Problem 27 - Prealgebra

Question

An electrician charges \(\$ 35\) for a house call and \(\$ 80\) per hour for each hour worked. If the total charge was \(\$ 915\) to wire a new house, which equation would you use to find the number of hours \(n\) that the electrician worked? A \(35 n+2 n(80)=915\) B \(80+35 n=915\) C \(35+(80-n)=915\) D\( 35+80 n=915\)

Step-by-Step Solution

Answer

The correct equation is Option D: \(35 + 80n = 915\).

Step 1: Understand the Problem

The task is to find out which equation correctly represents the scenario of calculating the total charge for an electrician's work, given a house call fee and an hourly rate.

Step 2: Define Components

Identify the known components: the electrician charges \(\\(35\) for the initial house call and \(\\)80\) per hour (\(n\) hours worked). The total charge should be \(\$915\).

Step 3: Formulate the Equation

To formulate the equation, consider the components: the fixed charge \(\\(35\) plus \(\\)80\) times the number of hours \(n\) worked. This leads to the equation \(35 + 80n = 915\).

Step 4: Check the Options

Compare the formulated equation to the provided options. The correct formulation is present in Option D: \(35 + 80n = 915\).

Step 5: Verify the Equation

Verify if the selected equation correctly sums the total cost components to equal the given total of \(\$915\). Indeed, the fixed fee and hourly fees are accounted for in the equation \(35 + 80n = 915\).

Key Concepts

Problem Solving

When approaching a problem, it's crucial to first clearly understand what is being asked. In this exercise, we needed to figure out how to find the number of hours an electrician worked, based on the fees charged. The problem involved a total cost, a fixed fee for a house call, and an hourly charge. By identifying what is known and what needs to be discovered, we took the first step in effective problem solving. Knowing how to break down the problem by recognizing different elements within the scenario helps make the task more manageable.
To find the solution, one must:

• Identify all relevant information, such as numbers and relationships.
• Understand what is being asked—in this case, the number of hours worked.
• Consider all potential approaches and logical sequences to tackle the problem.

By delineating these facets, we pave the way for crafting a step-by-step solution.

Cost Calculation

Cost calculation plays a pivotal role in many real-life scenarios, like hiring an electrician. In this case, the electrician charges two types of fees: a flat rate for a house call, and an additional hourly charge. Calculating the total cost requires combining these elements appropriately. Let's break it down:

• You have a base fee of \( \\(35 \), which you always pay, regardless of how long the work takes.
• In addition, there is an \( \\)80 \) charge for every hour of work performed.

To determine the total cost, sum the fixed fee and the product of the hourly rate and hours worked. Therefore, the more hours involved, the higher the total cost. It's all about correctly understanding and applying these individual cost components to find an accurate calculation.

Equation Formulation

Formulating the correct equation is about accurately translating the details of the scenario into mathematical terms. Here, we needed to set up an equation that relates all parts of the payment system into one equation that equals the total fee.The pattern to follow for equation formulation includes:

• Starting with what is fixed—in this instance, the \( \\(35 \) house call fee.
• Adding the repeated charge, which is \( \\)80 \) times the number of hours worked, represented by \( n \).
• Setting this total equal to the known total cost, \( \$915 \).

Thus, the equation needed was: \[35 + 80n = 915\]This kind of formulation requires attention to every piece of the problem to ensure all components are correctly represented mathematically.

Step by Step Solution

Breaking down the problem into smaller, manageable steps is often essential for finding the correct solution. Each step should build upon the previous, steadily advancing towards solving the equation.Here's how it was done in this exercise:

• Understand the Problem: First, clarify what the question is asking and what information is provided.
• Identify the Components: List the known figures—the fixed house call fee, the hourly rate, and the total amount.
• Formulate the Equation: Convert these components into a mathematical equation, capturing all cost elements and equating them to the total cost.
• Select and Verify: Examine all given options against your formulated equation, ensuring correctness, as demonstrated with option D, confirming everything adds up to \( \$915 \).

By tackling the problem incrementally, the final solution becomes clearer and verification becomes more straightforward.