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12.3 Fractions, Decimals, Percentages, and Proportional Problem Solving

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1 Chapter 1: The CSET Program and the Multiple Subjects Examination

1.1 The CSET Program: Which Tests a California Teacher Actually Needs 1.2 CSET Multiple Subjects: Test Structure, Scoring, and Test-Day Mechanics

2 Chapter 2: Language Structure, Linguistics, and Non-Written Communication

2.1 Linguistics and Language Structure: Phonology, Morphology, Syntax, Semantics, and Pragmatics 2.2 First and Second Language Acquisition and ELD Instruction 2.3 Non-Written Communication, Listening, Speaking, and Media Analysis

3 Chapter 3: Literacy Development, Reading Comprehension, and Text Complexity

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5 Chapter 5: Ancient Civilizations and Medieval World History

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6 Chapter 6: United States History and Government

6.1 Indigenous North America, Colonial Settlement, and the American Revolution 6.2 Constitutional Foundations, Federalism, and the American System of Government 6.3 Westward Expansion, Sectional Conflict, the Civil War, and Reconstruction 6.4 Industrialization, the Progressive Era, World Wars, and the Civil Rights Movement

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8 Chapter 8: Social Science Skills: Geography, Civics, Economics, and Source Analysis

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9 Chapter 9: Scientific Practices and Physical Science

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10 Chapter 10: Life Science and Ecology

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11 Chapter 11: Earth, Water, and Space Science

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12 Chapter 12: Number Sense, Operations, and Fractions

12.1 Number Classifications, Place Value, Properties of Operations, and Integers 12.2 Number Theory, Prime Factorization, GCF/LCM, and Exponents 12.3 Fractions, Decimals, Percentages, and Proportional Problem Solving

13 Chapter 13: Algebra, Functions, and Geometry

13.1 Ratios, Rates, Proportions, and Scaling Relationships 13.2 Algebraic Expressions, Linear Equations, Systems, and Functions 13.3 Geometric Properties, Two-Dimensional Polygons, and Three-Dimensional Solids

14 Chapter 14: Coordinate Geometry, Measurement, Statistics, and Mathematical Practices

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18 Chapter 18: Visual Art, Arts Integration, and Constructed-Response Mastery

18.1 Elements of Art, Principles of Design, and Visual Analysis 18.2 Art Media, Techniques, World Art History, and Art Criticism 18.3 California Arts Standards and Cross-Curricular Arts Integration 18.4 CSET Constructed-Response Strategy, Scoring Scale, and Domain-Specific Exemplars

12.3 Fractions, Decimals, Percentages, and Proportional Problem Solving

Key Takeaways Explain This Topic AI

Fractions embody four distinct conceptual constructs: Part-Whole (area/set), Measurement (iteration on a number line), Division/Quotient ( a / b = a ÷ b a/b = a \div b a / b = a ÷ b ), and Ratio/Operator (multiplicative scaling).

Fraction multiplication ( a b × c d = a c b d \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} b a ​ × d c ​ = b d a c ​ ) is visually represented by 2D area grid overlays, while fraction division is conceptually rooted in quotitive (measurement) and partitive (sharing) models and mathematically justified by the invert-and-multiply reciprocal algorithm.

A fully simplified fraction a b \frac{a}{b} b a ​ produces a terminating decimal if and only if the prime factors of denominator b b b consist exclusively of 2 2 2 s and/or 5 5 5 s ( b = 2 m 5 n b = 2^m 5^n b = 2 m 5 n ); otherwise, it produces a repeating periodic decimal.

Tri-way rational conversions link fractions, decimals, and percentages ( p % = p / 100 = 0.01 p p\% = p/100 = 0.01p p % = p /100 = 0.01 p ) through benchmark equivalencies such as 1 8 = 0.125 = 12.5 % \frac{1}{8} = 0.125 = 12.5\% 8 1 ​ = 0.125 = 12.5% and 1 3 = 0. 3 ‾ = 33. 3 ‾ % \frac{1}{3} = 0.\overline{3} = 33.\overline{3}\% 3 1 ​ = 0. 3 = 33. 3 % .

Proportional problem solving utilizes the percent proportion ( Part Whole = Percent 100 \frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100} Whole Part ​ = 100 Percent ​ ), percent change formula, and financial applications including discount, sales tax, commission, and simple interest ( I = P r t I = Prt I = P r t ).

