Direct and Indirect Proportion

This page covers the concepts of direct and indirect proportion, their equations, and practical applications in mathematics. It provides clear definitions, examples, and problem-solving techniques for both types of proportional relationships.

Direct proportion is explained as a relationship where two or more things are linked so that as one changes, the other changes in the same way. The equation y = kx represents this relationship, where k is the constant of proportionality.

Definition: Direct proportion is when two values are linked so that as one changes, the other changes in the same way, represented by the equation y = kx.

Indirect proportion is described as a relationship where two or more things are linked so that as one changes, the other changes in the opposite way. The equation y = k/x represents this relationship.

Definition: Indirect proportion is when two values are linked so that as one changes, the other changes in the opposite way, represented by the equation y = k/x.

The page emphasizes the importance of the constant of proportionality calculation in both types of proportion. For direct proportion, dividing the values yields this constant.

Highlight: The constant of proportionality $k$ is crucial in both direct and indirect proportion equations and can be calculated by dividing the corresponding values.

Two detailed examples are provided to illustrate problem-solving techniques for direct and indirect proportion:

1. Direct proportion example: Finding y when x changes, given initial values and the proportional relationship.

Example: y is directly proportional to x. When x = 3, y = 7.5. Find y when x = 4.4. Solution steps:

1. Write the equation: y = kx
2. Find k: 7.5 = k$3$, so k = 2.5
3. Use the new x value: y = 2.5$4.4$ = 11

2. Indirect proportion example: Finding y when x changes, given initial values and the inverse proportional relationship.

Example: y is indirectly proportional to x. When x = 6, y = 8. Find y when x = 9. Solution steps:

1. Write the equation: y = k/x
2. Find k: 8 = k/6, so k = 48
3. Use the new x value: y = 48/9 = 16/3

The page concludes by mentioning that proportionality can also involve variations of x, such as x², and that the problem-solving approach remains similar for these cases.

Highlight: Proportional relationships can involve variations of x $e.g., x², √x$, but the problem-solving approach remains consistent.