91Ó°ÊÓ

In inverse variation, the constant of proportionality is a key component in the relationship between two variables. When we say that a variable "varies inversely" as another, it means the product of these two variables equals a constant value. In our example, as one variable increases, the other decreases to maintain this constant product. This constant is denoted as \( k \).
The formula for inverse variation is given by \( y = \frac{k}{x} \). Here, \(y\) and \(x\) are the variables, and \(k\) is the constant of proportionality. It ensures that no matter what values \(y\) and \(x\) take, their product \(xy\) will always be \(k\).

• If you have information about either \(x\) or \(y\), you can easily find the other variable if \(k\) is known.
• In the exercise, we determined \(k=75\) using the given values of \(y=3\) and \(x=25\).

Understanding this constant is crucial because it allows us to predict and describe how one variable changes in response to another.

A mathematical model is a representation in mathematical terms of a real-world concept. These models help in predicting and understanding phenomena by establishing a relationship between variables using equations. In the context of inverse variation, the model describes how one variable changes concerning another through a constant relationship.
With our example, the derived mathematical model is \( y = \frac{75}{x} \). This equation tells us that for any value of \(x\), we can calculate the corresponding \(y\) using the constant of proportionality \(k = 75\).

• This model allows us to see how \(y\) will decrease as \(x\) increases, maintaining the inverse relationship.
• The model is predefined by the inverse variation formula \( y = \frac{k}{x} \), adaptable for different scenarios by changing the constant \(k\).

Mathematical models like this one provide a clear and calculated insight into relationships and dependencies between variables.

Cross-multiplication is a mathematical technique used to solve equations involving fractions or ratios. It is immensely useful in finding solutions for inverse variation problems, where variables typically form a fraction or ratio on opposite sides of an equation.
In our exercise, we use cross-multiplication to determine the constant of proportionality \(k\) from the equation \(3 = \frac{k}{25}\).

• To apply cross-multiplication, multiply both sides of the equation by 25 to isolate \(k\). This transforms the equation into \(k = 3 \times 25\).
• This straightforward method allows us to solve for \(k\), ensuring an easy and quick calculation to find the constant of proportionality.

Using cross-multiplication simplifies the process and helps in swiftly arriving at the solution, making it an invaluable tool for solving inverse variation problems efficiently.