Problem 52 - College Algebra

Question

What does it mean if two quantities vary inversely?

Step-by-Step Solution

Answer

Two quantities that vary inversely are inversely proportional. This means if one quantity increases, the other decreases proportionally. Their product is always constant.

Step 1: Definition of Inverse Variation

Inverse variation, or inverse proportionality, describes the relationship between two variables such that when one variable increases, the other decreases in proportion, and vice versa. Mathematically, if two variables \(x\) and \(y\) are inversely proportional, it means that the product of the two variables is a constant. This is written as \(xy = k\), where \(k\) is the constant of variation.

Step 2: Illustration with an Example

Let's illustrate this concept with an example: Suppose we have a car that travels at a constant speed. If we increase the speed of the car, the time it takes to travel a fixed distance decreases. Conversely, if we decrease the speed of the car, it will take longer time to travel the same distance. Here, the speed of the car and the time are inversely proportional. If we multiply the speed at which the car is traveling (speed = \(x\)) by the time it takes to cover a certain distance (time = \(y\)), we get a constant value, which is the distance the car covered.

Step 3: Key Takeaway

To summarize, when two quantities vary inversely, it means they are inversely proportional. An increase in one quantity results in a proportional decrease in the other quantity, so that their product remains constant.