Introduction to Ratios and Proportions

Introduction to Ratio

• A ratio compares two or more quantities of the same kind.
• Ratios are expressed as a fraction, a colon, or with the word "to".
• Example: The ratio of apples to oranges is 3:2 (or 3/2 or 3 to 2). This means for every 3 apples, there are 2 oranges.
• A ratio is a comparison, not a measurement, meaning the units are not crucial. If the ratio is of the same unit, units can be omitted. If it is apples to oranges, there is no need to state "apples to oranges".

Simplifying Ratios

• Ratios should always be simplified to their lowest terms, like fractions.
• This involves dividing both parts of the ratio by their greatest common divisor (GCD).
• Example: The ratio 6:8 can be simplified to 3:4 by dividing both sides by 2.

Proportion

• A proportion is an equation stating that two ratios are equal.
• Example: 2/3 = 4/6 is a proportion.
• Proportions are used to solve for an unknown value in a ratio.
• Often deals with two ratios containing one unknown, and shows relationship between values

Types of Proportions

• Direct Proportion: As one quantity increases, the other quantity increases by the same factor. The ratio between them remains constant.
  • Example: Speed = Distance / Time. Double the time, double the distance, and the ratio will hold.
• Inverse Proportion: As one quantity increases, the other quantity decreases by the same factor. The product of the two quantities remains constant.
  • Example: If there are more workers to complete a job, the time taken to complete the whole job is less.

Solving Proportions

• When solving for an unknown in a proportion, cross-multiply.
• For example, if x/5 = 12/20, then 20x = (5)(12), leading to x = 3.

Applications of Ratio and Proportion

• Scaling: Used in maps, blueprints, and resizing images.
• Recipes: Recipes require precise ratios of ingredients for consistent results.
• Similar Figures: Ratios are crucial for comparing lengths, areas, and volumes of similar shapes.
• Unit Conversions: Converting units from one system to another (e.g., kilometers to miles) often involves ratios.

Calculations using Ratios

• Finding a Part of a Whole Using a Ratio: Determine the amount of one quantity given the total and the ratio.
• Finding the Total Given a Part and a Ratio: Calculate the total quantity when a part and its ratio to the whole are known.

Key Concepts

• Equivalent Ratios: Ratios representing the same relationship between quantities (e.g., 1/2 = 2/4)
• Proportionality: The relationship between quantities based on ratios.
• Extremes and Means: In a proportion, the numbers on the outer positions are called extremes, and those in the inner positions are called means.
• Cross-multiplication: A method for solving proportions by multiplying the extremes and the means. Crucial tool for finding unknown quantities within proportions.