Problem 56 - College Algebra with Corequisite Support

Question

When hired at a new job selling electronics, you are given two pay options: Option A: Base salary of $$\$ 14,000$$ a year with a commission of \(10 \%\) of your sales Option 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?

Step-by-Step Solution

Answer

You need to sell more than $83,333.33 in electronics for Option A to provide a larger income.

Step 1: Understand the Pay Options

There are two pay options: - **Option A**: Base salary = $14,000, Commission = 10% - **Option B**: Base salary = $19,000, Commission = 4% Our goal is to find the sales amount for which Option A produces a larger income than Option B.

Step 2: Set Up Income Equations

For Option A, the income can be expressed as:\[ \text{Income from A} = 14000 + 0.10X \]For Option B, the income is:\[ \text{Income from B} = 19000 + 0.04X \]where \(X\) is the amount of electronics sold in dollars.

Step 3: Set the Inequality

To find when Option A produces a larger income than Option B, set the inequality:\[ 14000 + 0.10X > 19000 + 0.04X \]

Step 4: Solve the Inequality for X

Rearrange the inequality:1. Subtract \(14000\) from both sides: \[ 0.10X > 5000 + 0.04X \]2. Subtract \(0.04X\) from both sides:\[ 0.06X > 5000 \]3. Divide both sides by \(0.06\):\[ X > \frac{5000}{0.06} \]\[ X > 83333.33 \]Thus, more than $83,333.33 of electronics need to be sold for Option A to be more beneficial.

Key Concepts

Commission Calculations

When deciding between two pay structures that include commission, it's essential to understand how commission works. A commission is a percentage of sales that an employee earns in addition to a base salary. To calculate the commission:

• Identify the commission rate, which is the percentage of sales given as a bonus.
• Apply this rate to the sales amount. For example, if you sell items worth $10,000 with a 10% commission rate, your commission is $1,000.

In this problem, Option A provides a 10% commission rate while Option B gives 4%. This means for Option A: - If you sell $1 worth of electronics, you earn $0.10 as commission. - For Option B, selling the same amount earns you $0.04. Commissions can significantly impact overall income, depending on the rate and total sales made. In this scenario, a higher commission rate only makes Option A lucrative if sales are beyond a certain threshold.

Income Comparison

When faced with multiple job offers, comparing the income structures is vital to make an informed decision. This involves adding both the base salary and the potential commission to determine total earnings for each option.For this exercise:

• Option A starts with a base salary of \(14,000, plus 10% of sales
• Option B offers a baseline of \)19,000, plus 4% of sales

The core question is to identify at what sales point Option A results in higher income than Option B. You need to set up income equations for each option:- Income from A becomes \(14,000 + 0.10X\)- Income from B is \(19,000 + 0.04X\)Here, \(X\) stands for the dollar amount of electronics sold. By comprehending these equations, you can visualize at what level of sales each option becomes financially advantageous.

Linear Inequalities

Linear inequalities are a fundamental concept in algebra used to compare quantities and find conditions where one is greater or less than another. In this task, we used a linear inequality to determine at what point Option A yields more income than Option B.To set up the inequality:- Start with the income equations: \(14,000 + 0.10X > 19,000 + 0.04X\)- The goal is to isolate \(X\) to find the amount of sales necessary.Break down the steps:

• Subtract 14,000 from both sides: \(0.10X > 5,000 + 0.04X\)
• Eliminate the 0.04X on the right: \(0.06X > 5,000\)
• To isolate \(X\), divide both sides by 0.06, resulting in \(X > 83,333.33\)

This final result tells you more than $83,333.33 in sales is required for Option A to surpass Option B in income. Here, solving linear inequalities provided a clear pathway to the solution.