2.2 Fractions, Mixed Numbers & Decimals

Fraction Fundamentals, Types & Conversions

Core Concept: A fraction represents a numerical part of a whole quantity or a ratio between two numbers, written in the form $\frac{a}{b}$, where $a$ is the numerator (number of parts counted) and $b$ is the denominator (total equal parts into which the whole is divided, $b \neq 0$).

1. Classification of Fractions

Fractions are categorised into three primary structural types:

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2. Converting Between Improper Fractions and Mixed Numbers

Converting between improper fractions and mixed numbers is a fundamental prerequisite for performing fraction arithmetic.

Method A: Improper Fraction to Mixed Number

To convert an improper fraction $\frac{a}{b}$ into a mixed number:

1. Divide the numerator $a$ by the denominator $b$ using integer division.
2. The whole number quotient becomes the whole number part.
3. The remainder becomes the new numerator over the original denominator $b$.

$$\frac{a}{b} = \text{Quotient} \frac{\text{Remainder}}{b}$$

Worked Example: Convert $\frac{29}{6}$ to a mixed number.

• Divide $29 \div 6$: $6 \times 4 = 24$, so Quotient $= 4$, Remainder $= 29 - 24 = 5$.
• Result: $\mathbf{4\frac{5}{6}}$.

Method B: Mixed Number to Improper Fraction

To convert a mixed number $w\frac{a}{b}$ into an improper fraction:

1. Multiply the whole number $w$ by the denominator $b$.
2. Add the numerator $a$ to this product.
3. Place the total over the original denominator $b$.

$$\text{Improper Numerator} = (w \times b) + a \implies \frac{(w \times b) + a}{b}$$

Worked Example: Convert $5\frac{3}{7}$ to an improper fraction.

• Multiply whole number by denominator: $5 \times 7 = 35$.
• Add numerator: $35 + 3 = 38$.
• Result: $\mathbf{\frac{38}{7}}$.

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3. Equivalent Fractions & Simplifying to Lowest Terms

Fundamental Property of Fractions

Multiplying or dividing both the numerator and denominator of a fraction by the same non-zero number produces an equivalent fraction (a fraction representing the exact same numerical value):

$$\frac{a}{b} = \frac{a \times k}{b \times k} = \frac{a \div k}{b \div k} \quad (k \neq 0)$$

Simplifying (Reducing) to Lowest Terms

A fraction is in its simplest form (lowest terms) when the Highest Common Factor (HCF) of its numerator and denominator is $1$.

Step-by-Step Algorithm for Simplifying:

1. Find the HCF of the numerator and denominator.
2. Divide both numerator and denominator by the HCF.

Worked Example: Simplify $\frac{48}{72}$ to lowest terms.

• Step 1: Prime factorise $48$ and $72$:
  • $48 = 2^4 \times 3$
  • $72 = 2^3 \times 3^2$
  • $\text{HCF}(48, 72) = 2^3 \times 3 = 8 \times 3 = 24$.
• Step 2: Divide numerator and denominator by $24$:
  • $48 \div 24 = 2$
  • $72 \div 24 = 3$
• Result: $\mathbf{\frac{2}{3}}$.

Fraction Arithmetic, Decimals & Practical Applications

1. Addition and Subtraction of Fractions

Case A: Like Denominators

When denominators are identical, add or subtract the numerators and keep the denominator unchanged:

$$\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}$$

Case B: Unlike Denominators (Finding the LCD)

When denominators differ, follow these steps:

1. Determine the Least Common Denominator (LCD), which is the Least Common Multiple (LCM) of all denominators.
2. Convert each fraction into an equivalent fraction with the LCD as its denominator.
3. Add or subtract the numerators.
4. Simplify the resulting fraction or convert to a mixed number.

Worked Example 1 (Addition): Calculate $2\frac{3}{4} + 1\frac{5}{6}$.

