Fractions and Decimals

Fractions and Decimals

Fractions and decimals represent parts of a whole. Here's what you need to know:

Fractions - Core Concepts

A fraction consists of a numerator (top number) and denominator (bottom number): $$\frac{numerator}{denominator}$$

Types of Fractions

• Proper Fraction: Numerator < Denominator (e.g., $$\frac{3}{4}$$)
• Improper Fraction: Numerator ≥ Denominator (e.g., $$\frac{5}{3}$$)
• Mixed Number: Whole number + Proper fraction (e.g., $$1\frac{2}{3}$$)

Equivalent Fractions

Fractions that represent the same value (e.g., $$\frac{1}{2} = \frac{2}{4} = \frac{3}{6}$$)

Key insight: Multiply or divide both numerator and denominator by the same non-zero number to get equivalent fractions.

Operations with Fractions

Addition/Subtraction: Need common denominators

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

Example: $$\frac{1}{4} + \frac{2}{4} = \frac{3}{4}$$

With different denominators, find the least common multiple (LCM):

Example: $$\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$$

Multiplication: Multiply numerators and denominators

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

Example: $$\frac{2}{3} \times \frac{3}{5} = \frac{6}{15} = \frac{2}{5}$$

Division: Multiply by the reciprocal of the divisor

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

Example: $$\frac{2}{3} \div \frac{1}{4} = \frac{2}{3} \times \frac{4}{1} = \frac{8}{3}$$

Decimals - Core Concepts

Decimals are another way to represent fractions with denominators that are powers of 10.

Place Value

Each position represents a power of 10:

• Tenths: $$10^{-1}$$ or $$\frac{1}{10}$$
• Hundredths: $$10^{-2}$$ or $$\frac{1}{100}$$
• Thousandths: $$10^{-3}$$ or $$\frac{1}{1000}$$

Types of Decimals

• Terminating: The decimal ends (e.g., 0.25 = $$\frac{1}{4}$$)
• Repeating: A pattern repeats indefinitely (e.g., 0.333... = $$\frac{1}{3}$$)

Converting Between Fractions and Decimals

Fraction to Decimal: Divide numerator by denominator

Example: $$\frac{3}{4} = 3 \div 4 = 0.75$$

Decimal to Fraction:

• For terminating decimals: Place over appropriate power of 10 and simplify
• Example: $$0.75 = \frac{75}{100} = \frac{3}{4}$$
• For repeating decimals: Use algebraic methods
• Example: $$0.333... = \frac{1}{3}$$

Operations with Decimals

Operations with decimals follow the same rules as whole numbers, with attention to the decimal point:

Addition/Subtraction: Line up decimal points

Example: 3.14 + 2.5 = 3.14 + 2.50 = 5.64

Multiplication: Multiply normally, then count decimal places

Example: 0.3 × 0.4 = 0.12 (1 decimal place + 1 decimal place = 2 decimal places)

Division: Convert to equivalent problem with whole number divisor

Example: 0.8 ÷ 0.2 = 8 ÷ 2 = 4

Common Pitfalls and Tips

• Always simplify fractions to lowest terms
• When comparing fractions, convert to common denominators or to decimals
• Be careful with repeating decimals in calculations
• Remember that dividing by a fraction is the same as multiplying by its reciprocal