Zorah, a musician, pays $[tex]120[/tex] to have her instrument tuned and $[tex]10[/tex] per hour for a booth at a fair. She estimates that she earns $[tex]25[/tex] 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 .
Solve for x : x = 8 . The number of hours Zorah has to play in order to break even is 8 .
Explanation
Problem Analysis Let's analyze the problem. Zorah has initial costs of $120 for tuning her instrument. She also pays $10 per hour for a booth. Her earnings are $25 per hour. The break-even point is when her earnings equal her costs. The equation given is 120 + 10 x = 25 x , where x is the number of hours she needs to play.
Setting up the Equation To find the break-even point, we need to solve the equation for x . We start with the equation: 120 + 10 x = 25 x
Isolating the x term Subtract 10 x from both sides of the equation to isolate the x terms on one side: 120 + 10 x − 10 x = 25 x − 10 x 120 = 15 x
Solving for x Now, divide both sides by 15 to solve for x :
15 120 = 15 15 x x = 15 120 x = 8
Final Answer Therefore, Zorah needs to play for 8 hours to break even.
Examples
Imagine you're running a lemonade stand. You spend $20 on supplies (lemons, sugar, cups). You sell each cup for $1.50 , but it costs you $0.50 per cup for the lemonade mix. To find out how many cups you need to sell to cover your initial costs, you can use the break-even concept. In this case, your initial cost is $20 , your revenue per cup is $1.50 , and your cost per cup is $0.50 . So, your profit per cup is $1.50 − $0.50 = $1.00 . To break even, you need to sell 1 20 = 20 cups. This concept is useful in many business scenarios to determine the sales volume needed to cover costs and start making a profit.
Zorah has to play for 8 hours to break even. This is calculated using the equation 120 + 10 x = 25 x , which represents her costs and earnings. By simplifying the equation, we find that x = 8 .
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