Arderi Air Conditioning sells one primary model of air conditioner. Each unit of this model costs them $[tex]$61.25$[/tex] to produce, but it sells for $[tex]$142.50$[/tex] apiece. Arderi Air Conditioning employs three salespeople, each of whom earns a different commission per unit sold, as shown on the following table.
SalespersonCommission/SaleMary$[tex]$36.25$[/tex]Jay$[tex]$28.50$[/tex]Abby$[tex]$25.75$[/tex]
Last month, Arderi Air Conditioning had a total fixed overhead cost of $[tex]$7,622.25$[/tex]. It is known that one salesperson sold 51 units, one salesperson sold 43 units, and one salesperson sold 54 units. Which arrangement of sales would cause Arderi Air Conditioning to break even?
a. Mary sells 54 units; Jay sells 51 units: Abby sells 43 units
b. Mary sells 54 units; Jay sells 43 units: Abby sells 51 units
c. Mary sells 43 units; Jay sells 51 units. Abby sells 54 units
d. Mary sells 43 units: Jay sells 54 units: Abby sells 51 units
Answer (1)
Calculate the profit per unit for each salesperson: Mary ($45.00), Jay ($52.75), Abby ($55.50).
Calculate the total profit for each sales arrangement.
Arrangement c (Mary: 43, Jay: 51, Abby: 54) yields a total profit of $7,622.25, matching the fixed overhead cost.
The break-even arrangement is Mary sells 43 units, Jay sells 51 units, and Abby sells 54 units: c
Explanation
Calculate Profit Per Unit First, we need to calculate the profit per unit for each salesperson. This is the selling price minus the production cost and the salesperson's commission.
Determine Individual Profits Mary's profit per unit: $142.50 - $61.25 - $36.25 = $45.00 Jay's profit per unit: $142.50 - $61.25 - $28.50 = $52.75 Abby's profit per unit: $142.50 - $61.25 - $25.75 = $55.50
Calculate Total Profit for Each Arrangement Next, we calculate the total profit for each arrangement by multiplying the number of units sold by each salesperson by their respective profit per unit and summing the results. We are looking for the arrangement where the total profit equals the fixed overhead cost of $7,622.25.
Compute Total Profits Arrangement a: Mary sells 54, Jay sells 51, Abby sells 43 Total profit = (54 * $45.00) + (51 * $52.75) + (43 * $55.50) = $2430 + $2690.25 + $2386.50 = $7506.75
Arrangement b: Mary sells 54, Jay sells 43, Abby sells 51 Total profit = (54 * $45.00) + (43 * $52.75) + (51 * $55.50) = $2430 + $2268.25 + $2830.50 = $7528.75
Arrangement c: Mary sells 43, Jay sells 51, Abby sells 54 Total profit = (43 * $45.00) + (51 * $52.75) + (54 * $55.50) = $1935 + $2690.25 + $2997 = $7622.25
Arrangement d: Mary sells 43, Jay sells 54, Abby sells 51 Total profit = (43 * $45.00) + (54 * $52.75) + (51 * $55.50) = $1935 + $2848.50 + $2830.50 = $7614.00
Identify Break-Even Arrangement Comparing the total profit for each arrangement to the fixed overhead cost, we find that arrangement c results in a total profit of $7,622.25, which is equal to the fixed overhead cost. Therefore, arrangement c is the break-even arrangement.
Final Answer Confirmed The arrangement that causes Arderi Air Conditioning to break even is: Mary sells 43 units; Jay sells 51 units; Abby sells 54 units.
Examples
Understanding break-even points is crucial in business. For example, a bakery needs to know how many cakes they must sell each month to cover their rent, ingredients, and staff salaries. By calculating the cost to produce each cake and the selling price, they can determine the number of cakes needed to break even. This helps them make informed decisions about pricing, production levels, and potential cost-cutting measures to ensure profitability. Similarly, in manufacturing, knowing the break-even point helps in setting production targets and managing expenses effectively.
