Ratio and Proportion Chapter Notes | Year 6 Mathematics IGCSE (Cambridge) PDF Download

Ratio

• Objective: Find and use equivalent ratios and find a ratio in its simplest form.
• Key terms:
  • Ratio: A relationship between two or more numbers, showing how many times one quantity is contained in another (e.g., 6:3).
  • Equivalent ratio: Ratios that represent the same relationship, obtained by multiplying or dividing both parts by the same number (e.g., 6:3 = 2:1).
  • Simplest form: A ratio where the numbers share no common factors other than 1, achieved by dividing both parts by their greatest common divisor.
• Understanding ratios:
  • Example: A bowl contains 6 apples and 3 oranges.
  • The ratio of apples to oranges is 6:3.
  • Simplify by dividing both numbers by their greatest common divisor (3): 6 ÷ 3 = 2, 3 ÷ 3 = 1, so 6:3 = 2:1.
  • Equation: 6:3 = 2:1.
• Finding equivalent ratios:
  • Example: In a survey, the ratio of people preferring apples to bananas is 3:5, and 45 people prefer bananas.
  • Find an equivalent ratio where the bananas part is 45: 3:5 = □:45.
  • Determine the multiplier: 45 ÷ 5 = 9.
  • Multiply both parts by 9: 3 × 9 = 27, 5 × 9 = 45, so 3:5 = 27:45.
  • Total people = 27 (apples) + 45 (bananas) = 72.
  • Equation: 3:5 = 27:45.
• Simplifying ratios:
  • Similar to simplifying fractions, divide both parts of the ratio by their greatest common divisor.
  • Example: Simplify 24:8. Divide both by 8: 24 ÷ 8 = 3, 8 ÷ 8 = 1, so 24:8 = 3:1.
  • Equation: 24:8 = 3:1.
• Real-world application: Ratios are used in scenarios like planting crops (e.g., 4 carrots for every 3 onions) or dividing items (e.g., counters between people).

Direct proportion

• Objective: Understand what 'in proportion' means and learn that when one quantity increases (or decreases), the other quantities increase (or decrease) in the same ratio.
• Key terms:
  • Direct proportion: When two quantities increase or decrease such that their ratio remains constant.
  • Proportion: The relationship between quantities that scale together.
  • Enlarge: To increase the size of an object or shape while maintaining the same proportions (e.g., enlarging a photograph).
• Understanding direct proportion:
  • When two quantities are in direct proportion, their ratio stays the same.
  • Example: If 120 grams of butter makes 10 cookies, for 20 cookies, the amount of butter scales proportionally.
• Solving direct proportion problems:
  • Example: Dakarai needs 350 grams of pasta for 4 people. How much for 12 people?
  • Determine the multiplier: 12 ÷ 4 = 3.
  • Multiply the pasta amount by 3: 350 × 3 = 1050 grams.
  • Equation: 350 × (12/4) = 1050.
  • Example: A pasta sauce recipe for 6 people uses 300 grams of tomatoes. How much for 12 people?
  • Determine the multiplier: 12 ÷ 6 = 2.
  • Multiply the tomato amount by 2: 300 × 2 = 600 grams.
  • Equation: 300 × (12/6) = 600.
• Real-world applications:
  • Recipes: Scaling ingredients (e.g., butter for cookies, ingredients for ice cream).
  • Work pay: Earnings proportional to hours worked (e.g., a builder’s pay).
  • Scale models: Model cars with lengths proportional to real cars (e.g., scale 1:24 means the real car is 24 times longer).
  • Enlargements: Scaling shapes like rectangles or photographs while maintaining proportions.
• Scale in models:
  • Example: For a model Beetle car, length = 170 mm, real Beetle length = 4080 mm.
  • Calculate scale: 4080 ÷ 170 = 24, so the scale is 1:24.
  • Equation: 4080 ÷ 170 = 24.
• Proportional shapes:
  • Example: Rectangle A has length 5 cm and width 2 cm. If enlarged to rectangle B with a length ratio of 1:2, the new length is 5 × 2 = 10 cm.
  • Widths and perimeters also scale by the same ratio.
  • Equation: Length_B = 5 × 2 = 10.