Introduction to Ratios

Introduction to Ratios

• Ratios compare two or more quantities of the same kind.
• A ratio is a fraction that shows the relative sizes of two or more values.
• Ratios can be written in different forms: using the colon (e.g., 2:3), as a fraction (e.g., 2/3), or in words (e.g., 2 to 3).
• Ratios are used in various applications, including scaling, proportions, and comparisons.

Types of Ratios

• Part-to-part ratios: Compare one part of a whole to another part.
  • Example: The ratio of red marbles to blue marbles in a bag.
• Part-to-whole ratios: Compare one part of a whole to the entire whole.
  • Example: The ratio of red marbles to all marbles in a bag.

Simplifying Ratios

• Simplifying ratios means expressing the ratio in its lowest terms, similar to reducing fractions.
• To simplify, divide both parts of the ratio by their greatest common divisor (GCD).
• Example: The ratio 6:9 simplifies to 2:3.

Ratio Word Problems

• Ratio word problems often involve finding the unknown value in a proportion.
  • Example: If the ratio of boys to girls in a class is 2:3, and there are 12 boys, how many girls are there?

Ratio and Proportion

• A proportion states that two ratios are equal.
  • Example: 2/3 = 4/6
• Proportions are used to solve for unknown values in ratio problems.
  • Using cross-multiplication is a common method of solving proportions
  • Example: If 2/3 = x/9 Cross multiplying yields 3x = 18, so x = 6

Ratio and Scaling

• Ratios are crucial in scaling.
  • Example: Reducing or enlarging blueprints; enlarging or reducing photographs.
• If a ratio is given for a length, the same ratio will hold true for other sizes, such as width and height.

Applications of Ratios

• Cooking: Recipes often use ratios to adjust ingredient amounts for different batches.
• Construction: Architects and engineers often use ratios to design and build structures.
• Finance: Ratios are used to analyze financial performance, like comparing profit to revenue (often referred to as a profit margin).
• Maps: Maps use ratios to represent distances between places.

Ratio and Rates

• A rate is a special type of ratio that compares two quantities with different units.
  • Example: Miles per hour (mph), cost per item, etc.
• Rates frequently highlight speed or efficiency.

Equivalent Ratios

• Equivalent ratios are different ways of expressing the same relationship between quantities.
  • Example: 1/2 = 2/4 = 3/6 and so on.
• Finding equivalent ratios often involves multiplying or dividing both parts of the original ratio by the same number.