Problem 37 - Elementary Algebra

Question

For exercises 37-38, \(T=\frac{R}{A}\) represents the relationship of the asset turnover ratio, \(T\); the sales revenue of a company, \(R\); and the total revenues of a company, \(A\). Is the relationship of the given variables a direct variation or an inverse variation? $$ R \text { is constant; the relationship of } A \text { and } T \text {. } $$

Step-by-Step Solution

Answer

The relationship between \( A \) and \( T \) is an inverse variation.

Step 1: Identify the Formula

The given relationship is represented by the formula: \[ T = \frac{R}{A} \] where \( T \) is the asset turnover ratio, \( R \) is the sales revenue, and \( A \) is the total assets.

Step 2: Understand the Variable Constraints

Given that \( R \) (sales revenue) is constant, analyze the relationship between \( A \) (total assets) and \( T \) (asset turnover ratio).

Step 3: Analyze the Relationship

Since \( R \) is constant and the formula \( T = \frac{R}{A} \), if \( A \) increases, \( T \) (started as a ratio) must decrease and vice versa. This means as \( A \) varies, \( T \) varies inversely.

Step 4: Conclusion

Based on the analysis, the relationship between \( A \) and \( T \) given constant \( R \) is an inverse variation.

Key Concepts

Asset Turnover Ratio

The asset turnover ratio is a measure of how efficiently a company uses its total assets to generate sales revenue. It is calculated using the formula: \[ T = \frac{R}{A} \]where \( T \) represents the asset turnover ratio, \( R \) is the sales revenue, and \( A \) denotes the total assets.
The asset turnover ratio indicates the productivity of a company's assets in creating sales. A higher ratio suggests that the company is using its assets more effectively, while a lower ratio might indicate inefficiencies.
If asset turnover ratio increases, this means the company generates more sales per unit of asset. On the other hand, if the ratio decreases, the company is generating fewer sales per unit of asset.

Sales Revenue

Sales revenue is the total amount of money that a company earns from selling its goods or services before any costs or expenses are deducted.
The formula for calculating sales revenue is: \[ R = P \times Q \]where \( P \) is the price per unit, and \( Q \) is the quantity of units sold.
In the context of the given equation \[ T = \frac{R}{A} \], sales revenue \( R \) is kept constant in this scenario. When \( R \) is constant and does not vary with changes in either \( T \) (asset turnover ratio) or \( A \) (total assets), it simplifies the analysis of their relationship.
Sales revenue is crucial for evaluating a company’s performance as it directly impacts profitability. Understanding how effectively revenue is generated relative to assets gives insights into operational efficiency.

Total Assets

Total assets encompass all assets owned by a company, which include current assets (like cash, inventory) and non-current assets (like property, equipment).
The formula to understand their impact on the asset turnover ratio is: \[ T = \frac{R}{A} \]where we see that if \( R \) remains constant while \( A \) changes, an increase in total assets \( A \) will result in a decrease in the asset turnover ratio \( T \). Conversely, a decrease in \( A \) will lead to an increase in \( T \).
This balance reflects an inverse variation, meaning when one variable increases, the other decreases, given that the sales revenue \( R \) remains unchanged.
Total assets play a vital role in this inverse relationship, highlighting the importance of managing asset levels to maintain operational efficiency and optimize the asset turnover ratio.