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Chapter 4: Problem 56

When hired at a new job selling electronics, you are given two pay options: \(\cdot\) Option A: Base salary of \(\$ 14,000\) a year with a commission of 10\(\%\) of your sales \(\cdot\) Option \(\mathrm{B} :\) Base salary of \(\$ 19,000\) a year with a commission of 4\(\%\) of your sales How much electronics would you need to sell for option A to produce a larger income?

Short Answer

Expert verified
Sell over \$83,333.33 in electronics for Option A to give more income.

Step by step solution

01

Define Variables for Income Equations

Let's define the variable \( x \) as the dollar amount of electronics sold. The income from Option A, which includes a base salary of \\(14,000 and a commission of 10%, can be expressed as \( I_A = 14000 + 0.10x \). Similarly, the income from Option B, which includes a base salary of \\)19,000 and a commission of 4%, can be written as \( I_B = 19000 + 0.04x \).
02

Set Up the Inequality for Option A to be Greater than Option B

For Option A to produce a larger income compared to Option B, the equation \( I_A > I_B \) must hold. By substituting the expressions from Step 1, the inequality becomes \( 14000 + 0.10x > 19000 + 0.04x \).
03

Solve the Inequality

Subtract \( 14000 \) and \( 0.04x \) from both sides of the inequality: \( 0.10x - 0.04x > 19000 - 14000 \). Simplify the inequality to \( 0.06x > 5000 \). Next, divide both sides by 0.06 to find \( x \): \( x > \frac{5000}{0.06} \).
04

Compute the Value of x

Calculate \( x \) by dividing 5000 by 0.06: \( x > 83333.33 \). This implies that you need to sell more than \$83,333.33 worth of electronics for Option A to give you a higher income than Option B.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variable Definition
Before tackling any algebraic problem, especially those dealing with income inequality, it's essential to define your variables clearly. In this scenario, the key variable is the total sales in dollars, which we'll denote by \( x \). Understanding that this variable represents the amount of electronics sold allows us to formulate expressions for different income options. This step is crucial because it transforms the real-world problem into a mathematical one, allowing for analysis and solution. Keep in mind that variables like \( x \) serve as placeholders that can represent any number depending on the situation.
Inequality Solving
Inequalities are mathematical expressions that show the relationship between two values when they are not equal. In this problem, we want the income from Option A to be greater than the income from Option B.
This gives rise to the inequality \( I_A > I_B \). By substituting the income formulas, this is expressed as:
  • \( 14000 + 0.10x > 19000 + 0.04x \)

Solving the inequality starts with simplifying both sides. Eliminate terms to isolate \( x \) on one side of the inequality. Simplify the inequality to obtain:
\( 0.06x > 5000 \). Finally, solve for \( x \) by dividing both sides by \( 0.06 \), resulting in \( x > 83333.33 \). This tells you how much you need to sell for Option A to be more beneficial.
Commission Calculation
Commission is a portion of sales given to the salesperson and is a critical component of income equations in sales jobs. For Option A, the commission is 10% of the total sales \( x \), which can be represented algebraically as \( 0.10x \).
For Option B, the commission is 4% of sales, or \( 0.04x \). Calculating commission accurately is fundamental, as it adds significantly to the base salary.
Remember, commissions incentivize sales, and understanding how they affect income helps in choosing the best pay option based on expected sales.
Income Equations
Income equations model total earnings by combining base salaries with commissions. For this exercise, two equations are set up to reflect two salary options:
  • Option A: \( I_A = 14000 + 0.10x \)
  • Option B: \( I_B = 19000 + 0.04x \)
Income equations are vital in understanding the cumulative effect of a base salary and commission over sales. By comparing these equations, you can determine the break-even point or when one option yields a higher income than the other.
Carefully assessing these equations allows for strategic financial planning and decision-making in job selection based on projected sales outcomes.

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