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Difficulty: MediumLinear Equations in One Variable

A technical consultant charges a one-time setup fee of $150\$150 plus a standard rate of $45\$45 per hour for the first 2020 hours of work on a project. For any hours worked beyond 2020 hours, the consultant charges an increased hourly rate that is 20%20\% higher than the standard rate. If the total bill for a project was $1,320\$1,320, how many total hours did the consultant work on the project?

Answer: 25 hours

Answer

The consultant worked a total of 25 hours on the project.
To find the total hours worked, first subtract the setup fee (150)andthecostofthefirst20hours(20∗150) and the cost of the first 20 hours (20 * 45 = 900)fromthetotalbillof900) from the total bill of 1,320. This leaves 1,320−1,320 - 1,050 = 270.Thehourlyrateafter20hoursincreasesby20270. The hourly rate after 20 hours increases by 20% to 45 * 1.20 = 54perhour.Dividingtheremaining54 per hour. Dividing the remaining 270 by $54 gives 5 additional hours. Adding these 5 hours to the initial 20 hours gives a total of 25 hours.

Step-by-Step Solution

1
Determine the base cost for the setup fee and the initial 20 hours.
Base cost = 150+(20∗150 + (20 * 45) = $1,050.
The initial cost tier applies up to 20 hours of work.
2
Calculate the higher hourly rate applied to additional hours worked beyond 20.
Increased rate = 45∗1.20=45 * 1.20 = 54 per hour.
The rate increases by 20% over the standard rate of $45 per hour.
3
Formulate and solve a linear equation for total hours hh.
1050+54(h−20)=1320  ⟹  54(h−20)=270  ⟹  h−20=5  ⟹  h=251050 + 54(h - 20) = 1320 \implies 54(h - 20) = 270 \implies h - 20 = 5 \implies h = 25.
Subtracting the base cost leaves 270forovertimehours,whichdividesby270 for overtime hours, which divides by 54 per hour to yield 5 additional hours, for 25 hours total.

Key Concept

Linear Equations in One Variable
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Topics

Linear Equations in One Variable

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