Khybar Inc. manufactures dental x-ray machines. The company can sell an x-ray machine which cost $508.17 to produce for $1,295.75. Each of the company's two salespeople earns a different commission per sale, as shown in the table below.
| Salesperson | Commission/Sale |
| :---------- | :---------------- |
| Greg | $243.15 |
| Colleen | $288.75 |
Last year, Greg sold 11 fewer x-ray machines than Colleen did. Khybar Inc.'s total expenses last year, not counting production costs or commissions, came to $79,558.59. If Khybar Inc. broke even, how many x-ray machines were sold last year in total?
A. 149
B. 153
C. 157
D. 160
Answer (2)
Khybar Inc. sold a total of 153 x-ray machines last year. This was calculated by defining the sales of each salesperson, establishing a break-even equation, and solving for the number of machines sold. Greg sold 71 machines, while Colleen sold 82 machines.
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Define variables: Let g be the number of machines Greg sold and c be the number of machines Colleen sold, where g = c − 11 .
Set up the revenue and cost equations based on the given information and the break-even condition.
Substitute g = c − 11 into the revenue equals cost equation and solve for c , finding c = 82 .
Calculate g using g = c − 11 , which gives g = 71 . The total number of machines sold is 153 .
Explanation
Problem Analysis Let's analyze the problem. We are given the production cost and selling price of an x-ray machine. We also know the commission each salesperson earns per sale. Greg sold 11 fewer machines than Colleen. The total fixed expenses are given. The company broke even, meaning total revenue equals total cost. We need to find the total number of x-ray machines sold.
Define Variables and Equations Let g be the number of machines Greg sold and c be the number of machines Colleen sold. We know that Greg sold 11 fewer machines than Colleen, so we can write this as: g = c − 11
Set up Revenue and Cost Equations The revenue per machine is $1295.75 , and the cost to produce each machine is $508.17 . Greg's commission is $243.15 per machine, and Colleen's commission is $288.75 per machine. The fixed expenses are $79558.59 . Since the company broke even, the total revenue equals the total cost. The total revenue is the selling price times the total number of machines sold, which is 1295.75 ( g + c ) . The total cost is the sum of the production costs, commissions, and fixed expenses, which is 508.17 ( g + c ) + 243.15 g + 288.75 c + 79558.59 . Therefore, we have the equation: 1295.75 ( g + c ) = 508.17 ( g + c ) + 243.15 g + 288.75 c + 79558.59
Solve for c Now, substitute g = c − 11 into the equation: 1295.75 (( c − 11 ) + c ) = 508.17 (( c − 11 ) + c ) + 243.15 ( c − 11 ) + 288.75 c + 79558.59 Simplify the equation: 1295.75 ( 2 c − 11 ) = 508.17 ( 2 c − 11 ) + 243.15 ( c − 11 ) + 288.75 c + 79558.59 2591.5 c − 14253.25 = 1016.34 c − 5589.87 + 243.15 c − 2674.65 + 288.75 c + 79558.59 2591.5 c − 14253.25 = 1548.24 c + 71294.07 2591.5 c − 1548.24 c = 71294.07 + 14253.25 1043.26 c = 85547.32 c = 1043.26 85547.32 = 82
Solve for g Now that we have the value of c , we can find the value of g :
g = c − 11 = 82 − 11 = 71
Calculate Total Machines Sold The total number of machines sold is g + c = 71 + 82 = 153 .
Final Answer Therefore, the total number of x-ray machines sold last year is 153.
Examples
Understanding break-even points is crucial in business. For example, a small bakery can use this concept to determine how many cakes they need to sell each month to cover their costs (ingredients, rent, salaries, etc.). By calculating the break-even point, the bakery can set realistic sales goals and pricing strategies to ensure profitability. This same principle applies to larger companies like Khybar Inc., where understanding the break-even point helps them manage production, sales, and expenses effectively to achieve financial stability.
