Ratio and Proportion

1. Understanding Ratios

What is a Ratio?

A ratio is a way of comparing two or more quantities. It tells us how much of one thing there is compared to another thing.

For example, if a recipe uses 2 cups of flour and 1 cup of sugar, we can write this as:

• Flour : Sugar = 2 : 1

This means for every 2 parts of flour, we need 1 part of sugar.

Important rules about ratios:

• All quantities in a ratio must use the same units before you compare them
• The order matters — the first number relates to the first item mentioned
• A ratio compares part to part, not part to whole

How Ratios and Fractions are Different

Although ratios and fractions both compare quantities, they work differently:

• A fraction compares a part to the whole (e.g. 5/8 of a pizza)
• A ratio compares one part to another part (e.g. 5:3 means 5 slices to 3 slices)

Example: A pizza is cut into 8 slices. One person gets 5 slices, another gets 3 slices.

• As fractions: First person gets 5/8 of the pizza; second person gets 3/8
• As a ratio: The ratio is 5 : 3 (the 8 doesn't appear in the ratio, but it's the total: 5 + 3 = 8)

How to Simplify a Ratio

Method: Divide all parts of the ratio by their highest common factor (HCF).

Example 1: Simplify 20 : 30 : 40

Solution:

1. Find the highest common factor of 20, 30, and 40 → HCF = 10
2. Divide each part by 10:
   • 20 ÷ 10 = 2
   • 30 ÷ 10 = 3
   • 40 ÷ 10 = 4

Answer: 2 : 3 : 4

Example 2: Simplify 30 : 18

Solution:

1. Find the HCF of 30 and 18 → HCF = 6
2. Divide both parts by 6:
   • 30 ÷ 6 = 5
   • 18 ÷ 6 = 3

Answer: 5 : 3

Calculator Tip: You can check your answer by entering the ratio like a fraction (e.g. 30/18) and your calculator will simplify it for you.