5.2 Fractions, Mixed Numbers & Operations

5.2 Fractions, Mixed Numbers & Operations

Core Concept: Fractions represent rational numbers expressed as the quotient of two integers $\frac{a}{b}$ (where $b \neq 0$). Success on the ACCUPLACER requires total fluency with equivalent fraction transformations, common denominator conversions, mixed number regrouping, cross-canceling, and dividing by reciprocals without a calculator.

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1. Anatomy of Fractions, Classifications & Structural Conversions

A fraction $\frac{a}{b}$ consists of a numerator $a$ (representing the number of equal parts considered) and a denominator $b$ (representing the total number of equal parts comprising one whole unit).

+-------------------------------------------------------------------------------------------------------+
|                                    FRACTION CLASSIFICATION MATRIX                                     |
+-------------------+---------------------------+-----------------------------------+-------------------+
| Classification    | Mathematical Condition    | Defining Characteristic           | Examples          |
+-------------------+---------------------------+-----------------------------------+-------------------+
| Proper Fraction   | Numerator < Denominator   | Value strictly between 0 and 1    | 3/8, 5/12, 7/9    |
| Improper Fraction | Numerator >= Denominator  | Value greater than or equal to 1  | 11/4, 9/9, 43/6   |
| Mixed Number      | Integer + Proper Fraction | Explicit whole number plus part   | 2 3/4, 7 1/6      |
+-------------------+---------------------------+-----------------------------------+-------------------+

Bidirectional Conversion Algorithms

• Mixed Number to Improper Fraction: Multiply the whole integer $W$ by the denominator $d$, add the numerator $n$, and place the resulting sum over the original denominator $d$:

  $$W \frac{n}{d} = \frac{(W \times d) + n}{d}$$

  Example: Convert $5 \frac{3}{8}$ to an improper fraction:

  $$5 \frac{3}{8} = \frac{(5 \times 8) + 3}{8} = \frac{40 + 3}{8} = \frac{43}{8}$$

• Improper Fraction to Mixed Number: Divide the numerator $n$ by the denominator $d$ using integer division. The quotient $q$ becomes the whole number, and the remainder $r$ becomes the new numerator over $d$:

  $$\frac{n}{d} = q \frac{r}{d}, \quad \text{where } n = (q \times d) + r$$

  Example: Convert $\frac{79}{6}$ to a mixed number:

  $$79 \div 6 = 13 \text{ with a remainder of } 1 \implies \mathbf{13 \frac{1}{6}}$$

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2. Prime Factorization, Greatest Common Factor (GCF) & Least Common Multiple (LCM)

Simplifying fractions and creating common denominators both rely on the prime factorization of integers.

Simplifying Fractions via Greatest Common Factor (GCF)

A fraction is in simplest form (lowest terms) when its numerator and denominator share no common factors other than $1$ (i.e., $\gcd(a, b) = 1$). To reduce a fraction in a single step, divide both terms by their GCF.

• Example: Reduce $\frac{84}{140}$ to simplest terms:
  1. Find prime factorizations: $$84 = 2^2 \times 3 \times 7$$ $$140 = 2^2 \times 5 \times 7$$
  2. Extract common prime factors with lowest exponents: $$\text{GCF}(84, 140) = 2^2 \times 7 = 28$$
  3. Divide numerator and denominator by $28$: $$\frac{84 \div 28}{140 \div 28} = \mathbf{\frac{3}{5}}$$

Finding the Least Common Denominator (LCD) via LCM

To add or subtract fractions with unlike denominators, you must convert them to equivalent fractions sharing the Least Common Denominator (LCD), which is the Least Common Multiple (LCM) of the denominators.

• Example: Find the LCD for fractions with denominators $12$, $18$, and $30$:
  1. Prime factorizations: $$12 = 2^2 \times 3^1$$ $$18 = 2^1 \times 3^2$$ $$30 = 2^1 \times 3^1 \times 5^1$$
  2. Take the highest power of every unique prime factor: $$\text{LCM}(12, 18, 30) = 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = \mathbf{180}$$
  3. The LCD is $180$.

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3. Addition and Subtraction with Like and Unlike Denominators

Fractions with Unlike Denominators

Unlike denominators represent unequal partition sizes; they cannot be combined directly. You must first scale each fraction by a form of 1 (such as $\frac{k}{k}$) to match the LCD.

$$\frac{a}{b} \pm \frac{c}{d} = \frac{a \cdot (\text{LCD}/b)}{\text{LCD}} \pm \frac{c \cdot (\text{LCD}/d)}{\text{LCD}}$$

• Worked Example: Compute $\frac{7}{12} + \frac{5}{18} - \frac{1}{8}$.
  1. Find LCD: Denominators are $12 = 2^2 \times 3$, $18 = 2 \times 3^2$, $8 = 2^3$. $$\text{LCD} = 2^3 \times 3^2 = 8 \times 9 = 72$$
  2. Convert each fraction: $$\frac{7}{12} = \frac{7 \times 6}{12 \times 6} = \frac{42}{72}$$ $$\frac{5}{18} = \frac{5 \times 4}{18 \times 4} = \frac{20}{72}$$ $$\frac{1}{8} = \frac{1 \times 9}{8 \times 9} = \frac{9}{72}$$
  3. Combine Numerators: $$\frac{42 + 20 - 9}{72} = \mathbf{\frac{53}{72}}$$

Mixed Number Addition and Subtraction with Regrouping (Borrowing)

When subtracting mixed numbers, if the fractional part of the minuend is smaller than that of the subtrahend, you must borrow $1$ whole ($1 = \frac{d}{d}$) from the whole-number integer.

