A landscaping company charges a flat equipment fee plus a fixed hourly rate for lawn maint… — ACT

A landscaping company charges a flat equipment fee plus a fixed hourly rate for lawn maintenance. For a job that took $4$ hours, the company charged a total of $\$190$ . For a different job that took $7$ hours, the company charged a total of $\$295$ . The company also offers a package that includes a $15\%$ discount on the hourly rate, but the flat equipment fee remains the same. Under this discounted package, what is the total charge, in dollars, for a job that takes $8$ hours?

Answer: 288 dollars

Answer

The total charge for an

8

-hour job under the discounted package is

288

dollars.

By translating the problem into the equations

F + 4H = 190

and

F + 7H = 295

, we find that the hourly rate

H

35

dollars and the flat fee

F

50

dollars. Applying a

15\%

discount to the hourly rate yields a new rate of

29.75

dollars per hour. The total cost for

8

hours is then

50 + 8(29.75) = 288

dollars.

Step-by-Step Solution

Define variables for the flat fee and hourly rate, and translate the given information into a system of linear equations.

F + 4H = 190

and

F + 7H = 295

, where

F

is the flat fee and

H

is the hourly rate.

To represent the cost structure algebraically.

Solve the system of equations by elimination or substitution.

H = 35

and

F = 50

To determine the individual cost components (flat fee and hourly rate).

Apply the

15\%

discount to the hourly rate.

H_{\text{discounted}} = 35 \times (1 - 0.15) = 29.75

To find the new hourly rate under the discounted package.

Calculate the total cost for

8

hours of work with the flat fee and the discounted hourly rate.

\text{Total Cost} = 50 + 8(29.75) = 288

To answer the question asking for the total charge of an

8

-hour job under the discounted package.

Key Concept

Translating verbal statements into a system of linear equations and solving them to evaluate a modified expression.

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