GED Mathematical Reasoning Quantitative Reasoning Percentages Find Percent Percent of a Number Percent Change

What Is This?

Percentages: Find Percent, Percent of a Number, Percent Change is the ability to calculate and apply percentages in various contexts. This topic is essential for understanding how to express a proportion of a whole as a percentage and how to calculate changes in quantities using percentages.

Why It Matters

This topic appears in various exams, including math, finance, and business exams, and is a crucial skill for professionals in these fields. It typically carries a moderate to high weightage, around 20-30% of the total marks, and is often tested in the form of numerical problems, case studies, or scenario-based questions. The examiner is testing your ability to apply mathematical concepts to real-world situations, understand the implications of percentage changes, and make informed decisions.

Core Concepts

To master this topic, you need to understand the following core concepts:

• Percent: A proportion of a whole, expressed as a value out of 100.
• Percent of a Number: A calculation that involves finding a percentage of a given number.
• Percent Change: A calculation that involves finding the difference between two quantities, expressed as a percentage.

You need to be able to distinguish between these concepts and apply them correctly in different contexts.

Prerequisites

Before tackling this topic, you need to have a solid understanding of:

• Basic arithmetic operations, including addition, subtraction, multiplication, and division.
• Fractions and decimals.
• Ratios and proportions.

If you are missing these prerequisites, you may struggle to understand the underlying concepts and apply them correctly.

The Rule-Book (How It Works)

The Primary Rule: To find a percentage, divide the value by the total and multiply by 100.

Sub-Rules:

• To find a percentage of a number, multiply the number by the percentage.
• To find the percentage change, subtract the original value from the new value, divide by the original value, and multiply by 100.

Exceptions and Edge Cases:

• When dealing with negative percentages, remember that they represent a decrease.
• When dealing with zero, remember that it represents a percentage of zero.

Simple Visual Pattern or Mnemonic: Imagine a pie chart, where the percentage represents the proportion of the pie.

Exam / Job / Audit Weighting

Frequency: 30-40% Difficulty Rating: Intermediate Question Type or Real-World Task Type: Numerical problems, case studies, scenario-based questions

Difficulty Level

Intermediate

Must-Know Rules, Formulas, Standards, or Principles

The following rules and formulas are essential for this topic:

• Percent Formula: (Value ÷ Total) × 100
• Percent of a Number Formula: (Number × Percentage) ÷ 100
• Percent Change Formula: ((New Value - Original Value) ÷ Original Value) × 100

Worked Examples (Step-by-Step)

Example 1: Easy

What is 25% of 100? * Step 1: Multiply 100 by 25% (0.25) * Step 2: Calculate 100 × 0.25 = 25 * Answer: 25

Example 2: Medium

A shirt is originally priced at $50. It is discounted by 20%. What is the new price? * Step 1: Calculate the discount: (20% of $50) = (0.20 × $50) = $10 * Step 2: Subtract the discount from the original price: $50 - $10 = $40 * Answer: $40

Example 3: Hard

A company's revenue increases by 15% from $100,000 to $115,000. What is the percentage increase? * Step 1: Calculate the increase: $115,000 - $100,000 = $15,000 * Step 2: Divide the increase by the original value and multiply by 100: ($15,000 ÷ $100,000) × 100 = 15% * Answer: 15%

Common Exam Traps & Mistakes

Trap 1: Incorrect Calculation

• Mistake: (20% of $50) = (0.20 × $50) = $10 ( incorrect calculation)
• Correct Approach: (20% of $50) = (0.20 × $50) = $10 ( correct calculation, but incorrect application)
• Trap 2: Misinterpreting Negative Percentages
• Mistake: A negative percentage represents an increase ( incorrect interpretation)
• Correct Approach: A negative percentage represents a decrease ( correct interpretation)

Trap 3: Forgetting to Multiply by 100

• Mistake: (25% of 100) = 25 ( forgetting to multiply by 100)
• Correct Approach: (25% of 100) = 25 × 100 = 2500

Trap 4: Incorrectly Applying the Percent Change Formula

• Mistake: ((New Value - Original Value) ÷ Original Value) × 100 = ((115,000 - 100,000) ÷ 100,000) × 100 = 15% ( incorrect application)
• Correct Approach: ((New Value - Original Value) ÷ Original Value) × 100 = ((115,000 - 100,000) ÷ 100,000) × 100 = 15% ( correct application)

Trap 5: Misinterpreting the Percent Formula

• Mistake: (Value ÷ Total) × 100 = (100 ÷ 50) × 100 = 200 ( incorrect interpretation)
• Correct Approach: (Value ÷ Total) × 100 = (100 ÷ 50) × 100 = 200 ( correct interpretation, but incorrect application)

Shortcut Strategies & Exam Hacks

Memory Aid: The "3-Step" Formula

To find a percentage, divide the value by the total, multiply by 100, and simplify.

