Problem 42 - Algebra 1

Question

State whether the variables have direct variation, inverse variation, or neither. The number of hours \(h\) that you must work to earn \(\$ 480\) and your hourly rate of pay \(p\) are related by the equation \(p h=480\)

Step-by-Step Solution

Answer

The variables \(p\) and \(h\) have an inverse variation.

Step 1: Interpret the Equation

First of all, analyze and interpret the given equation, \(p h = 480\). Here \(p\) represents the hourly rate of pay, \(h\) represents the working hours, and 480 is the total earning. So, the equation shows the relation between the working hours and the hourly rate.

Step 2: Analyze the Variation

In the equation \(p h = 480\), the earning remains constant (480) when the product of the hourly rate (\(p\)) and the number of hours worked (\(h\)) is taken. If the hourly rate increases, the number of hours worked to earn 480 decreases, and vice versa. This indicates a clear inverse variation.

Step 3: Conclude the Variation Type

Based on the analysis in step 2, the equation \(p h = 480\) demonstrates an inverse variation, because as \(p\) increases, \(h\) decreases, and vice versa. In other words, \(h\) and \(p\) are inversely proportional to each other, since their product is constant (480).

Key Concepts

Direct Variation

Direct variation is a simple, yet important concept in algebra that describes a linear relationship between two variables. Two variables are said to be directly proportional if one variable (\(y\)) is a constant multiple of the other variable (\(x\)). In mathematical terms, this can be expressed as \(y = kx\), where \(k\) is the constant of variation. Here is a bit more about direct variation:

• As one variable increases, the other also increases.
• The graph of a direct variation is a straight line that passes through the origin (0,0).
• Direct variation is common in situations where one thing depends on another in a proportional manner.

An example of direct variation can be seen in the relationship between speed and distance. If a car travels at a constant speed, the distance it covers is directly proportional to the time it travels for.
For instance, if a car travels at 60 miles per hour, the distance covered is directly related to how long it maintains that speed. If the time doubles, so does the distance.

Proportional Relationships

Proportional relationships extend the idea of direct variation to include situations involving both direct and inverse variations. A relationship is proportional when two quantities maintain a consistent ratio or product. Here are the key types:

• **Direct Proportions:** As described in the previous section, these involve a consistent ratio and are expressed as \(y = kx\).
• **Inverse Proportions:** These involve a consistent product, meaning \(xy = k\), where \(k\) is a constant. As one variable increases, the other decreases.

Understanding proportional relationships can be incredibly helpful in real-life situations. For instance, consider the rates of fuel consumption. If a car gets 30 miles per gallon, the amount of fuel needed is directly proportional to the distance traveled. However, suppose the speed of travel affects fuel efficiency, and thus is inversely proportional; increasing speed may increase consumption, showing an inverse proportion.
Analyzing proportional relationships allows for better planning and decision-making, both mathematically and practically.

Algebraic Equations

Algebraic equations are essential tools for expressing relationships between variables in mathematics. These equations can represent various types of relationships, including direct and inverse variations. Here is a basic overview:

• Algebraic equations use symbols to represent unknown values or changing quantities, allowing complex ideas to be handled easily.
• Equations can be linear, representing straightforward relationships, or nonlinear, representing more complex ones.
• Different forms, like quadratic, polynomial, or rational equations, describe diverse phenomena and patterns.

In our given problem, \(ph = 480\) is an algebraic equation where the product of hourly rate \(p\) and work hours \(h\) is constant. This showcases inverse variation, which is a powerful concept in understanding how different factors can offset each other to maintain a balance.
Algebraic equations allow us to solve for unknowns, make predictions, and understand the underlying mechanics of real-world scenarios, acting as a foundation for further mathematical reasoning and exploration.