Maths Summary Notes

Significant Figures Table

• Type of Zero & Significance:
  • Leading zeros (before first non-zero digit): Not significant
    • Example: 0.0045 → 2 significant figures (4 and 5)
  • Captive zeros (between two non-zero digits): Significant
    • Example: 105 → 3 significant figures
  • Trailing zeros after a decimal point: Significant
    • Example: 2.300 → 4 significant figures
  • Trailing zeros before a decimal point (no decimal shown): Not significant
    • Example: 2300 → 2 significant figures
  • Trailing zeros with a decimal point: Significant
    • Example: 2300. → 4 significant figures

Fractions and Decimals

Writing Fractions as Decimals

• Example: 2 ⅛ ➡️ 2.125
• Method: Convert by performing division.

Writing Decimals as Fractions

• Example: 2.55 = 2 55/100 = 2 11/20 (Simplified)

Recurring Decimals

• Fraction to Recurring Decimal

  • Example: ⅙
  • Use long division, treating numerator inside and denominator outside.

• Recurring Decimal to Fraction

  • Example: 0.16 (recurring)
  • Method:
    • Let x = 0.16, then 10x = 1.6666…, 100x = 16.6666…
    • Solve for x: 90x = 15 → x = 15/90 = ⅙

Operations with Decimals

Multiplying Decimals

• Ignore decimals initially.
• Multiply as whole numbers.
• Count decimal places in the original numbers.
• Adjust the final answer by moving the decimal.

Dividing Decimals

• Ignore the decimal initially.
• Divide as whole numbers.
• Place decimal in the result based on original numbers.

Percentages

Writing Fractions as Percentages

1. Convert fraction to decimal (long division).
2. Multiply result by 100.
3. Add % sign.

Algebraic Concepts

Difference of Two Squares

• Formula: a² - b² = (a - b)(a + b)
• Example: Solving 23² - 21²
  • Apply formula: (23-21)(23+21) = 2 × 44 = 88

Finding the Midpoint

• Formula: (a + b) / 2

Perfect Squares

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²

Factorising Difference of Two Squares

1. Identify perfect squares.
2. Take square roots.
3. Apply: (a−b)(a+b)

Solving Ratio Problems

• Example: Zoe's savings ratio problem
  • Initial ratio: Zoe : Lucy = 9 : 4
  • Solve using cross-multiplication and substitutions
  • Result: Lucy’s savings = $108

Holiday Pay Calculation

• Person: Sarah, Weekly Salary: $1,200, Holiday Pay Loading: 17.5%
• Calculation: Total holiday pay for 4 weeks = $5,640

Non-Monic Quadratics

• Equation: Ax² + bx + c = 0
  • Multiply a and c
  • Find factors of ac that sum to b
  • Factor and solve

Factorising Algebraic Fractions

• Always factor before cancelling.

Expanding Three Brackets

1. Multiply first two brackets.
2. Multiply result by third bracket.
3. Distribute terms and combine like terms.

Assessment Results

• Part A + B = 68/70
• Mentals = 15/20