Last updated: August 2026

12.3 Fractions, Decimals, Percentages, and Proportional Problem Solving

CSET Focus: Rational number reasoning constitutes the core of upper-elementary and middle-school mathematics on CSET Subtest II. Candidates must master conceptual representations (area models, fraction strips, and number line iterations), provide mathematical justifications for arithmetic algorithms (such as 'invert and multiply' for fraction division), determine why certain decimals terminate while others repeat, and solve complex multi-step percent, discount, and simple interest problems.

1. Conceptual Meanings and Models of Fractions

A fraction a b \frac{a}{b} b a ​ (where a , b ∈ Z , b ≠ 0 a, b \in \mathbb{Z}, b \ne 0 a , b ∈ Z , b  = 0 ) represents multiple interconnected conceptual constructs in elementary mathematics: THE FOUR FRACTION CONSTRUCTS

  • ┌──────────────────┬──────────────────┬──────────────────┐
  • Part-Whole Measurement Quotient Operator
  • (Area & Set: (Distance on a (Equal Sharing: (Multiplicative

a equal parts number line: a divided by b, Scaling:

  • out of b total) a iterations of a/b = a ÷ b) a/b of a quantity)
  • unit fraction 1/b)
  • The Four Core Fraction Constructs

Part-Whole Construct: A whole region or discrete set is partitioned into b b b equal-sized parts, and a a a parts are selected.

Area Model: A rectangle or circle divided into equal areas (e.g., shading 3 3 3 out of 4 4 4 equal quadrants of a circle).

Set Model: A discrete collection of distinct objects partitioned into subsets (e.g., 3 3 3 red apples in a basket of 5 5 5 total apples represents 3 5 \frac{3}{5} 5 3 ​ of the set).

Measurement Construct: A fraction represents a specific distance from zero on a continuous number line. The fraction a b \frac{a}{b} b a ​ is formed by iterating the unit fraction 1 b \frac{1}{b} b 1 ​ exactly a a a times along the coordinate axis.

Quotient (Division) Construct: The fraction a b \frac{a}{b} b a ​ is the exact numerical value obtained when a a a units are divided equally among b b b recipients ( a ÷ b a \div b a ÷ b ). For example, dividing 3 3 3 pizzas equally among 4 4 4 students gives each student 3 4 \frac{3}{4} 4 3 ​ of a pizza.

Ratio / Operator Construct: The fraction functions as a multiplicative transformation or scaling factor (e.g., finding 2 3 \frac{2}{3} 3 2 ​ of a $90 budget = 2 3 × 90 = $ 60 \frac{2}{3} \times 90 = \text{\textdollar}60 3 2 ​ × 90 = $ 60 ).

Types of Fractions and Simplification

Proper Fraction: Numerator is strictly less than denominator ( ∣ a ∣ < ∣ b ∣ |a| < |b| ∣ a ∣ < ∣ b ∣ , value < 1 < 1 < 1 ).

Improper Fraction: Numerator is greater than or equal to denominator ( ∣ a ∣ ≥ ∣ b ∣ |a| \ge |b| ∣ a ∣ ≥ ∣ b ∣ , value ≥ 1 \ge 1 ≥ 1 ).

Mixed Number: A combination of a non-zero whole number and a proper fraction ( W a b = W + a b = W ⋅ b + a b W \frac{a}{b} = W + \frac{a}{b} = \frac{W \cdot b + a}{b} W b a ​ = W + b a ​ = b W ⋅ b + a ​ ).

Simplest Form (Lowest Terms): A fraction a b \frac{a}{b} b a ​ is in simplest form if and only if gcd ⁡ ( ∣ a ∣ , ∣ b ∣ ) = 1 \gcd(|a|, |b|) = 1 g cd ( ∣ a ∣ , ∣ b ∣ ) = 1 (the numerator and denominator are relatively prime/coprime). Fractions are simplified by dividing both numerator and denominator by gcd ⁡ ( a , b ) \gcd(a,b) g cd ( a , b ) : a b = a ÷ gcd ⁡ ( a , b ) b ÷ gcd ⁡ ( a , b ) \frac{a}{b} = \frac{a \div \gcd(a,b)}{b \div \gcd(a,b)} b a ​ = b ÷ g cd ( a , b ) a ÷ g cd ( a , b ) ​

  • 2. Fraction Arithmetic and Visual Pedagogical Representations
  • Addition and Subtraction of Fractions

Like Denominators: Combine numerators directly while keeping the common denominator constant: a d ± b d = a ± b d \frac{a}{d} \pm \frac{b}{d} = \frac{a \pm b}{d} d a ​ ± d b ​ = d a ± b ​ .

