Problem 5 - 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 How many sales must be made under option (a) to receive \(\$ 200\) a week?

Step-by-Step Solution

Answer

Answer: A salesperson must make 1,000 sales under option (a) to receive a salary of $200 a week.

Step 1: Write down option (a) salary formula

Option (a) salary formula is given as \(100+0.10s\) dollars. This formula will be used to set up an equation, where we equate the salary to \(200\) dollars and solve for the number of sales, \(s.\)

Step 2: Set up the equation

Using the formula in step 1, let's set up an equation that will equate the weekly salary to \(200:\) $$ 100 + 0.10s = 200 $$

Step 3: Solve for the number of sales, \(s\)

Now, let's solve the equation for \(s:\) \begin{align*} 100 + 0.10s &= 200 \\ 0.10s &= 200 - 100 \\ 0.10s &= 100 \\ s &= \frac{100}{0.10} \\ s &= 1000 \end{align*}

Step 4: Interpret the result

The number of sales that must be made under option (a) to receive \(\$200\) a week is \(s=1000\). This means that a salesperson needs to make 1000 sales in a week to receive a salary of \(\$200\) under the given formula.

Key Concepts

Salary Formulas

Many companies provide different salary formulas for their employees. These formulas help to calculate the earnings based on various factors, like sales, hours worked, or performance metrics. Understanding these formulas is crucial for employees who work on commission or have performance-based pay.

In the provided exercise, we see three different salary formulas:

• Option (a): \(100+0.10s\) dollars, where the employee earns a base salary of \(100 plus 10% of the total sales, denoted by \(s\).
• Option (b): \(150+0.05s\) dollars, which offers a higher base salary of \)150, but only 5% of the sales as commission.
• Option (c): a flat rate of $175, regardless of sales made.

Understanding different salary packages allows salespeople to evaluate which option might be more profitable based on their expected sales performance.

With salary formulas, calculating your expected earnings becomes straightforward by plugging your sales numbers into the equation.

Equation Solving

Equation solving is a fundamental math skill that allows us to find unknown values, like how many sales are needed for a specific salary.

In the exercise, solving the equation involves a few steps:- Start with the formula \(100 + 0.10s = 200\).- Subtract 100 from both sides to isolate the term with \(s\): \(0.10s = 100\).- Divide both sides by 0.10 to find \(s\): \(s = \frac{100}{0.10} = 1000\).

This step-by-step approach is crucial to solve linear equations efficiently. Make sure to always perform the same operation to both sides of the equation, maintaining balance. This concept is essential for solving any mathematical problem where values need to be discovered.

PRACTICE TIP: Try solving similar problems by writing out the steps. Consistent practice helps build confidence in equation-solving skills.

Linear Relationships

In mathematics, linear relationships are expressed as straight-line graphs. These relationships are defined by the equation \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.

In the salary formulas, the linear relationship between sales, \(s\), and salary is represented by equations such as \(100 + 0.10s\). This means the salary will increase linearly as the sales increase. Here:

• examining the term \(0.10s\) reveals how the salary depends on the number of sales.
• the constant term (100, 150, or 175) represents a fixed starting salary.

Linear equations help forecast financial outcomes based on varying input, such as sales numbers. Grasping linear relationships aids in predicting how changes in one variable can affect another. Hence, understanding this type of relationship is essential in many fields, especially in finance and business, to make informed decisions.