Proportional Relationships and Equivalent Ratios

Study Notes

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 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.

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 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.

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.

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.

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.