Fractions and Percentages Chapter Notes | Year 5 Mathematics IGCSE (Cambridge) PDF Download

Percentages

• A percentage represents a proportion of a whole, expressed as parts per hundred (percent means "per hundred").
• Each part of a whole divided into 100 equal parts is 1% = 1/100.
• Percentages can be written as fractions with a denominator of 100:
  • Example: 20% = 20/100 = 1/5.
  • Example: 25% = 25/100 = 1/4.
  • Example: 60% = 60/100 = 3/5.
  • Example: 15% = 15/100 = 3/20.
• Equivalence between fractions and percentages:
  • Example: 1/10 = 10/100 = 10%.
  • Example: 7/100 = 7% = 0.07.
• A 10x10 grid (100 squares) represents one whole:
  • Each small square is 1/100 = 1%.
  • Ten small squares are 10/100 = 10%.

Equivalent Fractions, Decimals and Percentages

• Fractions, decimals, and percentages can represent the same proportion of a whole.
• Example: In a picture with 100 symbols:
  • Each symbol is 1/100 = 1% = 0.01.
  • 10 symbols are 10/100 = 1/10 = 10% = 0.1.
• Converting between forms:
  • Fraction to decimal: 1/10 = 1 ÷ 10 = 0.1.
  • Percentage to fraction: 10% = 10/100 = 1/10.
  • Percentage to decimal: 10% = 10/100 = 0.1.
  • Example: 50% = 50/100 = 1/2 = 0.5.
  • Example: 30% = 30/100 = 3/10 = 0.3.

Comparing and Ordering Quantities

• Quantities can be compared using symbols: < (less than), > (greater than), or = (equal to).
• Comparing fractions, decimals, and percentages requires converting to a common form:
  • Example: Compare 0.9 and 50%:
    • 0.9 = 90/100 = 90%.
    • 50% = 50/100 = 0.5.
    • 0.9 > 50%.
  • Example: 3/10 < 50% (since 3/10 = 30/100 = 30%, and 30% < 50%).
  • Example: 0.4 > 1/10 (since 1/10 = 0.1, and 0.4 > 0.1).
  • Example: 0.8 > 0.7.
  • Example: 20% < 90% < 40% < 80% (ordering percentages directly).
• Ordering quantities from smallest to largest:
  • Example: Order 3/6, 4/6, 1/6, 5/6 becomes 1/6, 3/6, 4/6, 5/6.
  • Example: Order 3/8, 7/8, 4/8, 2/8 becomes 2/8, 3/8, 4/8, 7/8.
  • Example: Order 0.2, 0.5, 0.3, 0.7 becomes 0.2, 0.3, 0.5, 0.7.
  • Example: Order 1.6, 0.9, 1.3, 0.6, 1.9 becomes 0.6, 0.9, 1.3, 1.6, 1.9.
  • Example: Order 0.2, 4/10, 0.8, 50%, 1/10, 90%, 1/10requires converting to decimals or percentages:
    • 0.2 = 20%, 4/10 = 0.4 = 40%, 0.8 = 80%, 50% = 0.5, 1/10 = 0.1 = 10%, 90% = 0.9.
    • Order: 1/10 (10%), 0.2 (20%), 4/10 (40%), 50% (50%), 0.8 (80%), 90% (90%).

Fractions as Operators

• Fractions act as operators to find a portion of a quantity.
• Methods to calculate a fraction of a quantity:
  • Method 1: Divide by the denominator, then multiply by the numerator.
    • Example: 7/10 of 400 = (400 ÷ 10) × 7 = 40 × 7 = 280.
  • Method 2: Multiply by the numerator, then divide by the denominator.
    • Example: 7/10 of 400 = (400 × 7) ÷ 10 = 2800 ÷ 10 = 280.
• Finding the whole from a fraction:
  • Example: If 1/10 of a box = 60 cents, the whole box is 60 × 10 = 600 cents.
  • Example: If 1/2 of a kg of onions = 75 cents, 1 kg costs 75 × 2 = 150 cents.
  • Example: If 1/3 of a kg of carrots = 30 cents, 1 kg costs 30 × 3 = 90 cents.
  • Example: If 1/8 of a box of oranges = 60 cents, the whole box costs 60 × 8 = 480 cents.
• Using place value to find fractions of amounts:
  • Example: For 1/100 of 1 kg of spice costing $30:
    • $30 ÷ 100 = $0.30 = 30 cents.

Ratio and Proportion

• Proportion compares a part to the whole, expressed as a fraction or percentage.
• Example: In a group of 10 fruits with 6 oranges:
  • Proportion of oranges: 6/10 = 3/5 = 60%.
• Ratio compares two quantities using the phrase "for every."
• Example: For every 1 box of limes, there are 3 boxes of lemons (ratio 1:3).
• Example: For every 2 boxes of grapes, there are 3 boxes of oranges (ratio 2:3).
• Proportions in a market with 100 boxes of fruit:
  • Apples: 20/100 = 20% = 1/5.
  • Bananas: 50/100 = 50% = 1/2.
  • Cherries: 10/100 = 10% = 1/10.
  • Plums: 20/100 = 20% = 1/5.