Problem 50 - College Algebra

Question

What does it mean if two quantities vary directly?

Step-by-Step Solution

Answer

Direct variation means that two quantities change at the same rate. This means that as one quantity increases, the other quantity also increases at the same rate and as one decreases, the other decreases at the same rate. The mathematical formula for direct variation is \(y = kx\), where k is the constant ratio.

Step 1: Definition of Direct Variation

In mathematics, direct variation is a relationship between two variables where their ratio is constant. This means, when one variable increases, the other variable also increases at the same rate or when one variable decreases, the other also decreases at the same rate. The formula for direct variation is \(y = kx\), where k is the constant of variation.

Step 2: Example of Direct Variation

As an example, consider the relationship between the cost of a product and its quantity. If a pen costs $2, then two pens would cost $4, three pens would cost $6, and so on. Here we see that as the quantity of pens (x) increases, the total cost (y) also increases. In this case, the constant of variation (k) is 2, which is the cost of one pen. Hence the relationship between cost and quantity can be represented as \(y = 2x\). This fulfills the definition of direct variation, thus we can say that cost and quantity vary directly.