Ratio and Proportion

1. Ratios

• Definition: A ratio is a comparison of two or more quantities of the same kind. It expresses how many times one quantity is contained within another.

• Notation: Ratios are typically written in three ways:

  • Colon notation: a : b (read as "a to b")

  • Fraction notation: a/b

  • Using the word "to": a to b

Example:

• If there are 3 red marbles and 5 blue marbles in a bag, the ratio of red marbles to blue marbles is:

  • 3:5

  • 3/5

  • 3 to 5

2. Proportions

• Definition: A proportion is a statement that two ratios are equal.

• Notation:

  • a : b :: c : d

  • a/b = c/d

  • where '::' or '=' represents "is proportional to"

Example:

• If the ratio of boys to girls in a class is 2:3 and there are 10 boys, then the proportion can be written as:

  • 2 : 3 = 10 : x

  • where 'x' represents the number of girls.

Key Properties of Proportions

• Means and Extremes: In the proportion a : b :: c : d,

  • 'a' and 'd' are called the extremes

  • 'b' and 'c' are called the means

• Product of Means = Product of Extremes: In any proportion, the product of the means is equal to the product of the extremes.

  • a/b = c/d

  • a * d = b * c

3. Types of Proportions

• Direct Proportion: When two quantities increase or decrease in the same ratio.

  • Example: If you increase the amount of fuel, the distance traveled by a car also increases proportionally.

• Inverse Proportion: When one quantity increases, the other quantity decreases in the same ratio, and vice versa.

  • Example: If you increase the speed, the time taken to travel a certain distance decreases.

4. Applications of Ratios and Proportions

• Scaling: Used in maps, blueprints, and models to represent real-world objects.

• Mixing: Used in recipes, chemical solutions, and paint mixtures.

• Unit Conversions: Converting between different units of measurement (e.g., inches to centimeters, pounds to kilograms).

• Business and Finance: Calculating profits, losses, and interest rates.

• Everyday Life: Comparing prices, calculating tips, and dividing quantities.

Example:

• Recipe Scaling: A recipe calls for 2 cups of flour to make 12 cookies. How much flour is needed to make 30 cookies?

  • Set up a proportion:

    • 2 cups flour / 12 cookies = x cups flour / 30 cookies

  • Solve for 'x':

    • 12x = 2 * 30

    • x = 60 / 12

    • x = 5 cups of flour