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.