Unlike Denominators: Rename fractions into equivalent forms using the Least Common Denominator (LCD) , which is the LCM \text{LCM} LCM of the denominators: a b + c d = a ⋅ LCM ( b , d ) b LCM ( b , d ) + c ⋅ LCM ( b , d ) d LCM ( b , d ) = a d + b c b d \frac{a}{b} + \frac{c}{d} = \frac{a \cdot \frac{\text{LCM}(b,d)}{b}}{\text{LCM}(b,d)} + \frac{c \cdot \frac{\text{LCM}(b,d)}{d}}{\text{LCM}(b,d)} = \frac{ad + bc}{bd} b a ​ + d c ​ = LCM ( b , d ) a ⋅ b LCM ( b , d ) ​ ​ + LCM ( b , d ) c ⋅ d LCM ( b , d ) ​ ​ = b d a d + b c ​

Mixed Number Regrouping: When subtracting mixed numbers with regrouping (e.g., 5 1 4 − 2 3 4 5\frac{1}{4} - 2\frac{3}{4} 5 4 1 ​ − 2 4 3 ​ ), borrow 1 1 1 from the whole number and convert it into 4 4 \frac{4}{4} 4 4 ​ : 5 1 4 = 4 + 1 + 1 4 = 4 + 4 4 + 1 4 = 4 5 4 ⟹ 4 5 4 − 2 3 4 = 2 2 4 = 2 1 2 5\frac{1}{4} = 4 + 1 + \frac{1}{4} = 4 + \frac{4}{4} + \frac{1}{4} = 4\frac{5}{4} \implies 4\frac{5}{4} - 2\frac{3}{4} = 2\frac{2}{4} = 2\frac{1}{2} 5 4 1 ​ = 4 + 1 + 4 1 ​ = 4 + 4 4 ​ + 4 1 ​ = 4 4 5 ​ ⟹ 4 4 5 ​ − 2 4 3 ​ = 2 4 2 ​ = 2 2 1 ​

Multiplication of Fractions & Visual Area Models

Multiplying fractions is conceptually interpreted as finding a fractional part of another fraction (e.g., 2 3 × 4 5 \frac{2}{3} \times \frac{4}{5} 3 2 ​ × 5 4 ​ means finding 2 3 \frac{2}{3} 3 2 ​ of an area that is 4 5 \frac{4}{5} 5 4 ​ of a unit square). Area Grid Overlay Model: 2/3 × 4/5

  • ┌───────┬───────┬───────┬───────┬───────┐
  • Row 1 │ █████ │ █████ │ █████ │ █████ │ │ <-- Shaded 2 rows
  • ├───────┼───────┼───────┼───────┼───────┤ across
  • Row 2 │ █████ │ █████ │ █████ │ █████ │ │
  • ├───────┼───────┼───────┼───────┼───────┤
  • Row 3 │ │ │ │ │ │
  • └───────┴───────┴───────┴───────┴───────┘
  • Col 1 Col 2 Col 3 Col 4 Col 5
  • <───── 4 out of 5 columns shaded ─────>

• Unit square partitioned into 3 × 5 = 15 total equal sub-rectangles.

• Double-shaded intersection contains 2 × 4 = 8 sub-rectangles.

• Total Area = 8/15.

Algorithm: Multiply numerators together and denominators together, cross-simplifying common factors before multiplying: a b × c d = a ⋅ c b ⋅ d \frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d} b a ​ × d c ​ = b ⋅ d a ⋅ c ​

  • Division of Fractions and Mathematical Proofs
  • Conceptual Models of Division

Measurement (Quotitive) Division: "How many groups of size c d \frac{c}{d} d c ​ are contained in a b \frac{a}{b} b a ​ ?"

Example: 2 ÷ 1 4 2 \div \frac{1}{4} 2 ÷ 4 1 ​ means "How many 1 4 \frac{1}{4} 4 1 ​ -sized pieces are in 2 2 2 wholes?" Visualizing 2 2 2 strips divided into fourths yields 8 8 8 fourths ( 2 ÷ 1 4 = 8 2 \div \frac{1}{4} = 8 2 ÷ 4 1 ​ = 8 ).

Partitive (Sharing) Division: "If c d \frac{c}{d} d c ​ of a group equals a b \frac{a}{b} b a ​ , what is the value of one whole group?"

Mathematical Justifications for the "Invert and Multiply" Algorithm

Why does a b ÷ c d = a b × d c \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} b a ​ ÷ d c ​ = b a ​ × c d ​ ?

