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5.4 Percent Calculations, Increase, Decrease, and Applications

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ACCUPLACER Suite Study Guide Exam

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1 Chapter 1: ACCUPLACER Suite Overview & Testing Mechanics

1.1 The ACCUPLACER Testing System & Computer-Adaptive Mechanics 1.2 Placement Policies, Score Scales & Institutional Cut Scores 1.3 Test-Taking Strategies, Tool Mechanics & Pacing

2 Chapter 2: Next-Generation Reading

2.1 The Reading Test Blueprint: Set-Based and Discrete Items 2.2 Information and Ideas: Close Reading & Central Themes 2.3 Summarizing and Understanding Relationships 2.4 Rhetoric, Text Structure & Author's Purpose 2.5 Analyzing Point of View and Authorial Purpose 2.6 Analyzing Arguments: Claims, Counterclaims, and Evidence 2.7 Synthesis & Paired Text Comparative Analysis 2.8 Vocabulary in Context & Figurative Language 2.9 Passage Genre, Content Area, Mode, and Text Complexity 2.10 Answer-Choice Traps and Evidence Discipline

3 Chapter 3: Next-Generation Writing: Expression of Ideas

3.1 The Writing Test Blueprint: Passage Sets and Content Distribution 3.2 Development: Proposition and Supporting Evidence 3.3 Focus: Adding, Retaining, and Deleting for Relevance 3.4 Organization: Logical Sequence and Paragraph Order 3.5 Introductions, Conclusions, and Transitional Cohesion 3.6 Effective Language Use: Precision and Word Choice 3.7 Concision: Eliminating Wordiness and Redundancy 3.8 Style, Tone, and Voice Consistency 3.9 Syntax: Sentence Combining for Rhetorical Purpose

4 Chapter 4: Next-Generation Writing: Standard English Conventions

4.1 Sentence Boundaries: Fragments, Run-ons, and Comma Splices 4.2 Subordination and Coordination 4.3 Parallel Structure 4.4 Modifier Placement: Misplaced and Dangling Modifiers 4.5 Inappropriate Shifts in Verb Tense 4.6 Inappropriate Shifts in Verb Voice and Mood 4.7 Inappropriate Shifts in Pronoun Person and Number 4.8 Subject-Verb Agreement, Pronoun Case, and Verb Forms 4.9 Possessive Determiners, Contractions, and Confusable Adverbs 4.10 Pronoun Clarity and Pronoun-Antecedent Agreement 4.11 Frequently Confused Words, Noun Agreement, and Logical Comparison 4.12 Within-Sentence Punctuation: Semicolons, Colons, Dashes, and Apostrophes 4.13 Items in a Series and Nonrestrictive Elements 4.14 End-of-Sentence Punctuation, Hyphenation, and Unnecessary Punctuation

5 Chapter 5: Next-Generation Arithmetic

5.1 Whole Number Operations, Estimation & Multi-Step Applications 5.2 Fractions, Mixed Numbers & Operations 5.3 Decimal Operations & Place-Value Conversions 5.4 Percent Calculations, Increase, Decrease, and Applications 5.5 Number Comparisons and Equivalents Across Formats

6 Chapter 6: Next-Generation Quantitative Reasoning, Algebra, and Statistics (QAS)

6.1 Rational Numbers, Absolute Value, and Order of Operations 6.2 Exponents, Radicals, and Scientific Notation 6.3 Ratio, Proportional Relationships, and Unit Conversions 6.4 Creating and Evaluating Algebraic Expressions 6.5 Linear Equations, Inequalities, and Systems of Two Equations 6.6 Linear Applications, Functions, and Graphs 6.7 Probability and Sets 6.8 Descriptive Statistics: Center, Spread, and Graphical Displays 6.9 Geometry for Pre-Algebra and Algebra 1

7 Chapter 7: Next-Generation Advanced Algebra, Quadratics & Polynomials

7.1 Advanced Linear Equations, Systems, and Coordinate Geometry 7.2 Factoring Methods for Quadratics, Cubics, and Polynomials 7.3 Quadratic Equations, Vertex Form & Parabolic Functions 7.4 Polynomial Equations, Roots, and Graphs 7.5 Radical and Rational Equations and Functions 7.6 Geometry for Algebra 1 and Algebra 2

8 Chapter 8: Functions, Exponentials, Logarithms & Trigonometry

8.1 Function Notation, Transformations, Compositions & Inverses 8.2 Domain, Range, and Interpreting Functions in Context 8.3 Exponential & Logarithmic Functions, Laws & Modeling 8.4 Right-Triangle Trigonometry, the Unit Circle, and Radian Measure 8.5 Law of Sines, Law of Cosines, and Trigonometric Equations 8.6 Choosing Your Math Test: Arithmetic, QAS, or AAF

