5.3 Decimals and Fractions

Decimals and fractions are two ways to represent parts of a whole. Converting between them lets us compare values easily. This skill is crucial for everyday math, from cooking to finances.

Ordering decimals and fractions helps us rank things by size or value. It's useful in many situations, like comparing prices or test scores. Mastering these skills opens doors to more advanced math concepts.

Converting and Comparing Decimals and Fractions

Fraction and decimal conversions

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• Divide numerator by denominator to convert fraction to decimal
  • $\frac{3}{4} = 3 \div 4 = 0.75$
• Write decimal as fraction over power of 10, then simplify to convert decimal to fraction
  • $0.6 = \frac{6}{10} = \frac{3}{5}$
• Convert repeating decimals to fractions using algebraic method
  • Let $x$ equal repeating decimal
  • Multiply both sides of equation by power of 10 to shift decimal point
  • Subtract original equation from new equation
  • Solve for $x$ and simplify resulting fraction
• Convert fractions to percentages by multiplying by 100 (percent)

Ordering of decimals and fractions

• Compare decimals digit by digit from left to right
  • Decimal with more digits is larger
  • Equal decimals have same digits
• Convert fractions to decimals or find common denominator to compare
  • $\frac{3}{4}$ and $\frac{5}{6}$ converted to decimals: $0.75 < 0.833$
• Convert all values to same form (decimals or fractions) to order from least to greatest or greatest to least
• Use rounding to compare approximate values when exact comparison is not necessary

Simplifying Expressions and Calculating with Circles

Order of operations with decimals

• Follow PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)
• Simplify expressions within parentheses first
• Evaluate exponents (powers, roots, etc.)
• Perform multiplication and division from left to right
• Perform addition and subtraction from left to right
• $2 + (0.5 \times 3) - \frac{1}{4} = 2 + 1.5 - 0.25 = 3.25$

Circle calculations using decimals

• Calculate circumference using formula $C = 2\pi r$, where $r$ is radius
  • If $r = 2.5$, then $C = 2\pi(2.5) = 5\pi \approx 15.71$
• Calculate area using formula $A = \pi r^2$
  • If $r = \frac{3}{2}$, then $A = \pi(\frac{3}{2})^2 = \frac{9}{4}\pi \approx 7.07$
• Approximate $\pi$ as 3.14 or $\frac{22}{7}$ when needed
• Express final answer in terms of $\pi$ if exact value required
• Use significant figures to express appropriate level of precision in calculations

Ratios and Proportions

• A ratio compares two quantities and can be written as a fraction
• A proportion is an equation stating that two ratios are equal
• Use cross multiplication to solve proportions
• Apply ratios and proportions to solve real-world problems involving scaling, rates, and comparisons