Proof 1: Common Denominator Approach Convert both fractions to have a common denominator b d bd b d : a b ÷ c d = a d b d ÷ b c b d = a d ÷ b c b d ÷ b d = a d b c 1 = a d b c = a b × d c \frac{a}{b} \div \frac{c}{d} = \frac{ad}{bd} \div \frac{bc}{bd} = \frac{ad \div bc}{bd \div bd} = \frac{\frac{ad}{bc}}{1} = \frac{ad}{bc} = \frac{a}{b} \times \frac{d}{c} b a ​ ÷ d c ​ = b d a d ​ ÷ b d b c ​ = b d ÷ b d a d ÷ b c ​ = 1 b c a d ​ ​ = b c a d ​ = b a ​ × c d ​

Proof 2: Complex Fraction Multiplicative Inverse Approach Express the division as a complex fraction and multiply both the numerator and the denominator by the reciprocal of the denominator ( d c \frac{d}{c} c d ​ ): a b c d = a b × d c c d × d c = a b × d c 1 = a b × d c = a d b c \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{\frac{a}{b} \times \frac{d}{c}}{\frac{c}{d} \times \frac{d}{c}} = \frac{\frac{a}{b} \times \frac{d}{c}}{1} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} d c ​ b a ​ ​ = d c ​ × c d ​ b a ​ × c d ​ ​ = 1 b a ​ × c d ​ ​ = b a ​ × c d ​ = b c a d ​

3. Decimals and Decimal Operations

Decimals represent fractions whose denominators are implicit powers of ten ( 10 − 1 = 0.1 , 10 − 2 = 0.01 , 10 − 3 = 0.001 10^{-1} = 0.1, 10^{-2} = 0.01, 10^{-3} = 0.001 1 0 − 1 = 0.1 , 1 0 − 2 = 0.01 , 1 0 − 3 = 0.001 ).

Terminating vs. Repeating Decimals Criterion

A rational number written as a fully simplified fraction a b \frac{a}{b} b a ​ in lowest terms ( gcd ⁡ ( a , b ) = 1 \gcd(a,b) = 1 g cd ( a , b ) = 1 ) converts to a terminating decimal if and only if the prime factorization of its denominator b b b contains no prime factors other than 2 2 2 and/or 5 5 5 : b = 2 m ⋅ 5 n ( where m , n ≥ 0 ) b = 2^m \cdot 5^n \quad (\text{where } m, n \ge 0) b = 2 m ⋅ 5 n ( where m , n ≥ 0 )

Mathematical Reason: In base-10, 10 = 2 × 5 10 = 2 \times 5 10 = 2 × 5 . Any fraction whose denominator divides a power of 10 10 10 ( 10 k = 2 k 5 k 10^k = 2^k 5^k 1 0 k = 2 k 5 k ) terminates after at most max ⁡ ( m , n ) \max(m, n) max ( m , n ) decimal places.

Repeating Decimals: If denominator b b b contains any prime factor other than 2 2 2 or 5 5 5 (such as 3 , 7 , 11 , 13 3, 7, 11, 13 3 , 7 , 11 , 13 ), the decimal will repeat infinitely in a periodic block of digits (the repetend).

7 40 = 7 2 3 ⋅ 5 1 ⟹ \frac{7}{40} = \frac{7}{2^3 \cdot 5^1} \implies 40 7 ​ = 2 3 ⋅ 5 1 7 ​ ⟹ Terminating ( 0.175 0.175 0.175 ).

5 14 = 5 2 ⋅ 7 ⟹ \frac{5}{14} = \frac{5}{2 \cdot 7} \implies 14 5 ​ = 2 ⋅ 7 5 ​ ⟹ Repeating ( 0.3 571428 ‾ 0.3\overline{571428} 0.3 571428 ) because of the prime factor 7 7 7 .

Converting Repeating Decimals to Rational Fractions

Use algebraic subtraction to eliminate the repeating repetend:

Example: Convert x = 0. 45 ‾ = 0.454545 … x = 0.\overline{45} = 0.454545\dots x = 0. 45 = 0.454545 … into a fraction: 100 x = 45.454545 … − x = 0 0.454545 … 99 x = 45 ⟹ x = 45 99 = 5 11 \begin{aligned}

  • 100x &= 45.454545\dots \\
  • \quad x &= \phantom{0}0.454545\dots \\
  • \hline
  • 99x &= 45 \implies x = \frac{45}{99} = \frac{5}{11}
  • \end{aligned} 100 x − x 99 x ​ = 45.454545 … = 0 0.454545 … = 45 ⟹ x = 99 45 ​ = 11 5 ​ ​ ​
  • Standard Decimal Arithmetic Procedures

Addition & Subtraction: Vertically align decimal points to ensure identical positional place values are added/subtracted.

