8.1 Scaling & Optional Ratio Sharing

5. Optional extension: ratio notation

A ratio describes the relative quantities of two or more groups, showing how many times one number contains another. It is written using a colon notation ($a:b$).

Crucial Properties of Ratios

1. Order Matters: The order of terms in a ratio strictly mirrors the order stated in the text. The ratio of cats to dogs is not identical to dogs to cats.
2. Identical Units Required: For a ratio comparing amounts of the same kind, convert to the same unit. Rates comparing different quantities, such as grams per millilitre, retain different units. You cannot directly form a ratio of 50p to £2 without converting both to pence first: $50\text{p} : 200\text{p} = 1:4$.
3. Simplifying Ratios: Divide all terms by their Highest Common Factor (HCF). When asked for simplest whole-number form, use the lowest whole-number terms: $$18 : 24 = (18 \div 6) : (24 \div 6) = 3 : 4$$ $$1.5 : 2.5 = 15 : 25 = 3 : 5$$

┌───────────────────────────────────────────────────────────────┐
│                     Ratio vs. Fraction                        │
├───────────────────────────────────────────────────────────────┤
│ In a fruit bowl with 3 Apples and 5 Oranges:                  │
│ • The ratio of apples to oranges is  3 : 5                    │
│ • Total parts in the bowl = 3 + 5 = 8 parts                   │
│ • The fraction of apples is  3/8                              │
│ • The fraction of oranges is 5/8                              │
└───────────────────────────────────────────────────────────────┘

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6. Optional extension: ratio sharing

Dividing a whole quantity into a specified ratio (e.g., sharing money between two or three individuals) is a useful extension exercise when tackled using the Three-Step Unitary Method:

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Step-by-Step Problem: Standard Sharing

Divide £420 between Daniel and Maya in the ratio $2 : 5$.

• Step 1 (Find total parts): $2 + 5 = 7$ total equal parts.
• Step 2 (Find value of 1 part): $\text{Value of 1 part} = 420 \div 7 = £60$.
• Step 3 (Multiply to find individual shares):
  • Daniel ($2$ parts): $2 \times £60 = £120$.
  • Maya ($5$ parts): $5 \times £60 = £300$.
• Check: $120 + 300 = £420$ (Total matches original sum).

Advanced Ratio Problem: Sharing with a Difference Given

An extension puzzle can give the difference between shares rather than the total:

Worked Example: James and Oliver share a collection of marbles in the ratio $3 : 7$. Oliver has 32 more marbles than James. How many marbles does James have?

• Step 1 (Find difference in parts): $$\text{Difference} = 7 - 3 = 4 \text{ parts}$$
• Step 2 (Value of 1 part): These 4 parts represent 32 marbles: $$\text{Value of 1 part} = 32 \div 4 = 8 \text{ marbles}$$
• Step 3 (Calculate James's share): James has 3 parts: $$\text{James} = 3 \times 8 = 24 \text{ marbles}$$
• (Oliver has $7 \times 8 = 56$ marbles; $56 - 24 = 32$. Matches check!)

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7. Direct Proportion & Scaling

Two quantities are in direct proportion when multiplying or dividing one quantity by a constant factor causes the other quantity to be multiplied or divided by that same factor. The ratio between them remains constant.

Recipe Scaling Problems

To scale a recipe up or down for a different number of servings:

• The Unitary Approach: Calculate the amount required for 1 serving, then multiply by the target number of servings.

Worked Example: A pancake recipe for 6 people requires:

• 300 g flour
• 450 ml milk
• 3 eggs

How much flour and milk are required to make pancakes for 10 people?

• Flour Calculation:
  • Flour for 1 person: $300 \text{ g} \div 6 = 50\text{ g}$.
  • Flour for 10 people: $50 \text{ g} \times 10 = 500\text{ g}$.
• Milk Calculation:
  • Milk for 1 person: $450 \text{ ml} \div 6 = 75\text{ ml}$.
  • Milk for 10 people: $75 \text{ ml} \times 10 = 750\text{ ml}$.

Scale Factor Alternative: Notice that 10 people is $\frac{10}{6} = \frac{5}{3}$ times the original recipe. Multiply $300 \times \frac{5}{3} = 500$ g, and $450 \times \frac{5}{3} = 750$ ml.

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8. Common Exam Pitfalls & Trap Avoidance