Zorah, a musician, pays $120 to have her instrument tuned and $10 per hour for a booth at a fair. She estimates that she earns $25 per hour in tips. The equation can be used to represent the break-even point.
[tex]120+10 x=25 x[/tex]
How many hours, [tex]x[/tex], will Zorah have to play in order to break even?
A. 4
B. 5
C. 8
D. 12
Answer (2)
Set up the break-even equation: 120 + 10 x = 25 x .
Subtract 10 x from both sides: 120 = 15 x .
Divide both sides by 15: x = 15 120 .
Calculate the value of x : x = 8 . The final answer is 8 .
Explanation
Problem Analysis Let's analyze the problem. Zorah has an initial cost of $120 for tuning her instrument. She also pays $10 per hour for a booth. Her earnings are $25 per hour in tips. The equation 120 + 10 x = 25 x represents the break-even point, where x is the number of hours she needs to play.
Solving for x To find the number of hours Zorah needs to play to break even, we need to solve the equation 120 + 10 x = 25 x for x .
Isolating the variable term First, subtract 10 x from both sides of the equation:
120 + 10 x − 10 x = 25 x − 10 x
This simplifies to:
120 = 15 x
Solving for x Next, divide both sides of the equation by 15 to solve for x :
15 120 = 15 15 x
x = 8
Final Answer Therefore, Zorah needs to play for 8 hours to break even.
Examples
Understanding break-even points is crucial in many real-world scenarios. For example, if you're starting a small business, you need to know how much you need to sell to cover your costs. Let's say you're selling handmade jewelry at a craft fair. You have fixed costs like the booth fee and materials, and variable costs like the time it takes to make each piece. By calculating your break-even point, you can determine how many pieces of jewelry you need to sell to start making a profit. This helps you set realistic goals and manage your finances effectively. In Zorah's case, she needs to play 8 hours to cover her tuning and booth costs, after which she starts making a profit.
Zorah needs to play for 8 hours to break even. This is found by solving the equation 120 + 10 x = 25 x , leading to x = 8 . Therefore, she must perform for 8 hours to cover her costs and start earning tips.
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