Proportional Relationships and Equivalent Ratios

Proportional Relationships

• A proportional relationship between two quantities exists when their ratio remains constant.
• If the ratio of two quantities is constant, then one quantity is said to be proportional to the other.
• A proportional relationship can be represented by an equation in the form of y = kx, where k is the constant of proportionality.

Equivalent Ratios

• Equivalent ratios are ratios that have the same value.
• Two ratios are equivalent if they can be simplified to the same ratio.
• Equivalent ratios can be identified by multiplying or dividing both numbers in the ratio by the same non-zero value.
• For example:
  • 2:4 and 1:2 are equivalent ratios because 2:4 can be simplified to 1:2 by dividing both numbers by 2.
  • 3:6 and 1:2 are equivalent ratios because 3:6 can be simplified to 1:2 by dividing both numbers by 3.
• Equivalent ratios can be used to solve problems involving proportions and scaling.

Proportional Relationships

• A proportional relationship exists between two quantities when their ratio remains constant.
• In a proportional relationship, one quantity is said to be proportional to the other.
• The equation y = kx represents a proportional relationship, where k is the constant of proportionality.

Equivalent Ratios

• Equivalent ratios are ratios that have the same value.
• Two ratios are equivalent if they can be simplified to the same ratio.
• Multiplying or dividing both numbers in a ratio by the same non-zero value identifies equivalent ratios.
• Examples of equivalent ratios include:
  • 2:4 and 1:2, as 2:4 can be simplified to 1:2 by dividing both numbers by 2.
  • 3:6 and 1:2, as 3:6 can be simplified to 1:2 by dividing both numbers by 3.
• Equivalent ratios are useful for solving problems involving proportions and scaling.

Proportional Relationships

Definition

• A proportional relationship exists between two quantities when their ratio remains constant, meaning an increase or decrease in one quantity triggers a corresponding increase or decrease in the other by the same factor.

Characteristics

• Graphs of proportional relationships are straight lines that pass through the origin (0, 0).
• The ratio of coordinates for any point on the line remains constant.
• When the ratio of two quantities is constant, they are considered proportional.

Examples

• Cost of items is directly proportional to the number of items purchased, resulting in a constant cost-to-item ratio.
• Distance traveled is directly proportional to time taken, resulting in a constant distance-to-time ratio.

Representing Proportional Relationships

• Algebraic representation: y = kx, where k is the constant of proportionality.
• Graphical representation: a straight line passing through the origin (0, 0).
• Numerical representation: a table with a constant ratio between quantities.

Identifying Proportional Relationships

• Identify a constant ratio between quantities.
• Verify if the graph is a straight line passing through the origin (0, 0).
• Check if the equation can be written in the form y = kx.

Real-World Applications

• Cost and quantity of items
• Distance and time
• Area and perimeter of shapes
• Speed and distance