• Step 1: Convert mixed numbers to improper fractions: $2\frac{3}{4} = \frac{11}{4}$, $1\frac{5}{6} = \frac{11}{6}$.
• Step 2: Find LCD of $4$ and $6$: $\text{LCM}(4, 6) = 12$.
• Step 3: Convert fractions: $\frac{11 \times 3}{4 \times 3} = \frac{33}{12}$, $\frac{11 \times 2}{6 \times 2} = \frac{22}{12}$.
• Step 4: Add numerators: $\frac{33 + 22}{12} = \frac{55}{12}$.
• Step 5: Convert to mixed number: $55 \div 12 = 4$ remainder $7 \implies \mathbf{4\frac{7}{12}}$.

Worked Example 2 (Subtraction with Borrowing): Calculate $5\frac{1}{4} - 2\frac{2}{3}$.

• Convert to improper: $\frac{21}{4} - \frac{8}{3}$.
• LCD of $4$ and $3$ is $12$: $\frac{21 \times 3}{12} - \frac{8 \times 4}{12} = \frac{63}{12} - \frac{32}{12}$.
• Subtract: $\frac{63 - 32}{12} = \frac{31}{12} = \mathbf{2\frac{7}{12}}$.

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2. Multiplication and Division of Fractions

Multiplication of Fractions

Multiply numerators together and denominators together. Simplify before multiplying by cross-cancelling common factors:

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

Worked Example: Evaluate $\frac{15}{28} \times \frac{14}{25}$.

• Cross-cancel common factors:
  • Divide $15$ and $25$ by $5 \implies 3$ and $5$.
  • Divide $14$ and $28$ by $14 \implies 1$ and $2$.
• Multiply remaining numerators and denominators: $\frac{3 \times 1}{2 \times 5} = \mathbf{\frac{3}{10}}$.

Division of Fractions (Keep-Change-Flip)

To divide by a fraction, multiply by its reciprocal (invert numerator and denominator):

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

Worked Example: Evaluate $3\frac{3}{4} \div 1\frac{7}{8}$.

• Convert mixed numbers to improper: $3\frac{3}{4} = \frac{15}{4}$, $1\frac{7}{8} = \frac{15}{8}$.
• Apply Keep-Change-Flip: $\frac{15}{4} \times \frac{8}{15}$.
• Cancel $15$ with $15$, and divide $8$ by $4$: $\frac{1 \times 2}{1 \times 1} = \mathbf{2}$.

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3. Decimal Place Value & Conversion Benchmark Table

Decimals express fractions whose denominators are powers of ten ($10, 100, 1000$). The table below highlights critical CPEA conversions:

Conversion Algorithms:

• Fraction to Decimal: Divide numerator by denominator ($3 \div 8 = 0.375$).
• Decimal to Fraction: Place decimal digits over place value power of 10 and simplify ($0.65 = \frac{65}{100} = \frac{13}{20}$).
• Decimal to Percentage: Multiply decimal by $100\%$ ($0.375 \times 100\% = 37.5\%$).

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4. Comprehensive CPEA Word Problem Walkthrough

Problem: Farmer John owns a piece of land measuring $24$ acres. He plants sugarcane on $\frac{3}{8}$ of the land, vegetables on $\frac{1}{3}$ of the land, and reserves the remainder for cattle grazing. How many acres are reserved for cattle grazing?

Step-by-Step Solution:

1. Calculate total fraction used: $$\frac{3}{8} + \frac{1}{3} = \frac{3 \times 3}{24} + \frac{1 \times 8}{24} = \frac{9}{24} + \frac{8}{24} = \frac{17}{24}$$
2. Calculate remaining fraction: $$1 - \frac{17}{24} = \frac{24}{24} - \frac{17}{24} = \frac{7}{24}$$
3. Calculate exact acreage for cattle grazing: $$\text{Grazing Acres} = \frac{7}{24} \times 24 = \mathbf{7 \text{ acres}}$$