GED Mathematical Reasoning Quantitative Reasoning Fractions Add Subtract Multiply Divide Mixed Numbers

What Is This?

Quantitative Reasoning — Fractions: Add, Subtract, Multiply, Divide — Mixed Numbers is the ability to manipulate and solve problems involving fractions and mixed numbers with speed and accuracy. This topic appears in exams to assess your understanding of fundamental mathematical operations and your ability to apply them in various contexts.

Why It Matters

This topic is commonly tested in exams such as the GMAT, GRE, and SAT, and carries a significant weightage of 20-30% of the total marks. The examiner is looking for your ability to apply mathematical rules and formulas to solve problems, rather than just recalling formulas or procedures.

Core Concepts

To master this topic, you must own the following foundational ideas:

• Equivalent Fractions: Fractions that have the same value but different numerators and denominators.
• Mixed Numbers: A combination of a whole number and a fraction.
• Operations with Fractions: Adding, subtracting, multiplying, and dividing fractions, including mixed numbers.

You must be able to distinguish between equivalent fractions, mixed numbers, and improper fractions, and apply the correct operations to solve problems.

Prerequisites

Before tackling this topic, you must already understand:

• Basic arithmetic operations (addition, subtraction, multiplication, and division)
• Fractions and decimals
• Ratios and proportions

If you are missing these prerequisites, you will struggle to understand the rules and formulas for this topic.

The Rule-Book (How It Works)

Adding Fractions

• The primary rule is: Add fractions with the same denominator.
• Sub-rule: If the fractions have different denominators, find the least common multiple (LCM) of the denominators.
• Exception: When adding mixed numbers, convert the whole number to an improper fraction and add the numerators.

Subtracting Fractions

• The primary rule is: Subtract fractions with the same denominator.
• Sub-rule: If the fractions have different denominators, find the least common multiple (LCM) of the denominators.
• Exception: When subtracting mixed numbers, convert the whole number to an improper fraction and subtract the numerators.

Multiplying Fractions

• The primary rule is: Multiply the numerators and denominators separately.
• Sub-rule: When multiplying mixed numbers, multiply the whole number by the numerator and add the product to the numerator.

Dividing Fractions

• The primary rule is: Invert the second fraction and multiply.
• Sub-rule: When dividing mixed numbers, convert the whole number to an improper fraction and invert the second fraction.

Visual Pattern

Remember the phrase "Same Denominator, Same Way" to help you add and subtract fractions.

Exam / Job / Audit Weighting

Difficulty Level

Intermediate

Must-Know Rules, Formulas, Standards, or Principles

1. Adding Fractions: Add fractions with the same denominator.
2. Subtracting Fractions: Subtract fractions with the same denominator.
3. Multiplying Fractions: Multiply the numerators and denominators separately.

Worked Examples (Step-by-Step)

Example 1: Easy

Question: 1/2 + 1/4 = ?

Reasoning process:

• Find the least common multiple (LCM) of 2 and 4, which is 4.
• Convert 1/2 to 2/4.
• Add the fractions: 2/4 + 1/4 = 3/4.

Answer: 3/4

Example 2: Medium

Question: 3/4 - 1/6 = ?

Reasoning process:

• Find the least common multiple (LCM) of 4 and 6, which is 12.
• Convert 3/4 to 9/12 and 1/6 to 2/12.
• Subtract the fractions: 9/12 - 2/12 = 7/12.

Answer: 7/12

Example 3: Hard

Question: (3/4 - 1/6) × (2/3) = ?

Reasoning process:

• Evaluate the expression inside the parentheses: (3/4 - 1/6) = 7/12.
• Multiply the fractions: (7/12) × (2/3) = 14/36.
• Simplify the fraction: 14/36 = 7/18.

Answer: 7/18

Common Exam Traps & Mistakes

1. Mistaking Equivalent Fractions: Failing to recognize that 1/2 and 2/4 are equivalent fractions.
2. Incorrectly Subtracting Fractions: Subtracting fractions with different denominators without finding the LCM.
3. Forgetting to Invert the Second Fraction: Failing to invert the second fraction when dividing fractions.
4. Not Simplifying the Fraction: Failing to simplify the fraction after performing the operation.