9 Chapter 9: WritePlacer & WritePlacer ESL Essays

9.1 WritePlacer Prompt Format, Planning, and the Writing Process 9.2 The WritePlacer Scoring Rubric & 6 Core Dimensions 9.3 Reading the WritePlacer Score Scale: What Each Band Demonstrates 9.4 Essay Architecture: Thesis, Evidence Integration & Structure 9.5 WritePlacer ESL Scoring Standards & Synthesis

10 Chapter 10: ACCUPLACER ESL Placement Suite

10.1 The ESL Placement Suite: Structure, Scoring, and Score Interpretation 10.2 ESL Reading Skills & Passage Comprehension 10.3 ESL Sentence Meaning & Context Clues 10.4 ESL Language Use & Grammatical Mechanics 10.5 ESL Listening Comprehension & Spoken Contexts

5.4 Percent Calculations, Increase, Decrease, and Applications

Key Takeaways Explain This Topic AI

A percent represents a ratio per hundred (x% = x/100 = x × 0.01), requiring seamless three-way conversions between fractions, decimals, and percentages.

The fundamental percent equation Part = Percent (in decimal form) × Whole solves for any unknown variable when two are provided.

Percent change measures relative variation against the original baseline value using the formula Percent Change = (|New - Original| / Original) × 100%.

Real-world financial calculations apply cumulative multipliers for retail discounts, sales tax, tips, commissions, and simple interest (I = Prt).

Rational numbers spanning positive and negative values are compared and ordered by establishing common fractional denominators or standardized decimal expansions on a real number line, with absolute value |x| representing distance from zero.

Last updated: August 2026

5.4 Percents, Rate Applications & Number Comparisons

Core Concept: A percent is a ratio expressing a quantity per 100 100 100 parts (from Latin per centum ). On the ACCUPLACER, you must seamlessly convert between fractions, decimals, and percentages, calculate rates of increase and decrease, solve multi-step commercial finance applications, and evaluate the relative ordering and absolute values of signed rational numbers.

  • 1. The Tri-Modal Equivalence: Fractions, Decimals & Percents
  • Every rational quantity can be expressed in three interchangeable formats: TRI-MODAL CONVERSION PATHWAYS
  • [ FRACTION: a/b ] <========================> [ DECIMAL: d ]
  • ▲ ▲
  • │ │
  • │ [ PERCENT: p% ] │
  • └───────────────────┴────────────────────────┘

• Fraction to Decimal: Divide numerator by denominator (a ÷ b).

• Decimal to Percent: Multiply by 100 and append % (shift decimal 2 places right).

• Percent to Decimal: Divide by 100 and remove % (shift decimal 2 places left).

• Percent to Fraction: Write over 100 and simplify via GCF.

Examples:

Convert 7.5 % 7.5\% 7.5% to decimal: 7.5 ÷ 100 = 0.075 7.5 \div 100 = \mathbf{0.075} 7.5 ÷ 100 = 0.075 .

Convert 0.625 0.625 0.625 to percent: 0.625 × 100 % = 62.5 % 0.625 \times 100\% = \mathbf{62.5\%} 0.625 × 100% = 62.5% .

Convert 7 8 \frac{7}{8} 8 7 ​ to percent: 7 ÷ 8 = 0.875 ⟹ 0.875 × 100 % = 87.5 % 7 \div 8 = 0.875 \implies 0.875 \times 100\% = \mathbf{87.5\%} 7 ÷ 8 = 0.875 ⟹ 0.875 × 100% = 87.5% .

Convert 36 % 36\% 36% to a simplified fraction: 36 100 = 36 ÷ 4 100 ÷ 4 = 9 25 \frac{36}{100} = \frac{36 \div 4}{100 \div 4} = \mathbf{\frac{9}{25}} 100 36 ​ = 100 ÷ 4 36 ÷ 4 ​ = 25 9 ​ .

