Scaling with Ratios

Scaling is one of the most practical applications of ratios in QR. Whenever a recipe, formula, or pattern needs to be adjusted for a different quantity, you are scaling with ratios. This lesson covers the methods for scaling up and down, including handling non-integer scale factors.

The Scale Factor

The scale factor tells you how much to multiply each quantity by.

Formula: Scale Factor = New Quantity ÷ Original Quantity

Example: A recipe serves 4 people. You want to serve 6.

Scale factor = 6 ÷ 4 = 1.5

Multiply every ingredient by 1.5.

Scaling Up

Worked Example: Recipe Scaling

Data: A cake recipe for 8 servings requires:

Ingredient	Amount
Flour	400 g
Sugar	200 g
Butter	250 g
Eggs	4
Milk	300 ml

Question: What quantities are needed for 12 servings?

Step 1: Scale factor = 12 ÷ 8 = 1.5

Step 2: Multiply each ingredient by 1.5:

Ingredient	Original	× 1.5	New Amount
Flour	400 g	400 × 1.5 = 600	600 g
Sugar	200 g	200 × 1.5 = 300	300 g
Butter	250 g	250 × 1.5 = 375	375 g
Eggs	4	4 × 1.5 = 6	6 eggs
Milk	300 ml	300 × 1.5 = 450	450 ml

Mental shortcut for ×1.5: The original amount plus half of it. E.g., 250 + 125 = 375.

Scaling Down

Worked Example

Question: The same recipe for 8 servings. How much flour for 5 servings?

Scale factor: 5 ÷ 8 = 0.625

Flour: 400 × 0.625

This is awkward to calculate. Use a different approach:

400 g for 8 servings → 50 g per serving (400 ÷ 8)
5 servings → 50 × 5 = 250 g

Tip: When the scale factor is a non-round decimal, it is often easier to find the amount per single unit first, then multiply by the target.

Handling Awkward Scale Factors

Method 1: Find Per Unit, Then Multiply

Example: A recipe for 6 uses 270 g of flour. How much for 10?

Per person: 270 ÷ 6 = 45 g
For 10: 45 × 10 = 450 g

Method 2: Scale Factor as a Fraction

Example: A recipe for 4 uses 300 g of sugar. How much for 7?

Scale factor = 7/4
Sugar = 300 × 7/4 = (300 × 7) ÷ 4 = 2,100 ÷ 4 = 525 g

Method 3: Build from Known Multiples

Example: A recipe for 6 uses 240 g of butter. How much for 9?

For 6: 240 g
For 3 (half): 120 g
For 9 (6 + 3): 240 + 120 = 360 g

This avoids calculating the scale factor entirely.