Understanding proportional reasoning is crucial for solving real-world problems related to recipe adaptation, map scales, and value determination. This mathematical concept revolves around the relationship between ratios, allowing for the application of this understanding in various contexts.

Introduction to Proportional Reasoning

Proportional reasoning involves recognising and using the multiplicative relationship between quantities. It's a key skill in mathematics, facilitating the application of ratios to diverse problems such as adjusting recipes, interpreting map scales, and comparing values economically.

Applying Proportional Reasoning

Recipe Adaptation

Ratios in Recipes

• Concept: Recipes provide ingredient quantities in ratios, facilitating easy adjustments based on the number of servings.
  • Example Problem: A recipe for 4 servings requires 2 eggs. How many eggs for 6 servings?
    • Solution:
      • Original ratio: 4 servings : 2 eggs
      • New requirement: 6 servings
      • Step 1: Simplify original ratio $\frac{4}{2} = \frac{2}{1}$ (2 servings per egg)
      • Step 2: Calculate for 6 servings $6 \times \frac{1}{2} = 3$ eggs

Map Scales

Understanding and Using Map Scales

• Concept: Map scales express the ratio of a distance on the map to the actual distance on the ground.
  • Example Problem: On a 1:100,000 scale map, two towns are 5 cm apart. What is the actual distance?
    • Solution:
      • Scale: 1 cm on map = 100,000 cm in reality
      • Step 1: Actual distance $5 \times 100,000 = 500,000$ cm
      • Step 2: Convert to kilometres $500,000 \text{ cm} = 5$ km

Value Determination

Calculating Unit Price

• Concept: Determining the cost per unit of items helps in assessing the value for money.
  • Example Problem: A 12-pack of pens costs £3. What is the cost per pen?
    • Solution:
      • Total cost: £3 for 12 pens
      • Cost per pen: $\frac{£3}{12} = £0.25$ per pen

Worked Problems

Problem 1: Scaling a Recipe

If a soup recipe for 8 servings requires 400g of tomatoes, how much is needed for 10 servings?

Solution:

• Original ratio: 8 servings : 400g
• New servings: 10
• Find grams per serving $\frac{400 \ g}{8} = 50 \ g$ per serving
• Calculate for 10 servings $10 \times 50 \ g = 500 \ g$

Therefore, you need $500 \ g$ for 10 servings.

Problem 2: Interpreting a Map Scale

Given a 1:50,000 scale map, if two cities are 8 cm apart, what is their actual distance?

Solution:

• Scale: 1 cm = 50,000 cm
• Map distance: 8 cm
• Actual distance: $8 \times 50,000 = 400,000$ cm or 4 km

Problem 3: Comparing Unit Prices

Brand A sells 500g of flour for £1.50, and Brand B sells 1kg for £2.80. Which is more economical?

Solution:

• Brand A: $\frac{£1.50}{500 \ g} = £0.003$ per gram
• Brand B: $\frac{£2.80}{1000 \ g} = £0.0028$ per gram
• Conclusion: Brand B offers better value £0.0028/g < £0.003/g.