Problem 319 - Elementary Algebra

Question

In the following exercises, solve. Andre has been offered an entry-level job. The company offered him \(\$ 48,000\) per year plus \(3.5 \%\) of his total sales. Andre knows that the average pay for this job is \(\$ 62,000\). What would Andre's total sales need to be for his pay to be at least as high as the average pay for this job?

Step-by-Step Solution

Answer

Andre's total sales need to be at least \(\$ 400,000\).

Step 1: Identify the fixed salary

The fixed salary offered to Andre is given directly. It is \(\$ 48,000\).

Step 2: Determine the pay he wants to achieve

Andre wants to achieve at least the average pay for this job, which is \(\$ 62,000\).

Step 3: Set up the equation for the total salary

The total salary Andre would earn combines his fixed salary plus \3.5 \% of his total sales. We'll denote total sales as \(S\). Hence, Andre's total salary can be represented as: \[ 48,000 + 0.035 S \]

Step 4: Set up and solve the inequality

To find the total sales needed for Andre's pay to be at least \(\$ 62,000\), set up the following inequality: \[ 48,000 + 0.035 S \ge 62,000 \] Subtract \(48,000\) from both sides: \[ 0.035 S \ge 14,000 \] Divide both sides by \0.035\: \[ S \ge \frac{14,000}{0.035} \]

Step 5: Calculate the necessary sales

Perform the division to find \(S\): \[ S \ge 400,000 \]

Key Concepts

Solving Inequalities

In algebra, inequalities help us compare two values or expressions. When we solve inequalities, we are finding the range of values that satisfy a certain condition. In this problem, we need to determine the total sales (\(S\)) that Andre needs to achieve so that his total salary is at least as high as \$62,000\. Here are the steps you should follow to solve an inequality:

• Identify the given information: In this problem, Andre's fixed salary is \$48,000\, and he wants his total salary to be at least \$62,000\.
• Set up the inequality: The inequality representing his total salary can be set as follows: \[48,000 + 0.035S \ge 62,000\].
• Isolate the variable: Subtract \$48,000\ from both sides to simplify: \[0.035S \ge 14,000\].
• Solve for the variable: Divide both sides by \(0.035\): \[S \ge \frac{14,000}{0.035}\]
• Perform the calculation: This gives \[S \ge 400,000\].

So, to solve inequalities, you often perform operations similar to those used in equations, but you need to be careful with the direction of the inequality sign, especially when multiplying or dividing by negative numbers.

Percentage Calculations

Percentages are a way to express a number as a fraction of 100. In this problem, Andre's commission is \(3.5\%\) of his total sales. To work with this percentage, follow these steps:

• Convert the percentage to a decimal: To convert \(3.5\%\) to a decimal, divide by 100: \(3.5 \div 100 = 0.035\).
• Apply the decimal in a calculation: Multiply this decimal by the total sales (\(S\)) to find the commission amount: \[ 0.035S \].

This commission is then added to the fixed salary to determine Andre's total earnings. Understanding how to convert and use percentages in calculations is crucial for solving real-world problems like this one. It allows us to tackle situations involving sales commissions, discounts, interest rates, and more.

Linear Equations

Linear equations are equations between two variables that produce a straight line when graphed. In this problem, the equation for Andre's total salary can be written as a linear expression: \[ 48,000 + 0.035S \]. The components of a linear equation include:

• Constant term: This is the fixed salary, which is \(\text{48,000}\) in our equation.
• Variable term: This involves the sales (\(\text{S}\)), which is multiplied by the commission rate (\(0.035\)), forming the linear term \[0.035S\].

To solve for a specific value of \(S\), you rearrange the equation and isolate the variable, just like we did with the inequality. First, subtract the constant term, then divide by the coefficient of the variable. Understanding linear equations is essential as it forms the basis of many algebraic operations and is widely used in various fields including business, physics, and economics.