Multiplication: Multiply factors as whole numbers ignoring decimal points. The number of decimal places in the product equals the sum of the decimal places in the factors .

Justification: ( 0.03 ) × ( 0.4 ) = 3 100 × 4 10 = 12 1000 = 0.012 (0.03) \times (0.4) = \frac{3}{100} \times \frac{4}{10} = \frac{12}{1000} = 0.012 ( 0.03 ) × ( 0.4 ) = 100 3 ​ × 10 4 ​ = 1000 12 ​ = 0.012 ( 2 + 1 = 3 2 + 1 = 3 2 + 1 = 3 decimal places).

Division: If the divisor is a decimal, multiply both divisor and dividend by 10 k 10^k 1 0 k to transform the divisor into an integer before performing standard long division:

Example: 14.28 ÷ 0.4 = 14.28 × 10 0.4 × 10 = 142.8 ÷ 4 = 35.7 14.28 \div 0.4 = \frac{14.28 \times 10}{0.4 \times 10} = 142.8 \div 4 = 35.7 14.28 ÷ 0.4 = 0.4 × 10 14.28 × 10 ​ = 142.8 ÷ 4 = 35.7 .

4. Percentages and Proportional Equivalencies

The term percent originates from Latin per centum ("out of one hundred"). A percentage is a ratio with an implicit denominator of 100 100 100 : p % = p 100 = 0.01 ⋅ p p\% = \frac{p}{100} = 0.01 \cdot p p % = 100 p ​ = 0.01 ⋅ p

Benchmark Equivalencies Conversion Table

  • Fraction (Lowest Terms) Decimal Equivalent Percent Equivalent Visual Benchmark Anchor
  • 1 2 \frac{1}{2} 2 1 ​ 0.5 0.5 0.5 50 % 50\% 50% Exactly half

1 3 \frac{1}{3} 3 1 ​ 0. 3 ‾ 0.\overline{3} 0. 3 33. 3 ‾ % 33.\overline{3}\% 33. 3 % or 33 1 3 % 33\frac{1}{3}\% 33 3 1 ​ % One-third

2 3 \frac{2}{3} 3 2 ​ 0. 6 ‾ 0.\overline{6} 0. 6 66. 6 ‾ % 66.\overline{6}\% 66. 6 % or 66 2 3 % 66\frac{2}{3}\% 66 3 2 ​ % Two-thirds

  • 1 4 \frac{1}{4} 4 1 ​ 0.25 0.25 0.25 25 % 25\% 25% One quarter
  • 3 4 \frac{3}{4} 4 3 ​ 0.75 0.75 0.75 75 % 75\% 75% Three quarters
  • 1 5 \frac{1}{5} 5 1 ​ 0.2 0.2 0.2 20 % 20\% 20% One-fifth

2 5 , 3 5 , 4 5 \frac{2}{5}, \frac{3}{5}, \frac{4}{5} 5 2 ​ , 5 3 ​ , 5 4 ​ 0.4 , 0.6 , 0.8 0.4, 0.6, 0.8 0.4 , 0.6 , 0.8 40 % , 60 % , 80 % 40\%, 60\%, 80\% 40% , 60% , 80% Fifths increments

1 8 \frac{1}{8} 8 1 ​ 0.125 0.125 0.125 12.5 % 12.5\% 12.5% or 12 1 2 % 12\frac{1}{2}\% 12 2 1 ​ % Half of a quarter ( 25 % ÷ 2 25\% \div 2 25% ÷ 2 )

  • 3 8 \frac{3}{8} 8 3 ​ 0.375 0.375 0.375 37.5 % 37.5\% 37.5% 3 × 12.5 % 3 \times 12.5\% 3 × 12.5%
  • 5 8 \frac{5}{8} 8 5 ​ 0.625 0.625 0.625 62.5 % 62.5\% 62.5% 5 × 12.5 % 5 \times 12.5\% 5 × 12.5%
  • 7 8 \frac{7}{8} 8 7 ​ 0.875 0.875 0.875 87.5 % 87.5\% 87.5% 7 × 12.5 % 7 \times 12.5\% 7 × 12.5%
  • 1 10 \frac{1}{10} 10 1 ​ 0.1 0.1 0.1 10 % 10\% 10% One tenth (shift decimal left 1 place)
  • 5. Real-World Percent Problem Solving & Financial Mathematics
  • The Two Primary Percent Frameworks

