In which type of variation does one variable increase while the other decreases in a multiplicative relationship?

The type of variation where one variable increases while the other decreases in a multiplicative relationship is known as inverse variation. In this scenario, as the value of one variable goes up, the corresponding value of the other variable goes down, maintaining a constant product. Mathematically, this relationship can be expressed as ( y = \frac{k}{x} ), where ( k ) is a non-zero constant.

In practical terms, if you think of a scenario such as speed and time for a fixed distance, as speed increases, time decreases; if you double the speed, the time taken to cover the same distance is halved. This illustrates the fundamental nature of inverse variation.

The other variations mentioned have different characteristics:

• Direct variation involves both variables increasing or decreasing together, maintaining a constant ratio.

• Linear variation represents a straight-line relationship on a graph, where changes occur equally without one variable diminishing while the other increases.

• Quadratic variation involves a squared relationship, producing a parabolic shape rather than the multiplicative decrease described in inverse variation.

Thus, the correct answer reflects the unique characteristic of how the variables relate in inverse variation.