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.
Salesperson & Commission/Sale
Mary & $[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 (2)
Arrangement c, where Mary sells 43 units, Jay sells 51 units, and Abby sells 54 units, leads to a total profit that exactly breaks even with the fixed costs of $7622.25. This arrangement results in the company covering all its overhead costs without making a loss. Therefore, the answer is option c.
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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 by multiplying the profit per unit by the number of units sold and summing the results.
Compare the total profit for each arrangement to the fixed overhead cost ($7622.25) to find the break-even point.
The arrangement where Mary sells 43 units, Jay sells 51 units, and Abby sells 54 units results in a total profit of 7622.25 , w hi c hi s t h e b re ak − e v e n p o in t . T h ere f ore , t h e an s w er i s \boxed{c}$.
Explanation
Problem Analysis Let's analyze the problem. We need to determine which sales arrangement results in Arderi Air Conditioning breaking even, meaning their total profit equals their fixed overhead costs. We'll calculate the profit for each salesperson and then test each arrangement to see which one matches the overhead cost of $7622.25.
Calculate Profit Per Unit First, we calculate the profit per unit for each salesperson. The selling price is $142.50, and the production cost is $61.25. Therefore, the gross profit per unit is $142.50 - $61.25 = $81.25. Now we subtract each salesperson's commission to find their individual profit per unit.
Mary's profit per unit: $81.25 - $36.25 = $45.00 Jay's profit per unit: $81.25 - $28.50 = $52.75 Abby's profit per unit: $81.25 - $25.75 = $55.50
Calculate Total Profit for Each Arrangement Now, we test each arrangement to see which one results in a total profit of $7622.25.
a. Mary sells 54 units, Jay sells 51 units, Abby sells 43 units: Total Profit = ($45.00 * 54) + ($52.75 * 51) + ($55.50 * 43) = $2430 + $2690.25 + $2386.50 = $7506.75
b. Mary sells 54 units, Jay sells 43 units, Abby sells 51 units: Total Profit = ($45.00 * 54) + ($52.75 * 43) + ($55.50 * 51) = $2430 + $2268.25 + $2830.50 = $7528.75
c. Mary sells 43 units, Jay sells 51 units, Abby sells 54 units: Total Profit = ($45.00 * 43) + ($52.75 * 51) + ($55.50 * 54) = $1935 + $2690.25 + $2997 = $7622.25
d. Mary sells 43 units, Jay sells 54 units, Abby sells 51 units: Total Profit = ($45.00 * 43) + ($52.75 * 54) + ($55.50 * 51) = $1935 + $2848.50 + $2830.50 = $7614.00
Determine Break-Even Arrangement Comparing the total profit for each arrangement to the fixed overhead cost of $7622.25, we find that arrangement c results in the company breaking even.
Final Answer Therefore, 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 small bakery needs to know how many cakes they must sell each month to cover their rent, ingredients, and staff salaries. By calculating their fixed costs (rent, salaries) and variable costs (ingredients per cake), they can determine the number of cakes needed to be sold to break even. This helps them set sales targets and pricing strategies to ensure profitability. Similarly, understanding sales commissions and their impact on overall profit helps businesses optimize their sales force and commission structures.
