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Chapter 7: Problem 60

You receive two sales job offers. One company offers a straight commission of \(6 \%\) of sales. The other company offers a salary of \(\$ 500\) per week plus \(3 \%\) of sales. How much would you have to sell in a week in order to make the straight commission job offer better?

Short Answer

Expert verified
You would have to sell approximately \$16666.67 in a week for the straight 6% commission job offer to become better.

Step by step solution

01

Set up the equations

Let s be the sales in a week. For the straight commission job, the earnings from a week's sales can be represented as \(0.06s\). For the salary job with commission, the earnings can be represented as \(500 + 0.03s\).
02

Equate the two earnings equations

The point where the straight commission job becomes better is when the earnings from the straight commission job are greater than the earnings from the salary job. Therefore, set the two equations equal to each other: \(0.06s = 500 + 0.03s\).
03

Solve for s

First, isolate s by subtracting \(0.03s\) from both sides, getting \(0.03s = 500\). Then divide both sides by \(0.03\) to solve for s: \(s = 500 / 0.03 = 16666.67\).

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Key Concepts

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

Commission Calculation
When faced with job offers that include commissions, it's essential to understand how commissions are calculated. In commission-based jobs, earnings are typically a percentage of sales made. For instance, receiving a 6% commission means you earn 6% of the total sales as your income. If your weekly sales total \(10,000, your earnings would be calculated as follows:

  • Commission rate: 0.06 (or 6%)
  • Sales amount: \)10,000
  • Commission earnings: \(0.06 \times 10000 = 600 \)
Commission calculations are straightforward once you know the percentage and your total sales. Remember to convert the commission percentage to a decimal when calculating.
Sales Comparison
In job comparisons involving commission and salary, you often need to know at what point one offer becomes more advantageous than the other. Consider two job offers: one that pays only commission and another that offers a base salary plus a smaller commission.

In such cases, it's helpful to set up equations to determine the sales required for one offer to become better. Here's how it works:
  • For a full commission job: Earnings = \(\text{commission rate} \times \text{sales} \)
  • For a salary plus commission job: Earnings = \(\text{base salary} + (\text{commission rate} \times \text{sales}) \)
The aim is to identify the sales figures where both earnings are equal, helping to know when the commission offer might surpass the salary one.
Problem-Solving
Problem-solving with linear equations, like comparing job offers, requires clear steps to find a solution. Here, we aim to find the sales amount that makes a commission-only job more lucrative than a salary-plus-commission job.

First, define your variables and set up the equations based on the problem. Use the example:
  • Straight commission: \(0.06s\)
  • Salary with commission: \(500 + 0.03s\)
Next, initiate a comparison by setting these equal. This enables you to isolate and solve for 's' (sales values). In our exercise, it involves subtracting \(0.03s\) from both sides, leading to \(0.03s = 500\). Divide both sides by 0.03 to find \(s = 16666.67\).

By solving this equation, you determine the sales needed to make one job more appealing than the other. Understanding each step of this process is vital in managing real-world financial decisions.

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Most popular questions from this chapter

A company has budgeted a maximum of \(1,000,000\) for national advertising of an allergy medication. Each minute of television time costs \(100,000\) and each one-page newspaper ad costs \(20,000 .\) Each television ad is expected to be viewed by 20 million viewers, and each newspaper ad is expected to be seen by 5 million readers. The company's market research department recommends that at most \(80 \%\) of the advertising budget be spent on television ads. What is the optimal amount that should be spent on each type of ad? What is the optimal total audience?

Acid Mixture A chemist needs 10 liters of a \(25 \%\) acid solution. The solution is to be mixed from three solutions whose concentrations are \(10 \%, 20 \%,\) and \(50\%\). How many liters of each solution will satisfy each condition? (a) Use 2 liters of the \(50 \%\) solution. (b) Use as little as possible of the \(50 \%\) solution. (c) Use as much as possible of the \(50 \%\) solution.

A wildlife management team studied the reproductive rates of deer in three tracts of a wildlife preserve. Each tract contained 5 acres. In each tract, the number of females \(x,\) and the percent of females \(y\) that had offspring the following year were recorded. The table shows the results. $$\begin{array}{|l|c|c|c|} \hline \text { Number, } x & 100 & 120 & 140 \\\ \hline \text { Percent, } y & 75 & 68 & 55 \\ \hline \end{array}$$ (a) Use the data to create a system of linear equations. Then find the least squares regression parabola for the data by solving the system. (b) Use a graphing utility to graph the parabola and the data in the same viewing window. (c) Use the model to estimate the percent of females that had offspring when there were 170 females. (d) Use the model to estimate the number of females when \(40 \%\) of the females had offspring.

Briefly explain whether it is possible for a consistent system of linear equations to have exactly two solutions.

A humanitarian agency can use two models of vehicles for a refugee rescue mission. Each model A vehicle costs \(1000\) and each model B vehicle costs \(1500 .\) Mission strategies and objectives indicate the following constraints. " The agency must use a total of at least 20 vehicles. \(\cdot \mathrm{A}\) model \(\mathrm{A}\) vehicle can hold 45 boxes of supplies. \(\mathrm{A}\) model B vehicle can hold 30 boxes of supplies. The agency must deliver at least 690 boxes of supplies to the refugee camp. \(\cdot \mathrm{A}\) model \(\mathrm{A}\) vehicle can hold 20 refugees. A model \(\mathrm{B}\) vehicle can hold 32 refugees. The agency must rescue at least 520 refugees. What is the optimal number of vehicles of each model that should be used? What is the optimal cost?
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