12.3 Fractions, Decimals, Percentages, and Proportional Problem Solving

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.

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1. Conceptual Meanings and Models of Fractions

A fraction $\frac{a}{b}$ (where $a, b \in \mathbb{Z}, b \ne 0$) 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

1. Part-Whole Construct: A whole region or discrete set is partitioned into $b$ equal-sized parts, and $a$ parts are selected.
   • Area Model: A rectangle or circle divided into equal areas (e.g., shading $3$ out of $4$ equal quadrants of a circle).
   • Set Model: A discrete collection of distinct objects partitioned into subsets (e.g., $3$ red apples in a basket of $5$ total apples represents $\frac{3}{5}$ of the set).
2. Measurement Construct: A fraction represents a specific distance from zero on a continuous number line. The fraction $\frac{a}{b}$ is formed by iterating the unit fraction $\frac{1}{b}$ exactly $a$ times along the coordinate axis.
3. Quotient (Division) Construct: The fraction $\frac{a}{b}$ is the exact numerical value obtained when $a$ units are divided equally among $b$ recipients ($a \div b$). For example, dividing $3$ pizzas equally among $4$ students gives each student $\frac{3}{4}$ of a pizza.
4. Ratio / Operator Construct: The fraction functions as a multiplicative transformation or scaling factor (e.g., finding $\frac{2}{3}$ of a $90 budget = $\frac{2}{3} \times 90 = \text{\textdollar}60$).

Types of Fractions and Simplification

• Proper Fraction: Numerator is strictly less than denominator ($|a| < |b|$, value $< 1$).
• Improper Fraction: Numerator is greater than or equal to denominator ($|a| \ge |b|$, value $\ge 1$).
• Mixed Number: A combination of a non-zero whole number and a proper fraction ($W \frac{a}{b} = W + \frac{a}{b} = \frac{W \cdot b + a}{b}$).
• Simplest Form (Lowest Terms): A fraction $\frac{a}{b}$ is in simplest form if and only if $\gcd(|a|, |b|) = 1$ (the numerator and denominator are relatively prime/coprime). Fractions are simplified by dividing both numerator and denominator by $\gcd(a,b)$:

$$\frac{a}{b} = \frac{a \div \gcd(a,b)}{b \div \gcd(a,b)}$$

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2. Fraction Arithmetic and Visual Pedagogical Representations

Addition and Subtraction of Fractions

• Like Denominators: Combine numerators directly while keeping the common denominator constant: $\frac{a}{d} \pm \frac{b}{d} = \frac{a \pm b}{d}$.
• Unlike Denominators: Rename fractions into equivalent forms using the Least Common Denominator (LCD), which is the $\text{LCM}$ of the denominators:

$$\frac{a}{b} + \frac{c}{d} = \frac{a \cdot \frac{\text{LCM}(b,d)}{b}}{\text{LCM}(b,d)} + \frac{c \cdot \frac{\text{LCM}(b,d)}{d}}{\text{LCM}(b,d)} = \frac{ad + bc}{bd}$$

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

$$5\frac{1}{4} = 4 + 1 + \frac{1}{4} = 4 + \frac{4}{4} + \frac{1}{4} = 4\frac{5}{4} \implies 4\frac{5}{4} - 2\frac{3}{4} = 2\frac{2}{4} = 2\frac{1}{2}$$

Multiplication of Fractions & Visual Area Models

Multiplying fractions is conceptually interpreted as finding a fractional part of another fraction (e.g., $\frac{2}{3} \times \frac{4}{5}$ means finding $\frac{2}{3}$ of an area that is $\frac{4}{5}$ 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:

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$

Division of Fractions and Mathematical Proofs

Conceptual Models of Division

1. Measurement (Quotitive) Division: "How many groups of size $\frac{c}{d}$ are contained in $\frac{a}{b}$?"
   • Example: $2 \div \frac{1}{4}$ means "How many $\frac{1}{4}$-sized pieces are in $2$ wholes?" Visualizing $2$ strips divided into fourths yields $8$ fourths ($2 \div \frac{1}{4} = 8$).
2. Partitive (Sharing) Division: "If $\frac{c}{d}$ of a group equals $\frac{a}{b}$, what is the value of one whole group?"