2. The Core Percent Equation & Three Question Archetypes

All basic percentage problems relate three quantities via the fundamental formula: Part = Percent (in decimal form) × Whole ⟺ Is Of = P 100 \text{Part} = \text{Percent (in decimal form)} \times \text{Whole} \quad \Longleftrightarrow \quad \frac{\text{Is}}{\text{Of}} = \frac{P}{100} Part = Percent (in decimal form) × Whole ⟺ Of Is ​ = 100 P ​ +-------------------------------------------------------------------------------------------------------+

  • | THE THREE PERCENT QUESTION ARCHETYPES |
  • +-------------------+-------------------------------+-----------------------+---------------------------+
  • | Archetype | Question Phrasing | Algebraic Setup | Worked Solution |
  • +-------------------+-------------------------------+-----------------------+---------------------------+
  • | 1. Find the Part | "What is 18% of 450?" | Part = 0.18 × 450 | Part = 81 |
  • | 2. Find Percent | "42 is what percent of 280?" | P = 42 / 280 | P = 0.15 = 15% |
  • | 3. Find the Whole | "78 is 65% of what number?" | Whole = 78 / 0.65 | Whole = 120 |
  • +-------------------+-------------------------------+-----------------------+---------------------------+
  • 3. Percent Increase, Percent Decrease & Successive Percentage Traps
  • The Universal Percent Change Formula

Percent change measures relative variation against the initial (original) baseline value : Percent Change = ∣ New Value − Original Value ∣ Original Value × 100 % = Δ Original × 100 % \text{Percent Change} = \frac{|\text{New Value} - \text{Original Value}|}{\text{Original Value}} \times 100\% = \frac{\Delta}{\text{Original}} \times 100\% Percent Change = Original Value ∣ New Value − Original Value ∣ ​ × 100% = Original Δ ​ × 100%

Critical Exam Rule: The denominator is always the original starting value , never the updated final value.

Worked Example (Percent Decrease): A jacket originally priced at $ 160 \text{\textdollar}160 $ 160 is marked down to $ 112 \text{\textdollar}112 $ 112 . What is the percent decrease? Percent Decrease = 160 − 112 160 × 100 % = 48 160 × 100 % = 0.30 × 100 % = 30 % \text{Percent Decrease} = \frac{160 - 112}{160} \times 100\% = \frac{48}{160} \times 100\% = 0.30 \times 100\% = \mathbf{30\%} Percent Decrease = 160 160 − 112 ​ × 100% = 160 48 ​ × 100% = 0.30 × 100% = 30%

The Successive Percent Trap

Percentages cannot be added directly across successive steps because the base changes after each step.

Question: If a stock increases by 20 % 20\% 20% on Monday and decreases by 20 % 20\% 20% on Tuesday, what is the net percent change from its initial price?

Solution: Let the initial price be $ 100 \text{\textdollar}100 $ 100 .

After Monday ( + 20 % +20\% + 20% ): $ 100 × 1.20 = $ 120 \text{\textdollar}100 \times 1.20 = \text{\textdollar}120 $ 100 × 1.20 = $ 120 .

After Tuesday ( − 20 % -20\% − 20% ): $ 120 × ( 1 − 0.20 ) = $ 120 × 0.80 = $ 96 \text{\textdollar}120 \times (1 - 0.20) = \text{\textdollar}120 \times 0.80 = \text{\textdollar}96 $ 120 × ( 1 − 0.20 ) = $ 120 × 0.80 = $ 96 .

Net change: 96 − 100 100 × 100 % = − 4 % \frac{96 - 100}{100} \times 100\% = \mathbf{-4\%} 100 96 − 100 ​ × 100% = − 4% (a net 4 % 4\% 4% decrease, not 0 % 0\% 0% ).

4. Commercial & Financial Arithmetic Applications +-------------------------------------------------------------------------------------------------------+

  • | COMMERCIAL ARITHMETIC FORMULA MATRIX |
  • +-----------------------+---------------------------------------+---------------------------------------+
  • | Application | Mathematical Formula | Multiplier Format |
  • +-----------------------+---------------------------------------+---------------------------------------+
  • | Sales Tax & Total | Total = Base Price + (Price × Tax) | Total = Base × (1 + r_tax) |
  • | Retail Discount | Sale Price = Retail - (Retail × Disc) | Sale Price = Retail × (1 - r_disc) |
  • | Sales Commission | Commission = Sales Volume × Rate | Comm = Sales × r_comm |
  • | Simple Interest | Interest = Principal × Rate × Time | I = P · r · t (t in years) |
  • | Total Balance (Loans) | Total Accumulated Amount = P + I | A = P(1 + rt) |
  • +-----------------------+---------------------------------------+---------------------------------------+

Worked Example (Simple Interest): A student deposits $ 2 , 400 \text{\textdollar}2,400 $ 2 , 400 into a fixed certificate of deposit earning 4.5 % 4.5\% 4.5% simple annual interest for a period of 30 30 30 months. What is the total accumulated balance at maturity?

