Lecture Notes: Mathematical Concepts and Problem Solving

Rounding and Estimation

• Rounding to Significant Figures
  • 6.32 rounds to 6 (to one significant figure)
  • 6.51 rounds to 7
  • 0.503 rounds to 0.5
• Estimation Strategy
  • Multiply and divide after rounding
  • Example: (6.32 x 6.51) ÷ 0.503 ≈ 84 (rounding and estimating)

Prime Factorization

• Process: Create a factor tree
  • Example: 180 → 18 and 10
  • Prime factors: 2, 3, 5
  • Express in index form: $2^2 \times 3^2 \times 5$

Least Common Multiple (LCM) and Greatest Common Factor (GCF)

• LCM of 15 and 40
  • LCM = 120
  • Requires 8 packs of pencils and 3 boxes of pens
• GCF of 72 and 90
  • Prime factor breakdown
  • GCF = 18

Solving Inequalities

• Drawing on a Number Line
  • Example: $-2 \leq x < 4$
  • Close circles for inequalities with 'equal to'

Percentage Calculations

• VAT Calculation
  • Calculate 20% VAT for a washing machine
  • Monthly payment after deposit and division
• Reverse Percentages
  • Example: Find original price from a sale price after percentage reduction
  • Techniques for reverse calculations

Interest Rates

• Simple Interest
  • Calculating interest over a period without compounding
• Compound Interest
  • Accumulated interest over several periods
  • Formula: principal × (1 + rate)^years

Depreciation

• Calculating Depreciated Value
  • Falls in value each year based on new price
  • Similar calculation to compound interest

Fractions

• Multiplying and Dividing Fractions
  • Times numerators and denominators
  • Simplify results
• Adding and Subtracting Fractions
  • Convert mixed numbers to improper fractions
  • Find common denominators

Standard Form

• Converting Numbers
  • Large numbers: positive powers of 10
  • Small numbers: negative powers of 10
• Operations in Standard Form
  • Multiplication and division strategies

Negative and Fractional Powers

• Understanding Powers
  • Negative powers: reciprocal
  • Fractional powers: roots

Combinations

• Calculating Possibilities
  • Multiplication principle for combinations
  • Example with padlocks and non-repeating digits

Error Intervals

• Rounding and Truncation
  • Determining possible values within bounds

Using a Calculator

• Functionality
  • Locating mathematical functions on a calculator
  • Understanding cube roots, sine, cosine, and tangent buttons

Additional Mathematical Concepts

• Surds
  • Rationalizing denominators
  • Simplifying expressions involving surds
• Recurring Decimals to Fractions
  • Using algebraic methods to find fractions
• Percentage Increase and Decrease
  • Solving problems involving profit/loss and value depreciation

These notes cover various mathematical topics, providing strategies and examples for solving problems related to rounding, percentages, interest rates, fractions, and more. Each section outlines key methods and formulas needed for exams and practical applications.