The Percent Proportion: Part Whole = Percent 100 ⟺ is of = % 100 \frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100} \quad \Longleftrightarrow \quad \frac{\text{is}}{\text{of}} = \frac{\%}{100} Whole Part ​ = 100 Percent ​ ⟺ of is ​ = 100 % ​

The Percent Equation: Part = Rate (as decimal) × Whole \text{Part} = \text{Rate (as decimal)} \times \text{Whole} Part = Rate (as decimal) × Whole

Percent of Change Formula

Percent of change measures relative change with respect to the initial (original) baseline: Percent Change = ∣ New Value − Original Value ∣ Original Value × 100 % \text{Percent Change} = \frac{|\text{New Value} - \text{Original Value}|}{\text{Original Value}} \times 100\% Percent Change = Original Value ∣ New Value − Original Value ∣ ​ × 100%

CSET Warning Trap: Always divide by the Original (Base) Value , never by the New Value! If an item increases from $80 to $100, the percent increase is 100 − 80 80 = 20 80 = 25 % \frac{100 - 80}{80} = \frac{20}{80} = 25\% 80 100 − 80 ​ = 80 20 ​ = 25% . If it decreases from $100 to $80, the percent decrease is 100 − 80 100 = 20 100 = 20 % \frac{100 - 80}{100} = \frac{20}{100} = 20\% 100 100 − 80 ​ = 100 20 ​ = 20% .

Percent Problem-Solving Applications & Formula Matrix

Financial / Applied Context Mathematical Formula Step-by-Step Procedure Exemplar Problem

Discount / Sale Price Sale Price = Original × ( 1 − d ) \text{Sale Price} = \text{Original} \times (1 - d) Sale Price = Original × ( 1 − d ) Multiply original price by the complement of discount rate d d d A $120 jacket at 30 % 30\% 30% off costs $ 120 × ( 1 − 0.30 ) = 120 × 0.70 = $ 84.00 \text{\textdollar}120 \times (1 - 0.30) = 120 \times 0.70 = \text{\textdollar}84.00 $ 120 × ( 1 − 0.30 ) = 120 × 0.70 = $ 84.00

Sales Tax & Total Cost Total = Subtotal × ( 1 + t ) \text{Total} = \text{Subtotal} \times (1 + t) Total = Subtotal × ( 1 + t ) Multiply purchase subtotal by ( 1 + tax rate t ) (1 + \text{tax rate } t) ( 1 + tax rate t ) An $84 purchase with 8 % 8\% 8% tax costs $ 84 × 1.08 = $ 90.72 \text{\textdollar}84 \times 1.08 = \text{\textdollar}90.72 $ 84 × 1.08 = $ 90.72

Tip / Gratuity Tip = Meal Cost × r \text{Tip} = \text{Meal Cost} \times r Tip = Meal Cost × r Multiply food total by gratuity rate r r r An 18 % 18\% 18% tip on a $65 dinner bill is $ 65 × 0.18 = $ 11.70 \text{\textdollar}65 \times 0.18 = \text{\textdollar}11.70 $ 65 × 0.18 = $ 11.70

Commission Earnings = Total Sales × c \text{Earnings} = \text{Total Sales} \times c Earnings = Total Sales × c Multiply gross sales volume by commission rate c c c A 6 % 6\% 6% commission on a $350,000 home sale is $ 350 , 000 × 0.06 = $ 21 , 000 \text{\textdollar}350,000 \times 0.06 = \text{\textdollar}21,000 $ 350 , 000 × 0.06 = $ 21 , 000

Simple Interest I = P r t I = Prt I = P r t ; Balance A = P ( 1 + r t ) \text{Balance } A = P(1 + rt) Balance A = P ( 1 + r t ) P = Principal , r = Annual Rate (dec) , t = Years P = \text{Principal}, r = \text{Annual Rate (dec)}, t = \text{Years} P = Principal , r = Annual Rate (dec) , t = Years Deposit $2,000 at 5 % 5\% 5% for 3 3 3 years: I = 2000 × 0.05 × 3 = $ 300 I = 2000 \times 0.05 \times 3 = \text{\textdollar}300 I = 2000 × 0.05 × 3 = $ 300 ; A = $ 2 , 300 A = \text{\textdollar}2,300 A = $ 2 , 300

Loading diagram... Mathematical Pathways for Converting Between Fractions, Decimals, and Percents