Problem 40 - College Algebra

Question

What does it mean if two quantities vary directly?

Step-by-Step Solution

Answer

When two quantities vary directly, it means that the ratio between the two is constant. When one quantity increases or decreases, the other does as well at the same rate. This can be represented mathematically by the formula \(y = kx\), where `k` is a constant.

Step 1: Define Direct Variation

Direct variation describes a situation where the ratio of two variables is constant. In other words, if one variable increases or decreases, the other variable will also increase or decrease at the same rate. It is expressed in the equation, \(y = kx\), where `k` is the constant of variation.

Step 2: Illustrate with Example

For instance, if we conceive an example where a car travels at a constant speed. Here, the distance travelled by the car (y) and the time taken (x) form a direct variation relation. Because the speed is constant, for every unit of time that passes, the car will always travel the same distance. So, the ratio of the distance to time would always remain constant.

Step 3: Understand Changes in Direct Variation

In the context of direct variation, if one variable is multiplied by a factor, the other variable will be multiplied by the same factor. Using the car example, if the time is doubled, the distance travelled would also double given a constant speed.