Essential Concepts of Ratios, Proportions, and Percentages

Fundamentals of Ratios and Proportions

Defining Ratios and Proportions

• Ratio: A ratio of two numbers is their quotient.
• Proportion: A proportion exists when we have two ratios whose quotients are equal.

Key Properties of Proportions

The terms in a proportion are often referred to as means and extremes (or ends).

Example: $2/4 = 3/6$

The Fundamental Property of Proportions states that the product of the extremes is equal to the product of the means. This property is essential for finding an unknown term.

Constant of Proportionality

The constant of proportionality is the ratio (quotient) of any of the corresponding terms in the proportion.

Understanding Proportionality

Magnitudes and Measurement

A magnitude is a measurable characteristic, such as:

• Length
• Volume
• Weight
• Mass
• Temperature

Direct Proportionality

Two magnitudes are directly proportional when an increase in one magnitude causes a corresponding increase in the other, and vice versa (a decrease causes a decrease).

To find the constant ratio, divide the corresponding values of both magnitudes.

Inverse Proportionality

Two magnitudes are inversely proportional when an increase in one magnitude causes a corresponding decrease in the other.

To find the inverse constant of proportionality, multiply the corresponding values of the magnitudes.

Calculation Methods

Solving Proportionality Problems

Common methods used to solve problems involving proportionality include:

1. Rule of Three (Simple): Used to find any unknown quantity.
2. Reduction to the Unit: Used to calculate the value corresponding to a single unit.

Inverse Proportionality Calculations

For inverse proportionality problems, we use:

1. Inverse Rule of Three: Used for finding an unknown quantity.
2. Reduction to the Unit: Used to find the value corresponding to the unit.

Percentages

Percentage Definition

A percentage (%) represents parts per 100. We consider the whole amount as 100 parts.

Calculating and Expressing Percentages

To calculate the percentage amount, multiply the total amount by the percentage rate and divide by 100.

A percentage is expressed as a fraction of 100 or a decimal:

Example: $44\% = 44/100 = 0.44$