Ratio and Proportion - Direct and Inverse Variation

Variation is the functional relationship between two quantities. If a change in one quantity causes a corresponding change in another, the two quantities are said to be in variation.

Direct Variation: Two quantities $x$ and $y$ are in direct variation if they increase or decrease together such that the ratio $\frac{y}{x}$ remains constant. Visually, if you plot $x$ and $y$ on a coordinate plane, the points will lie on a straight line that passes through the origin $(0,0)$, representing a constant rate of growth.

The Constant of Variation ($k$) for direct variation is defined by the equation $y = kx$. This $k$ represents the slope of the line in a visual graph; the steeper the line, the larger the value of $k$.

Inverse Variation: Two quantities $x$ and $y$ are in inverse variation if an increase in $x$ leads to a proportional decrease in $y$ (and vice versa). Visually, the graph of an inverse variation is a smooth curve called a hyperbola that gets closer and closer to the $x$ and $y$ axes but never actually touches them.

In Inverse Variation, the product of the two variables remains constant: $x \times y = k$. This means that as one value gets very large, the other must get very small to keep the product the same.

To solve variation problems, you can use the Proportion Method. For direct variation, we use the proportion $\frac{x_1}{y_1} = \frac{x_2}{y_2}$, and for inverse variation, we use the equation $x_1 y_1 = x_2 y_2$.

Identifying Variation: Always check the relationship first. For example, the number of articles and their total cost is Direct Variation (more articles = more cost), while the number of workers and the time taken to finish a task is Inverse Variation (more workers = less time).