Shortcut Strategies & Exam Hacks

1. Mnemonic Device: Remember the phrase "Same Denominator, Same Way" to help you add and subtract fractions.
2. Elimination Strategy: Eliminate options that are clearly incorrect and focus on the remaining options.
3. Pattern Recognition: Recognize patterns in the numerators and denominators to simplify the calculation.

Question-Type Taxonomy

Practice Set (MCQs)

1. Question: 1/2 + 1/4 = ?

A) 1/4 B) 3/4 C) 1/2 D) 2/4

Correct Answer: B) 3/4

Explanation: Add fractions with the same denominator.

Why the Distractors Are Tempting:

• A) 1/4 is a plausible answer, but it is incorrect.
• C) 1/2 is a plausible answer, but it is incorrect.

• D) 2/4 is a plausible answer, but it is incorrect.

• Question: 3/4 - 1/6 = ?

A) 1/2 B) 7/12 C) 5/6 D) 3/4

Correct Answer: B) 7/12

Explanation: Subtract fractions with the same denominator and simplify the fraction.

Why the Distractors Are Tempting:

• A) 1/2 is a plausible answer, but it is incorrect.
• C) 5/6 is a plausible answer, but it is incorrect.

• D) 3/4 is a plausible answer, but it is incorrect.

• Question: (3/4 - 1/6) × (2/3) = ?

A) 7/18 B) 5/12 C) 3/4 D) 2/3

Correct Answer: A) 7/18

Explanation: Evaluate the expression inside the parentheses, multiply the fractions, and simplify the fraction.

Why the Distractors Are Tempting:

• B) 5/12 is a plausible answer, but it is incorrect.
• C) 3/4 is a plausible answer, but it is incorrect.

• D) 2/3 is a plausible answer, but it is incorrect.

• Question: 1/2 + 1/4 = ?

A) 3/4 B) 2/4 C) 1/2 D) 1/4

Correct Answer: A) 3/4

Explanation: Add fractions with the same denominator.

Why the Distractors Are Tempting:

• B) 2/4 is a plausible answer, but it is incorrect.
• C) 1/2 is a plausible answer, but it is incorrect.

• D) 1/4 is a plausible answer, but it is incorrect.

• Question: 3/4 - 1/6 = ?

A) 7/12 B) 5/6 C) 3/4 D) 1/2

Correct Answer: A) 7/12

Explanation: Subtract fractions with the same denominator and simplify the fraction.

Why the Distractors Are Tempting:

• B) 5/6 is a plausible answer, but it is incorrect.
• C) 3/4 is a plausible answer, but it is incorrect.
• D) 1/2 is a plausible answer, but it is incorrect.

30-Second Cheat Sheet

• Add fractions with the same denominator.
• Subtract fractions with the same denominator.
• Multiply fractions by multiplying the numerators and denominators separately.
• Divide fractions by inverting the second fraction and multiplying.
• Simplify fractions after performing the operation.
• Use the least common multiple (LCM) to add and subtract fractions with different denominators.
• Remember the phrase "Same Denominator, Same Way" to help you add and subtract fractions.

Learning Path

1. Begin by reviewing the basics of fractions and decimals.
2. Learn the rules for adding and subtracting fractions with the same denominator.
3. Practice adding and subtracting fractions with different denominators using the least common multiple (LCM).
4. Learn the rules for multiplying and dividing fractions.
5. Practice multiplying and dividing fractions using the rules.
6. Review the rules and practice problems to reinforce your understanding.
7. Take timed drills and mock tests to simulate the exam experience.

Related Topics

1. Decimals: Understanding decimals is essential for working with fractions.
2. Ratios and Proportions: Understanding ratios and proportions is essential for working with fractions.
3. Percentages: Understanding percentages is essential for working with fractions in real-world applications.