OCR GCSE Maths: Proportion and Direct Relationships

Direct Proportion and Linear Relationships

What is Direct Proportion?

Two quantities are in direct proportion when they increase or decrease at the same rate. This means if one quantity doubles, the other quantity doubles too. If one quantity is halved, the other quantity is halved too.

Recognizing Direct Proportion

• Look for a constant ratio: If you divide one quantity by the other, you always get the same answer.
• Look for a linear relationship: When plotted on a graph, direct proportion creates a straight line passing through the origin (0,0).

Setting up and Solving Proportion Problems

1. Identify the quantities in proportion: Decide which two quantities are directly proportional.
2. Set up a proportion: Write a proportion using the two quantities. This can be represented as:
   • Quantity 1 / Quantity 2 = Constant
   • Quantity 1 / Quantity 2 = Quantity 3 / Quantity 4
3. Solve for the unknown: Use the information you have to solve for the unknown quantity.

Example 1: Simple Direct Proportion

• Problem: If 3 apples cost £1.50, how much would 7 apples cost?
• Solution:
  • Let x be the cost of 7 apples.
  • Set up the proportion: 3 / 1.50 = 7 / x
  • Cross-multiply: 3 * x = 7 * 1.50
  • Solve for x: x = (7 * 1.50) / 3 = £3.50

Example 2: Direct Proportion in Algebraic Contexts

• Problem: The cost of a taxi journey is directly proportional to the distance travelled. If a 5km journey costs £8, how much would a 12km journey cost?
• Solution:
  • Let y be the cost of a 12km journey.
  • Set up the proportion: 5 / 8 = 12 / y
  • Cross-multiply: 5 * y = 8 * 12
  • Solve for y: y = (8 * 12) / 5 = £19.20

Example 3: Currency Conversions

• Problem: If £1 is equivalent to €1.15, how many euros would you get for £200?
• Solution:
  • Let x be the equivalent amount in euros.
  • Set up the proportion: 1 / 1.15 = 200 / x
  • Cross-multiply: 1 * x = 1.15 * 200
  • Solve for x: x = 1.15 * 200 = €230

Formulating Equations Representing Proportional Changes

Direct proportion can be represented by a linear equation of the form:

• y = kx

Where:

• y is the dependent variable (the quantity that changes in response to the other quantity)
• x is the independent variable (the quantity that is changed)
• k is the constant of proportionality (the constant ratio between the two quantities)

Example:

• Problem: The number of cakes baked is directly proportional to the amount of flour used. If 2 cakes can be baked with 150g of flour, what is the equation representing this relationship?
• Solution:
  • Identify the variables:
    • y = number of cakes
    • x = amount of flour
  • Find the constant of proportionality: k = y / x = 2 / 150 = 1/75
  • Write the equation: y = (1/75)x

Practice Problems

1. If 4 litres of paint cover 12 square metres, how many litres of paint would you need to cover 30 square metres?
2. The cost of hiring a car is directly proportional to the number of days you hire it for. If a 3-day hire costs £90, how much would a 7-day hire cost?
3. The number of people attending a concert is directly proportional to the number of tickets sold. If 150 tickets are sold, 120 people attend. What is the equation representing this relationship?

Remember to practice and apply the principles of direct proportion to various problems. Mastering this concept will enhance your understanding of linear relationships and help you solve a wide range of real-world applications.