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Difficulty: EasyAlgebraic Word Problems and Equation Modeling

Company A charges a one-time fixed fee of $20\$20 plus $5\$5 per hour to rent a piece of machinery. Company B charges a one-time fixed fee of $35\$35 plus $2\$2 per hour to rent the same machinery. For how many hours of rental will the total cost charged by both companies be equal?

  1. A
    3
  2. 5Answer
  3. C
    7
  4. D
    11
  5. E
    15

Answer

The total cost at both companies will be equal for a rental duration of 5 hours.
The cost charged by Company A is modeled by 20+5h20 + 5h and Company B by 35+2h35 + 2h. Setting these equal yields 20+5h=35+2h20 + 5h = 35 + 2h, which simplifies to 3h=153h = 15, giving h=5h = 5. Thus, 5 hours is the correct rental duration.

Step-by-Step Solution

1
Define the variable and write total cost equations for each company.
Let hh represent the number of hours. Company A's cost is CA=20+5hC_A = 20 + 5h and Company B's cost is CB=35+2hC_B = 35 + 2h.
Linear cost models combine a fixed base cost with a variable hourly component.
2
Set the two cost equations equal to solve for hh.
20+5h=35+2h20 + 5h = 35 + 2h
The total costs are equal when CA=CBC_A = C_B.
3
Isolate the variable term hh on one side of the equation.
5h−2h=35−20⇒3h=15⇒h=55h - 2h = 35 - 20 \Rightarrow 3h = 15 \Rightarrow h = 5
Subtracting 2h2h and 2020 from both sides simplifies the equation to solve directly for hh.

Key Concept

Setting up and solving single-variable linear equations from real-world cost contexts.
Estimated Time:45s
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Topics

Algebraic Word Problems and Equation Modeling

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