Question

Larry's Lawn Service charges w/hour for the first x hours of grass trimming, then w + 2 dollars for every hour of work over x hours. How much more will a homeowner be charged for a grass trimming job that took z hours if z > x than for a job which took only w hours if x < w < z? 
 

• A. x(z + w)
• B. (w + 2) - zx
• C. (w + 2) (z-w)
• D. xw + 2 - (z-w)
• E. w(x + z) + x

Hint:

When a problem uses the same variable for different quantities (like 'w' for rate and 'w' for time), be extremely careful. Write down the cost expressions for each scenario step-by-step. When finding the difference, many terms will often cancel out, simplifying the final expression.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is an algebraic word problem that requires setting up expressions for the cost of two different jobs based on a tiered pricing structure. The goal is to find the difference between these two costs. The variable 'w' is used for both a rate and a duration, which requires careful reading.

Step 2: Key Formula or Approach:
The cost of a job that exceeds \(x\) hours is calculated in two parts:
Cost = (Cost for the first \(x\) hours) + (Cost for the hours beyond \(x\)).
Cost = \( (w \times x) + ((w+2) \times (\text{total hours} - x)) \).

Step 3: Detailed Explanation:
Let's calculate the cost for each job separately.
Job 1: Duration = z hours
We are given that \( z > x \). The cost (\(C_z\)) is calculated as follows:
- Cost for the first \(x\) hours: \( w \times x \)
- Hours worked beyond \(x\): \( z - x \)
- Cost for the additional hours: \( (w+2) \times (z-x) \)
Total cost for Job 1:
\[ C_z = wx + (w+2)(z-x) \] Job 2: Duration = w hours
We are given that \( x < w < z \). Since the duration \(w\) is greater than \(x\), the tiered pricing also applies here. The cost (\(C_w\)) is calculated as:
- Cost for the first \(x\) hours: \( w \times x \)
- Hours worked beyond \(x\): \( w - x \)
- Cost for the additional hours: \( (w+2) \times (w-x) \)
Total cost for Job 2:
\[ C_w = wx + (w+2)(w-x) \] Find the difference in cost.
The question asks for "how much more" the first job costs than the second job. This is the difference: \( C_z - C_w \).
\[ \text{Difference} = [wx + (w+2)(z-x)] - [wx + (w+2)(w-x)] \] The \(wx\) terms cancel out:
\[ \text{Difference} = (w+2)(z-x) - (w+2)(w-x) \] Factor out the common term \( (w+2) \):
\[ \text{Difference} = (w+2) [ (z-x) - (w-x) ] \] Simplify the expression inside the brackets:
\[ \text{Difference} = (w+2) [ z - x - w + x ] \] The \(-x\) and \(+x\) terms cancel out:
\[ \text{Difference} = (w+2)(z-w) \]

Step 4: Final Answer:
The homeowner will be charged (w + 2)(z-w) more for the longer job. This matches option (C).