Elimination Strategy: Eliminate Impossible Answers

If an answer choice is clearly incorrect, eliminate it and focus on the remaining options.

Pattern Recognition Tip: Recognize the Percent Formula

The percent formula is a simple division and multiplication operation.

Question-Type Taxonomy

Type 1: Numerical Problems

• Example: What is 25% of 100?
• Exams that favor this type: Math, Finance, Business

Type 2: Case Studies

• Example: A shirt is originally priced at $50. It is discounted by 20%. What is the new price?
• Exams that favor this type: Business, Finance

Type 3: Scenario-Based Questions

• Example: A company's revenue increases by 15% from $100,000 to $115,000. What is the percentage increase?
• Exams that favor this type: Business, Finance

Type 4: Multiple-Choice Questions

• Example: What is the value of x in the equation (x ÷ 100) × 100 = 25?
• Exams that favor this type: Math, Finance, Business

Practice Set (MCQs)

Question 1: Easy

What is 25% of 100? * A) 20 * B) 25 * C) 50 * D) 75 * Correct Answer: B) 25 * Explanation: Multiply 100 by 25% (0.25) and simplify.
* Why the Distractors Are Tempting: A) 20 is close to 25, but incorrect; C) 50 is half of 100, but incorrect; D) 75 is three-quarters of 100, but incorrect.

Question 2: Medium

A shirt is originally priced at $50. It is discounted by 20%. What is the new price? * A) $30 * B) $40 * C) $50 * D) $60 * Correct Answer: B) $40 * Explanation: Calculate the discount: (20% of $50) = (0.20 × $50) = $10; subtract the discount from the original price: $50 - $10 = $40.
* Why the Distractors Are Tempting: A) $30 is too low; C) $50 is the original price; D) $60 is too high.

Question 3: Hard

A company's revenue increases by 15% from $100,000 to $115,000. What is the percentage increase? * A) 10% * B) 12.5% * C) 15% * D) 20% * Correct Answer: C) 15% * Explanation: Calculate the increase: $115,000 - $100,000 = $15,000; divide the increase by the original value and multiply by 100: ($15,000 ÷ $100,000) × 100 = 15%.
* Why the Distractors Are Tempting: A) 10% is too low; B) 12.5% is close to 15%, but incorrect; D) 20% is too high.

Question 4: Easy

What is the value of x in the equation (x ÷ 100) × 100 = 25? * A) 20 * B) 25 * C) 50 * D) 75 * Correct Answer: B) 25 * Explanation: Simplify the equation: x = 25.
* Why the Distractors Are Tempting: A) 20 is close to 25, but incorrect; C) 50 is half of 100, but incorrect; D) 75 is three-quarters of 100, but incorrect.

Question 5: Medium

A shirt is originally priced at $50. It is discounted by 20%. What is the discount amount? * A) $5 * B) $10 * C) $15 * D) $20 * Correct Answer: B) $10 * Explanation: Calculate the discount: (20% of $50) = (0.20 × $50) = $10.
* Why the Distractors Are Tempting: A) $5 is too low; C) $15 is too high; D) $20 is too high.

30-Second Cheat Sheet

• Percent Formula: (Value ÷ Total) × 100
• Percent of a Number Formula: (Number × Percentage) ÷ 100
• Percent Change Formula: ((New Value - Original Value) ÷ Original Value) × 100
• Negative Percentages: Represent a decrease
• Zero: Represents a percentage of zero
• Multiplying by 100: Essential for finding percentages

Learning Path

1. Beginner Foundation: Understand basic arithmetic operations, fractions, and decimals.
2. Core Rules: Learn the percent formula, percent of a number formula, and percent change formula.
3. Practice: Practice solving numerical problems, case studies, and scenario-based questions.
4. Timed Drills: Practice solving questions under timed conditions.
5. Mock Tests: Take mock tests to assess your knowledge and identify areas for improvement.

Related Topics

• Fractions and Decimals: Essential for understanding percentages.
• Ratios and Proportions: Essential for understanding percentages.
• Financial Math: Includes topics such as interest rates, compound interest, and annuities.