of an integer is its distance from zero on the number line.
• Order of operations: PEMDAS/BODMAS
• Integer properties: Recognize patterns and apply rules for integer operations.

Learning Path

1. Beginner foundation: Review basic arithmetic operations, fractions, and decimals.
2. Core rules: Learn the rules for integer operations, including addition, subtraction, multiplication, and division.
3. Practice: Practice integer operations using online resources or worksheets.
4. Timed drills: Practice integer operations under timed conditions to improve speed and accuracy.
5. Mock tests: Take mock tests to assess your understanding and identify areas for improvement.

Related Topics

1. Fractions and decimals: Understanding fractions and decimals is essential for integer operations.
2. Algebraic expressions and equations: Algebraic expressions and equations are used to solve problems involving integer operations.
3. Geometry and trigonometry: Geometry and trigonometry are used to solve problems involving integer operations in real-world scenarios.