Problem 61 - Prealgebra

Question

Suppose a telephone company charges \(\$ 0.41\) for the first minute and then \(\$ 0.32\) for each additional minute for a long-distance call. If a long- distance call costs \(\$ 2.33,\) how many minutes was the call?

Step-by-Step Solution

Answer

The call lasted 7 minutes.

Step 1: Understand the Problem

We need to determine how many minutes a call lasted based on the pricing structure. The first minute is charged at $0.41, and each additional minute is charged at $0.32. The total cost given is $2.33.

Step 2: Subtract the Cost of the First Minute

Calculate the cost remaining after the first minute is accounted for: $2.33 (total cost) - $0.41 (first minute cost) = $1.92.

Step 3: Calculate the Number of Additional Minutes

Each additional minute costs $0.32. To find out how many additional minutes were charged, divide the remaining cost by the cost per additional minute: $1.92 ÷ $0.32 per minute = 6 minutes.

Step 4: Total Number of Minutes

Add the first minute to the total additional minutes calculated: 1 (first minute) + 6 (additional minutes) = 7 minutes.

Key Concepts

Step-by-Step Problem Solving

Solving problems in a step-by-step manner is essential in mathematics, especially for complex problems that require clear and logical reasoning. Let's break down the process:

• **Understanding the Problem**: Before diving into calculations, identify what you need to find. Here, the task was to determine the duration of a phone call given a specific cost structure.
• **Identifying Key Information**: Pinpoint crucial data. In our scenario, we note the different charges for the first and subsequent minutes.
• **Sequential Execution**: Solve the problem by tackling each part of it sequentially. This prevents confusion and errors.
• **Verification**: Always review your solution. Make sure the final answer satisfies the conditions set by the problem.

By following these steps, you can approach problems methodically, ensuring accuracy and understanding at each stage.

Arithmetic Operations

Arithmetic operations are the foundation of problem solving in mathematics. They involve basic calculations that include addition, subtraction, multiplication, and division, each of which serves a pivotal role.
In our problem, we utilized subtraction first to find the remaining cost of a phone call after the first minute was accounted for. Specifically, we subtracted the cost of the first minute from the total call cost: \[2.33 - 0.41 = 1.92\]Next, division was used to determine the number of additional minutes by dividing the remaining cost by the cost per additional minute:\[1.92 \div 0.32 = 6\]Finally, addition was applied to find the total number of minutes by adding the first minute to the additional minutes:\[1 + 6 = 7\]These operations simplify complex problems into manageable calculations, ensuring clarity at each step of the problem-solving process.

Mathematical Reasoning

Mathematical reasoning is the ability to analyze, make decisions, and solve problems based on mathematical concepts. It involves understanding relationships between numbers and the logical flow of operations.
In our phone call cost problem, reasoning helps you recognize that

• Each operation has a purpose, such as initial subtraction to isolate the variable component of the cost.
• Division to determine repeated costs, showing a relationship between remaining charges and additional minutes.
• Finally, addition to bring all the parts together for a comprehensive answer.

Understanding these relationships is foundational to problem-solving in mathematics. This enables one to apply the right operations efficiently and logically, ensuring no aspect of the problem is overlooked. Mathematical reasoning turns simple calculations into a cohesive process, leading to accurate solutions.