🔢 Fractions, Decimals, Percents — Reference

Essential concepts and definitions for working with rational numbers in MATHCOUNTS.

Fractions

Basic Concepts

Definition: A fraction represents parts of a whole, written as $\frac{a}{b}$ where $a$ is the numerator and $b$ is the denominator.

Types of fractions:

• Proper fraction: Numerator < denominator (e.g., $\frac{3}{4}$)
• Improper fraction: Numerator ≥ denominator (e.g., $\frac{7}{4}$)
• Mixed number: Whole number + proper fraction (e.g., $1\frac{3}{4}$)
• Unit fraction: Numerator = 1 (e.g., $\frac{1}{2}$, $\frac{1}{3}$)

Fraction Operations

Addition: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$ Subtraction: $\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$ Multiplication: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$ Division: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

Common Fractions

Decimals

Decimal Places

Tenths: First place after decimal (0.1) Hundredths: Second place after decimal (0.01) Thousandths: Third place after decimal (0.001)

Decimal Operations

Addition: Line up decimal points Subtraction: Line up decimal points Multiplication: Count decimal places in factors Division: Move decimal point in divisor and dividend

Scientific Notation

Format: $a \times 10^n$ where $1 \leq a < 10$ Examples:

• $3,400 = 3.4 \times 10^3$
• $0.0007 = 7 \times 10^{-4}$

Percents

Basic Concepts

Definition: A percent is a fraction with denominator 100 Symbol: % means “per hundred” Examples: 50% = $\frac{50}{100} = \frac{1}{2} = 0.5$

Percent Operations

Finding percent of a number: $a%$ of $b = \frac{a}{100} \times b$ Finding what percent one number is of another: $\frac{a}{b} \times 100%$ Percent increase/decrease: $\frac{\text{change}}{\text{original}} \times 100%$

Common Percents

Conversions

Fraction to Decimal

Method 1: Divide numerator by denominator

• $\frac{3}{4} = 3 \div 4 = 0.75$

Method 2: Use equivalent fractions with powers of 10

• $\frac{3}{4} = \frac{75}{100} = 0.75$

Decimal to Fraction

Method 1: Write as fraction with appropriate denominator

• $0.75 = \frac{75}{100} = \frac{3}{4}$

Method 2: Use place value

• $0.75 = \frac{7}{10} + \frac{5}{100} = \frac{70}{100} + \frac{5}{100} = \frac{75}{100} = \frac{3}{4}$

Fraction to Percent

Method 1: Convert to decimal, then multiply by 100

• $\frac{3}{4} = 0.75 = 75%$

Method 2: Use equivalent fractions

• $\frac{3}{4} = \frac{75}{100} = 75%$

Percent to Fraction

Method 1: Write as fraction with denominator 100, then simplify

• $75% = \frac{75}{100} = \frac{3}{4}$

Method 2: Convert to decimal first

• $75% = 0.75 = \frac{75}{100} = \frac{3}{4}$

Decimal to Percent

Method: Multiply by 100 and add % symbol

• $0.75 = 0.75 \times 100% = 75%$

Percent to Decimal

Method: Divide by 100 and remove % symbol

• $75% = 75 \div 100 = 0.75$

Properties

Fraction Properties

Equivalent fractions: $\frac{a}{b} = \frac{ac}{bc}$ (when $c \neq 0$) Reciprocal: $\frac{a}{b}$ and $\frac{b}{a}$ are reciprocals Zero: $\frac{0}{a} = 0$ (when $a \neq 0$) Undefined: $\frac{a}{0}$ is undefined

Decimal Properties

Place value: Each position represents a power of 10 Rounding: Use standard rounding rules Significant digits: Count non-zero digits from left

Percent Properties

100%: Represents the whole 0%: Represents nothing Over 100%: Represents more than the whole Negative percents: Represent decreases

Common Applications

Word Problems

Part-whole problems: Find part given whole and percent Percent change problems: Find increase or decrease Mixture problems: Combine different percents Interest problems: Calculate simple interest

Real-world Examples

Sales tax: Add percent to original price Discounts: Subtract percent from original price Tips: Add percent to bill Markups: Add percent to cost

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