Mathematical Justifications for the "Invert and Multiply" Algorithm

Why does $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$?

Proof 1: Common Denominator Approach Convert both fractions to have a common denominator $bd$:

$$\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bd} \div \frac{bc}{bd} = \frac{ad \div bc}{bd \div bd} = \frac{\frac{ad}{bc}}{1} = \frac{ad}{bc} = \frac{a}{b} \times \frac{d}{c}$$

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 ($\frac{d}{c}$):

$$\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{\frac{a}{b} \times \frac{d}{c}}{\frac{c}{d} \times \frac{d}{c}} = \frac{\frac{a}{b} \times \frac{d}{c}}{1} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$$

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3. Decimals and Decimal Operations

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

Terminating vs. Repeating Decimals Criterion

A rational number written as a fully simplified fraction $\frac{a}{b}$ in lowest terms ($\gcd(a,b) = 1$) converts to a terminating decimal if and only if the prime factorization of its denominator $b$ contains no prime factors other than $2$ and/or $5$:

$$b = 2^m \cdot 5^n \quad (\text{where } m, n \ge 0)$$

• Mathematical Reason: In base-10, $10 = 2 \times 5$. Any fraction whose denominator divides a power of $10$ ($10^k = 2^k 5^k$) terminates after at most $\max(m, n)$ decimal places.
• Repeating Decimals: If denominator $b$ contains any prime factor other than $2$ or $5$ (such as $3, 7, 11, 13$), the decimal will repeat infinitely in a periodic block of digits (the repetend).
  • $\frac{7}{40} = \frac{7}{2^3 \cdot 5^1} \implies$ Terminating ($0.175$).
  • $\frac{5}{14} = \frac{5}{2 \cdot 7} \implies$ Repeating ($0.3\overline{571428}$) because of the prime factor $7$.

Converting Repeating Decimals to Rational Fractions

Use algebraic subtraction to eliminate the repeating repetend:

• Example: Convert $x = 0.\overline{45} = 0.454545\dots$ into a fraction:

$$\begin{aligned} 100x &= 45.454545\dots \\ - \quad x &= \phantom{0}0.454545\dots \\ \hline 99x &= 45 \implies x = \frac{45}{99} = \frac{5}{11} \end{aligned}$$

Standard Decimal Arithmetic Procedures

1. Addition & Subtraction: Vertically align decimal points to ensure identical positional place values are added/subtracted.
2. 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: $(0.03) \times (0.4) = \frac{3}{100} \times \frac{4}{10} = \frac{12}{1000} = 0.012$ ($2 + 1 = 3$ decimal places).
3. Division: If the divisor is a decimal, multiply both divisor and dividend by $10^k$ to transform the divisor into an integer before performing standard long division:
   • Example: $14.28 \div 0.4 = \frac{14.28 \times 10}{0.4 \times 10} = 142.8 \div 4 = 35.7$.

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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 $100$:

$$p\% = \frac{p}{100} = 0.01 \cdot p$$

Benchmark Equivalencies Conversion Table

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5. Real-World Percent Problem Solving & Financial Mathematics

The Two Primary Percent Frameworks

1. The Percent Proportion:

$$\frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100} \quad \Longleftrightarrow \quad \frac{\text{is}}{\text{of}} = \frac{\%}{100}$$

1. The Percent Equation:

$$\text{Part} = \text{Rate (as decimal)} \times \text{Whole}$$

Percent of Change Formula

Percent of change measures relative change with respect to the initial (original) baseline:

$$\text{Percent Change} = \frac{|\text{New Value} - \text{Original Value}|}{\text{Original Value}} \times 100\%$$

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 $\frac{100 - 80}{80} = \frac{20}{80} = 25\%$. If it decreases from $100 to $80, the percent decrease is $\frac{100 - 80}{100} = \frac{20}{100} = 20\%$.

Percent Problem-Solving Applications & Formula Matrix