Problem 51 - Intermediate Algebra

Question

Express each variation model in words. In each equation, \(k\) is the constant of variation. $$ P=\frac{k m}{n} $$

Step-by-Step Solution

Answer

\(P\) is directly proportional to \(m\) and inversely proportional to \(n\).

Step 1: Identify the Type of Variation

The given equation is \(P = \frac{k m}{n}\). This indicates a direct variation of \(P\) with \(m\) and an inverse variation with \(n\).

Step 2: Express the Direct Variation in Words

Since \(P\) is directly proportional to \(m\), as \(m\) increases, \(P\) will also increase, assuming \(n\) remains constant.

Step 3: Express the Inverse Variation in Words

Since \(P\) is inversely proportional to \(n\), as \(n\) increases, \(P\) will decrease, assuming \(m\) remains constant.

Step 4: Combine the Variations in Words

Combining these facts, the equation means that \(P\) varies directly with the product of \(k\) and \(m\) and inversely with \(n\). This can be expressed as "\(P\) is directly proportional to \(m\) and inversely proportional to \(n\)".

Key Concepts

Direct Variation

Direct variation is a fundamental concept in algebra where two variables change in the same direction. If one variable increases, the other also increases, and if one decreases, the other does the same. This happens because the ratio between the variables remains constant.
Imagine a situation where your earnings are directly proportional to the number of hours you work. The more hours you work, the higher your earnings will be, assuming the pay rate is constant. Mathematically, this is represented as:

• Formula: If variable 'y' varies directly as 'x', then the relationship is given by \(y = kx\) where 'k' is a non-zero constant.
• Explanation: As 'x' increases, 'y' increases proportionally, as long as 'k' remains constant.

In the case of the equation \(P = \frac{k m}{n}\), it indicates that 'P' is directly proportional to 'm'. As 'm' increases, 'P' increases proportionally if 'n' stays the same. This relationship is straightforward and is a very common mechanism in mathematical models.

Inverse Variation

Inverse variation describes a relationship where one variable increases while the other decreases. This type of relationship is common in real-world scenarios, such as speed and travel time. If you travel at a higher speed, the time to reach a destination decreases, as long as the distance is unchanged.
In inverse variation, the product of the two variables is constant even as they change.

• Formula: When 'y' varies inversely as 'x', it is expressed as \(y = \frac{k}{x}\), where 'k' is the constant of variation.
• Explanation: As one variable 'x' increases, 'y' will decrease in such a way that their product is always 'k'.

For the given equation \(P = \frac{k m}{n}\), it reveals that 'P' is inversely proportional to 'n'. Thus, as 'n' rises, 'P' will decrease, provided 'm' is constant. This insight allows you to predict one variable's behavior in relation to another.

Constant of Variation

The constant of variation is a crucial element in understanding relationships between variables in both direct and inverse variations. It serves as the consistent factor that binds the variables together in a proportionate manner.
In the context of standard equations, the constant of variation gives insight into the rate or ratio at which one variable changes concerning another.

• Direct Variation: In the equation \(y = kx\), 'k' is the constant that defines how much 'y' will change with a unit increase in 'x'.
• Inverse Variation: For \(y = \frac{k}{x}\), 'k' is the product of 'y' and 'x', which remains unchanged across different values of the variables.

In our equation \(P = \frac{k m}{n}\), 'k' holds together the dual relationship between 'm' and 'n'. It helps determine the rate at which 'P' varies directly with 'm' and inversely with 'n'. Recognizing the role of 'k' allows one to harness the predictive power of the equation effectively.