• Worked Example: Evaluate $8 \frac{1}{6} - 3 \frac{3}{4}$.
  1. Establish Common Denominator ($12$): $$8 \frac{1}{6} = 8 \frac{2}{12}, \quad 3 \frac{3}{4} = 3 \frac{9}{12}$$
  2. Regroup (Borrow $1$ whole): Because $\frac{2}{12} < \frac{9}{12}$, borrow $1$ from $8$: $$8 \frac{2}{12} = 7 + \frac{12}{12} + \frac{2}{12} = 7 \frac{14}{12}$$
  3. Subtract Whole Numbers and Fractions Separately: $$(7 - 3) + \left(\frac{14}{12} - \frac{9}{12}\right) = 4 + \frac{5}{12} = \mathbf{4 \frac{5}{12}}$$

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4. Multiplication, Cross-Cancellation & Division Algorithms

+-------------------------------------------------------------------------------------------------------+
|                                 FRACTION MULTIPLICATION & DIVISION RULES                              |
+----------------+--------------------------------------+-----------------------------------------------+
| Operation      | Algorithmic Formula                  | Critical Step                                 |
+----------------+--------------------------------------+-----------------------------------------------+
| Multiplication | (a/b) × (c/d) = (a × c) / (b × d)    | Cross-cancel common factors BEFORE multiplying|
| Division       | (a/b) ÷ (c/d) = (a/b) × (d/c)        | Invert divisor (reciprocal) and multiply      |
| Mixed Numbers  | Convert all to improper fractions    | Never multiply mixed whole numbers and        |
|                | before computing                     | fractions separately                          |
+----------------+--------------------------------------+-----------------------------------------------+

Cross-Cancellation in Multiplication

Never multiply large raw numerators and denominators without checking for shared diagonal factors first. Cross-canceling eliminates common prime factors early and keeps arithmetic manageable.

• Worked Example: Calculate $\frac{14}{25} \times \frac{15}{21} \times \frac{5}{8}$.
  1. Pair $\frac{14}{21}$: Divide both by $7 \implies 14 \to 2$ and $21 \to 3$.
  2. Pair $\frac{15}{25}$: Divide both by $5 \implies 15 \to 3$ and $25 \to 5$.
  3. The expression becomes: $\frac{2}{5} \times \frac{3}{3} \times \frac{5}{8} = \frac{2}{5} \times 1 \times \frac{5}{8}$.
  4. Pair the $5$ in the denominator with the $5$ in the numerator: $\frac{2}{1} \times \frac{1}{8} = \frac{2}{8}$.
  5. Reduce $\frac{2}{8} = \mathbf{\frac{1}{4}}$.

Division by Multiplying by the Reciprocal

Division by a non-zero fraction is mathematically identical to multiplication by its multiplicative inverse (reciprocal):

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

• Worked Example: Evaluate $4 \frac{2}{3} \div 2 \frac{4}{5}$.
  1. Convert mixed numbers to improper fractions: $$4 \frac{2}{3} = \frac{14}{3}, \quad 2 \frac{4}{5} = \frac{14}{5}$$
  2. Invert the divisor and change $\div$ to $\times$: $$\frac{14}{3} \div \frac{14}{5} = \frac{14}{3} \times \frac{5}{14}$$
  3. Cross-cancel the $14$ in numerator and denominator: $$\frac{1}{3} \times \frac{5}{1} = \mathbf{\frac{5}{3} = 1 \frac{2}{3}}$$

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5. Applied Fraction Word Problems

Real-World Carpentry & Measurement Problem

• Problem: A carpenter has a raw wooden timber measuring $14 \frac{1}{2}$ feet in length. The carpenter cuts $4$ identical structural braces from this timber, each measuring $2 \frac{3}{4}$ feet long. Each saw cut destroys $\frac{1}{8}$ foot of wood as sawdust kerf waste (requiring a total of $4$ cuts). What is the exact length of the remaining wood piece?
• Step-by-Step Solution:
  1. Calculate Length of 4 Braces: $$4 \times 2 \frac{3}{4} = 4 \times \frac{11}{4} = 11 \text{ feet}$$
  2. Calculate Total Kerf Sawdust Waste: $$4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \text{ foot}$$
  3. Sum Total Lumber Consumed: $$11 + \frac{1}{2} = 11 \frac{1}{2} \text{ feet}$$
  4. Subtract from Original Timber: $$14 \frac{1}{2} - 11 \frac{1}{2} = \mathbf{3 \text{ feet}}$$
  5. Conclusion: Exactly $3$ feet of lumber remains.