Problem 27 - Introductory Algebra for College Students

Question

What does it mean if two quantities vary inversely?

Step-by-Step Solution

Answer

When two quantities vary inversely, it means that the product of the two quantities is constant. Hence, the increase in one quantity results in a proportional decrease in the other one, and vice versa.

Step 1: Understand Inverse Variation

In mathematics, 'inverse variation' refers to a relationship between two variables such that when one variable increases, the other decreases proportionally, and vice versa. In other words, the two variables are said to vary inversely if their product is constant. This relationship can be represented by the formula \(xy = k\), where \(x\) and \(y\) are the two variables and \(k\) is the constant of variation.

Step 2: Illustrate Inverse Variation with an Example

Suppose we have two quantities, 'A' and 'B'. If 'A' increases while 'B' decreases in such a manner that the product \(AB = k\) stays constant, then 'A' and 'B' are inversely proportional to each other. For example, if we have \(A = 2\) and \(B = 5\) (so that \(AB = 10\)), and we then increment 'A' to 4, 'B' will have to decrease to 2.5 to ensure the product remains constant (i.e., \(4 * 2.5 = 10\)). Therefore, A and B are said to vary inversely.

Step 3: Conclusion

In summary, when we say that two quantities vary inversely, we mean that their product is constant i.e., the increase in one is compensated by a decrease in the other, and vice versa. It is a fundamental concept in mathematics giving the relationship between two varying quantities.