Identify parameters: P = $ 2 , 400 P = \text{\textdollar}2,400 P = $ 2 , 400 , r = 0.045 r = 0.045 r = 0.045 , t = 30 12 = 2.5 t = \frac{30}{12} = 2.5 t = 12 30 ​ = 2.5 years.

Compute simple interest: I = P ⋅ r ⋅ t = 2 , 400 × 0.045 × 2.5 I = P \cdot r \cdot t = 2,400 \times 0.045 \times 2.5 I = P ⋅ r ⋅ t = 2 , 400 × 0.045 × 2.5 2 , 400 × 0.045 = 108 ⟹ 108 × 2.5 = 270 2,400 \times 0.045 = 108 \implies 108 \times 2.5 = 270 2 , 400 × 0.045 = 108 ⟹ 108 × 2.5 = 270 I = $ 270 I = \$270 I = $270

Total accumulated balance: A = P + I = $ 2 , 400 + $ 270 = $ 2 , 670 A = P + I = \$2,400 + \$270 = \mathbf{\$2,670} A = P + I = $2 , 400 + $270 = $2 , 670

  • 5. Comparing and Ordering Rational Numbers & Absolute Value
  • Number Line Geometry & Signed Comparisons

On a real number line, values increase from left to right. For negative numbers, a larger numerical magnitude represents a value further to the left (i.e., smaller): − 0.8 < − 0.6 -0.8 < -0.6 − 0.8 < − 0.6 . <---|-------|-------|-------|-------|-------|-------|-------|-------|--->

  • 1.0 -0.8 -0.6 -0.4 -0.2 0.0 +0.2 +0.4 +0.6
  • Systematic Protocol for Ordering Mixed Rational Numbers

Convert all numbers (fractions, mixed numbers, percentages) into standard decimal format to equal decimal precision.

Separate negative values from positive values.

Order negative values from most negative (smallest) to least negative.

Order positive values from smallest to largest.

Worked Example: Order from least to greatest: − 5 8 , − 0.65 , 38 % , 1 3 , − 0.6 , 0.35 -\frac{5}{8}, \; -0.65, \; 38\%, \; \frac{1}{3}, \; -0.6, \; 0.35 − 8 5 ​ , − 0.65 , 38% , 3 1 ​ , − 0.6 , 0.35 .

Convert all elements to decimals (3 decimal places):

  • − 5 8 = − 0.625 -\frac{5}{8} = -0.625 − 8 5 ​ = − 0.625
  • − 0.65 = − 0.650 -0.65 = -0.650 − 0.65 = − 0.650
  • 38 % = + 0.380 38\% = +0.380 38% = + 0.380

1 3 = + 0.333... \frac{1}{3} = +0.333... 3 1 ​ = + 0.333...

  • − 0.6 = − 0.600 -0.6 = -0.600 − 0.6 = − 0.600
  • 0.35 = + 0.350 0.35 = +0.350 0.35 = + 0.350

Order negative numbers: − 0.650 < − 0.625 < − 0.600 ⟹ − 0.65 < − 5 8 < − 0.6 -0.650 < -0.625 < -0.600 \implies -0.65 < -\frac{5}{8} < -0.6 − 0.650 < − 0.625 < − 0.600 ⟹ − 0.65 < − 8 5 ​ < − 0.6 .

Order positive numbers: + 0.333... < + 0.350 < + 0.380 ⟹ 1 3 < 0.35 < 38 % +0.333... < +0.350 < +0.380 \implies \frac{1}{3} < 0.35 < 38\% + 0.333... < + 0.350 < + 0.380 ⟹ 3 1 ​ < 0.35 < 38% .

Unified ascending order: − 0.65 < − 5 8 < − 0.6 < 1 3 < 0.35 < 38 % \mathbf{-0.65 < -\frac{5}{8} < -0.6 < \frac{1}{3} < 0.35 < 38\%} − 0.65 < − 8 5 ​ < − 0.6 < 3 1 ​ < 0.35 < 38%

Absolute Value Properties

The absolute value ∣ x ∣ |x| ∣ x ∣ represents the non-negative geometric distance of a number x x x from 0 0 0 on the number line:

  • ∣ x ∣ = x |x| = x ∣ x ∣ = x if x ≥ 0 x \ge 0 x ≥ 0
  • ∣ x ∣ = − x |x| = -x ∣ x ∣ = − x if x < 0 x < 0 x < 0
  • ∣ a − b ∣ = ∣ b − a ∣ |a - b| = |b - a| ∣ a − b ∣ = ∣ b − a ∣ (distance between points a a a and b b b )