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Difficulty: MediumLinear Equations in One Variable

A company manufactures custom travel mugs. The daily cost CC, in dollars, to manufacture nn mugs is given by the formula C=1.25n+250C = 1.25n + 250. The company sells each mug for 4.50.Tomakeaprofitofexactly4.50. To make a profit of exactly 400$ in a day, how many mugs must the company manufacture?

Answer: 200 mugs

Answer

The company must manufacture 200 mugs to make a profit of exactly $400 in a day.
Subtracting the cost expression from the revenue expression gives the profit equation: 4.50n−(1.25n+250)=4004.50n - (1.25n + 250) = 400. Distributing the negative sign results in 4.50n−1.25n−250=4004.50n - 1.25n - 250 = 400, which simplifies to 3.25n−250=4003.25n - 250 = 400. Adding 250 to both sides gives 3.25n=6503.25n = 650. Dividing both sides by 3.25 yields n=200n = 200.

Step-by-Step Solution

1
Set up the profit equation
4.50n−(1.25n+250)=4004.50n - (1.25n + 250) = 400
Profit is calculated as total revenue (4.50n4.50n) minus total cost (1.25n+2501.25n + 250).
2
Distribute the negative sign and simplify the equation
3.25n−250=4003.25n - 250 = 400
Subtracting the entire cost expression requires distributing the subtraction to both 1.25n1.25n and 250250, yielding 4.50n−1.25n−2504.50n - 1.25n - 250.
3
Isolate the variable term by adding 250 to both sides
3.25n=6503.25n = 650
Adding 250 to both sides of the equation isolates the term containing nn.
4
Solve for nn by dividing both sides by 3.25
n=200n = 200
Dividing 650 by 3.25 gives the number of mugs needed to achieve the target profit.

Key Concept

Solving linear equations in one variable derived from a real-world context.
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Topics

Linear Equations in One Variable

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