arderi air conditioning sells one primary model of air - conditioner. each unit of this model costs them…

arderi air conditioning sells one primary model of air - conditioner. each unit of this model costs them $61.25 to produce, but it sells for $142.50 apiece. arderi air conditioning employs three salespeople, each of whom earns a different commission per unit sold, as shown on the following table.\n|salesperson|commission/sale|\n|--|--|\n|mary|$36.25|\n|jay|$28.50|\n|abby|$25.75|\nlast month, arderi air conditioning had a total fixed overhead cost of $7,622.25. 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?\n a. mary sells 54 units; jay sells 51 units; abby sells 43 units\n b. mary sells 54 units; jay sells 43 units; abby sells 51 units\n c. mary sells 43 units; jay sells 51 units; abby sells 54 units\n d. mary sells 43 units; jay sells 54 units; abby sells 51 units

arderi air conditioning sells one primary model of air - conditioner. each unit of this model costs them $61.25 to produce, but it sells for $142.50 apiece. arderi air conditioning employs three salespeople, each of whom earns a different commission per unit sold, as shown on the following table.\n|salesperson|commission/sale|\n|--|--|\n|mary|$36.25|\n|jay|$28.50|\n|abby|$25.75|\nlast month, arderi air conditioning had a total fixed overhead cost of $7,622.25. 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?\n a. mary sells 54 units; jay sells 51 units; abby sells 43 units\n b. mary sells 54 units; jay sells 43 units; abby sells 51 units\n c. mary sells 43 units; jay sells 51 units; abby sells 54 units\n d. mary sells 43 units; jay sells 54 units; abby sells 51 units

Answer

Explanation:

Step1: Calculate profit per unit before commission

The selling - price per unit is $142.50 and the production cost per unit is $61.25. So the profit per unit before commission is $142.50−61.25 = 81.25$.

Step2: Set up the break - even equation

Let the number of units sold by Mary be $m$, by Jay be $j$, and by Abby be $a$. The total fixed cost is $C = 7622.25$. The profit from selling the units is $P=(81.25m + 81.25j+81.25a)-(36.25m + 28.50j + 25.75a)$. Simplify the profit formula: $P=(81.25 - 36.25)m+(81.25 - 28.50)j+(81.25 - 25.75)a=45m + 52.75j+55.5a$. We want to find when $P = 7622.25$.

Step3: Test option a

If $m = 54$, $j = 51$, and $a = 43$. Then $P=45\times54+52.75\times51 + 55.5\times43$. $45\times54=2430$, $52.75\times51 = 52.75\times(50 + 1)=52.75\times50+52.75=2637.5+52.75 = 2690.25$, $55.5\times43=(50 + 5.5)\times43=50\times43+5.5\times43=2150+236.5 = 2386.5$. $P=2430+2690.25+2386.5=7506.75\neq7622.25$.

Step4: Test option b

If $m = 54$, $j = 43$, and $a = 51$. Then $P=45\times54+52.75\times43+55.5\times51$. $45\times54 = 2430$, $52.75\times43=52.75\times(40 + 3)=52.75\times40+52.75\times3=2110+158.25 = 2268.25$, $55.5\times51=(50 + 5.5)\times51=50\times51+5.5\times51=2550+280.5 = 2830.5$. $P=2430+2268.25+2830.5=7528.75\neq7622.25$.

Step5: Test option c

If $m = 43$, $j = 51$, and $a = 54$. Then $P=45\times43+52.75\times51+55.5\times54$. $45\times43 = 1935$, $52.75\times51=2690.25$, $55.5\times54=(50 + 5.5)\times54=50\times54+5.5\times54=2700+297 = 2997$. $P=1935+2690.25+2997=7622.25$.

Step6: Test option d

If $m = 43$, $j = 54$, and $a = 51$. Then $P=45\times43+52.75\times54+55.5\times51$. $45\times43 = 1935$, $52.75\times54=52.75\times(50 + 4)=52.75\times50+52.75\times4=2637.5+211 = 2848.5$, $55.5\times51=2830.5$. $P=1935+2848.5+2830.5=7614\neq7622.25$.

Answer:

C. Mary sells 43 units; Jay sells 51 units; Abby sells 54 units