Question

Sales Commission A used car salesperson is paid a commission of \(\$ 25\) plus \(40 \%\) of the selling price in excess of owner's cost. The owner claims that used cars typically sell for at least owner's cost plus \(\$ 200\) and at most owner's cost. plus \(\$ 3000 .\) For each sale made, over what range can the salesperson expect the commission to vary?

Short answer

The commission ranges from \(105 to \)1225.

Step-by-step solution

Understand the Problem

The problem involves calculating the range of commission a used car salesperson receives based on the selling price exceeding the owner's cost by between \(200 and \)3000.

Define Variables

Let the owner's cost be denoted as \(C\). The selling price then ranges from \(C + 200\) to \(C + 3000\).

Formulate the Commission Formula

The commission given is \(\text{Commission} = 25 + 0.4 \times (\text{Selling Price} - C)\).

Calculate Minimum Commission

Substitute the minimum selling price \(C + 200\) into the commission formula: \(\text{Commission}_{\text{min}} = 25 + 0.4 \times (C + 200 - C) = 25 + 0.4 \times 200 = 25 + 80 = 105\).

Calculate Maximum Commission

Substitute the maximum selling price \(C + 3000\) into the commission formula: \(\text{Commission}_{\text{max}} = 25 + 0.4 \times (C + 3000 - C) = 25 + 0.4 \times 3000 = 25 + 1200 = 1225\).

Determine Range

From the calculations, the commission can vary from \(\$105\) to \(\$1225\).

Key concepts

Percentage of Selling Price

Understanding the percentage of the selling price is crucial for calculating commission.
In this problem, the salesperson receives 40% of the selling price that goes above the owner’s cost.
This percentage shows how much of the extra money (above the cost) the salesperson keeps as commission.
To find this, use the formula:
\(\text{Commission} = 25 + 0.4 \times (\text{Selling Price} - C)\)
This means every dollar above the owner's cost, the salesperson gets 40 cents.
Knowing percentages helps you see how small changes in the selling price affect the commission.

Commission Range

The range of commission tells us the least and most a salesperson can earn.
Here, the owner claims cars sell between \$200 and \$3000 more than they cost.
By applying these limits to the formula, we can see how commission changes.
Let's substitute these extreme values:
For the minimum increase of \$200,\(\text{Commission Min} = 25 + 0.4 \times 200 = 25 + 80 = 105\)
For the maximum increase of \$3000, \(\text{Commission Max} = 25 + 0.4 \times 3000 = 25 + 1200 = 1225\)
With these calculations, commission varies from \$105 to \$1225.
The range helps in setting expectations for earnings.

Cost-Price Relationship

The cost-price relationship is the difference between the selling price and the owner's cost.
This difference is crucial as it becomes the base for calculating the commission percentage.
Essentially, if a car sells for \(C + 200\), the excess is \$200.
If it sells for \(C + 3000\), the excess is \$3000.
This way, depending on how much more than the cost the car sells, the commission changes proportionally.
Understanding this relationship helps see how much more profit (and commission) comes from higher selling prices.

Variable Definition

Clear variable definitions make solving the problem easier.
In this problem, the owner's cost is denoted as \(C\).
The selling price ranges from \(C + 200\) to \(C + 3000\).
Defining these helps set up the commission formula accurately.
Always start by defining your variables, so you know exactly what you're calculating.
Clarity in variable definitions avoids confusion and errors during calculations.