Chapter 8.6, Problem 31E

To determine

To tell:

Whether annual sales and annual earnings show direct variation. Justify your answer mathematically.

Answer to Problem 31E

For John annual sales and annual earnings show direct variation, while for Lisa annual sales and annual earnings doesn’t show direct variation.

Explanation of Solution

Given:

Lisa and john work in different department stores. Lisa earns a salary of 18,000 plus 2% commission on sales. John earns no salary but earns a 6% commissionon his sales.

Concept used:

We know that two directly proportional quantities are in form $y=kx$ , where k represents constant of proportionality.

Calculation:

Let x represent annual sales and y represent annual earnings.

Lisa’s annual earnings will be salary of $18,000 plus 2% commission on annual sales ( x ). We can represent this information in an equation as:

  $\begin{array}{l}y=\frac{2}{100}\cdot x+18000\\ y=0.02x+18000\end{array}$

Upon comparing this equation with direct variation equation, we can see that our equation is not in form $y=kx$ . We have a constant (18,000) added to our equation, therefore, Lisa’s annual sales and annual earnings doesn’t show direct variation.

John’s annual earnings will be 6% commission on his annual sales ( x ). We can represent this information in an equation as:

  $\begin{array}{l}y=\frac{6}{100}\cdot x\\ y=0.06x\end{array}$

Upon comparing this equation with direct variation equation, we can see that our equation is in form $y=kx$ , therefore, John’s annual sales and annual earnings show direct variation.