Ratios and proportions are fundamental mathematical concepts that describe relationships between quantities. A ratio compares two or more quantities, showing how many times one value contains or is contained within another. A proportion states that two ratios are equal. These concepts are essential in cooking, map reading, scale models, medicine dosages, financial analysis, and countless other real-world applications.

What is a Ratio?

A ratio is a comparison of two or more quantities, showing the relative size of one quantity to another. Ratios can be expressed in several ways:

• Using a colon: 3:4 (read as "3 to 4")
• Using the word "to": 3 to 4
• As a fraction: 3/4
• In words: "the ratio of 3 to 4"

For example, if a recipe calls for 2 cups of flour to 1 cup of sugar, the ratio of flour to sugar is 2:1. This means for every 2 parts of flour, you need 1 part of sugar.

Simplifying Ratios

Like fractions, ratios should be expressed in simplest form. Divide all numbers in the ratio by their greatest common factor (GCF). For example:

• 12:8 simplifies to 3:2 (divided by GCF of 4)
• 15:25:10 simplifies to 3:5:2 (divided by GCF of 5)

What is a Proportion?

A proportion is an equation stating that two ratios are equal. It's written as:
a:b = c:d or a/b = c/d
For example: 2/3 = 4/6 is a proportion because both ratios simplify to the same value.

Cross-Multiplication Method

To solve proportions (find a missing value), use cross-multiplication:
If a/b = c/d, then a × d = b × c

Example: Solve for x: 3/5 = x/20
Cross-multiply: 3 × 20 = 5 × x
60 = 5x
x = 12

Unit Rates

A unit rate is a ratio with a denominator of 1, showing how much of one quantity corresponds to 1 unit of another. Examples include miles per hour (mph), price per pound, or dollars per hour. To find a unit rate, divide the numerator by the denominator.

Example: If 12 apples cost $6, the unit rate is $6 ÷ 12 = $0.50 per apple.