How to Simplify and Use Ratios — Scaling, Proportions & Examples

What Is a Ratio?

A ratio compares two or more quantities of the same kind. It can be written in three common forms:

• Colon notation: 3:5
• Word form: "3 to 5"
• Fraction form: 3/5 (though note: a ratio is a comparison, not necessarily a part-to-whole like a fraction)

Ratios can involve two terms (e.g., 2:3) or more (e.g., 1:2:5 for cement:sand:gravel in concrete).

Simplifying Ratios: Finding the Simplest Form

Just like fractions, ratios can be simplified by dividing all terms by their greatest common divisor (GCD).

Steps to Simplify:

1. List all numbers in the ratio.
2. Find the GCD of all numbers.
3. Divide each term by the GCD.

Example: Simplify 18:24

• GCD of 18 and 24 is 6
• 18 ÷ 6 = 3, 24 ÷ 6 = 4
• Simplified ratio: 3:4

For three or more terms, the process is the same:
12:18:30 → GCD = 6 → 2:3:5

Scaling Ratios: Enlarging or Reducing Proportionally

Ratios maintain their relationship when all terms are multiplied or divided by the same number.

• To scale up: Multiply all terms by a factor
  Example: Double 2:3 → 4:6
• To scale down: Divide all terms by a factor
  Example: Halve 10:15 → 5:7.5 (or multiply by 2 to eliminate decimal: 10:15)

This is essential in recipes, maps, and model building.

Solving Proportions: Finding the Missing Value

A proportion states that two ratios are equal: a:b = c:d

This can be solved using cross-multiplication:

a × d = b × c

Example: If 3:4 = x:20, find x

• 3 × 20 = 4 × x → 60 = 4x → x = 15

Converting Ratios to Other Forms

• Ratio to Fraction: For a:b, the fraction of the first part is a/(a+b)
  Example: In a 2:3 mix, the first component is 2/(2+3) = 2/5 of the total.
• Ratio to Percentage: Convert the fraction to a percentage
  Example: 2/5 = 0.4 = 40%

Pro Tips & Common Mistakes

• Always use the same units: Convert 2 hours : 30 minutes to 120 min : 30 min before simplifying.
• Don’t confuse ratio with fraction: 3:5 is not the same as 3/5 in meaning (though mathematically related).
• Simplify before scaling: It’s easier to work with 1:2 than 15:30.
• Use ratios for consistency: In baking, a 5:3 flour-to-water ratio ensures consistent dough regardless of batch size.

Practical Applications

• Cooking & Baking: Scale recipes for more or fewer servings
• Mixing Paint or Concrete: Maintain colour or strength consistency
• Finance: Analyse debt-to-equity or price-to-earnings ratios
• Photography: Understand aspect ratios (e.g., 16:9 for video)
• Education: Solve word problems involving sharing or distribution