Problem 7 - Algebra Form and Function

Question

Company offers three formulas for the weekly salary of its sales people, depending on the number of sales, \(s,\) made each week: (a) \(100+0.10 s\) dollars (b) \(150+0.05 s\) dollars (c) 175 dollars At what sales level do options (a) and (b) produce the same salary?

Step-by-Step Solution

Answer

Answer: The same salary is produced at a sales level of 1000.

Step 1: Write down the equations for the salaries in both options (a) and (b)

For option (a), the weekly salary is given by the equation \(W_a = 100 + 0.10s\). For option (b), the weekly salary is given by the equation \(W_b = 150 + 0.05s\).

Step 2: Set the two equations equal to each other

To find the sales level where both options produce the same salary, we need to set \(W_a\) equal to \(W_b\), so we get the equation: \(100 + 0.10s = 150 + 0.05s\).

Step 3: Subtract 0.05s from both sides

Subtracting 0.05s from both sides of the equation, we obtain \(0.05s = 50\).

Step 4: Divide by 0.05 to find s

To solve for \(s\), we can divide both sides of the equation by 0.05: \(\frac{0.05s}{0.05} = \frac{50}{0.05}\), which simplifies to \(s = 1000\). Thus, options (a) and (b) produce the same salary at a sales level of 1000.

Key Concepts

Weekly Salary Calculation

When calculating a weekly salary for a sales job, it's important to understand how different components of the pay formula affect your earnings. Many jobs include a base salary and a commission based on sales.
The equations given in our exercise represent these components nicely. For example:

• Option (a): Start with a base pay of \(100 and add a 10% commission on total sales, represented as \(100 + 0.10s\).
• Option (b): This begins with a higher base pay of \)150 but includes a smaller commission rate of 5% on sales, described by \(150 + 0.05s\).
• Option (c): A fixed salary of $175, which remains constant regardless of sales.

Understanding how these components work is crucial, as it allows employees to estimate their potential earnings for varying levels of sales. It's also valuable for comparing different pay structures to see which might offer better financial rewards based on your expected sales efforts.

Setting Equations Equal

One vital concept in algebra is setting equations equal to find a point where two different expressions give the same result. This is used extensively in pay comparison scenarios.
For example, when trying to find when two salary options provide the same earnings, you set their equations equal:

• Equation for option (a): \(W_a = 100 + 0.10s\)
• Equation for option (b): \(W_b = 150 + 0.05s\)

To find out when these two pay structures yield the same salary, set them equal: \(100 + 0.10s = 150 + 0.05s\). This step requires solving for the sales level \(s\) that will make both equations balance. This type of problem-solving is common in scenarios where comparison of different offers or options is necessary, helping to determine which is the most advantageous in specific circumstances.

Sales Level Comparison

Once the equations are set equal, the next step is to determine the sales level at which those pay options converge. Simplifying the equality \(100 + 0.10s = 150 + 0.05s\), subtracting \(0.05s\) from both sides simplifies the equation to \(0.05s = 50\).

To find \(s\), divide both sides by 0.05, resulting in \(s = 1000\). This means when a sales person sells products worth $1,000, the earnings from both pay options are the same.

This type of comparison not only allows businesses to create competitive pay structures but also empowers employees with knowledge to navigate the best pay options based on their performance. In practical terms, understanding these calculations helps prioritize strategies aimed at surpassing sales benchmarks, ultimately leading to increased overall earnings.