Problem 47 - Algebra 1

Question

Use the following information. You are moving to Houston, Texas, and are switching your cellular phone company. Your new peak air time rate in Houston is \(\$ .23\) per minute. Your bill also includes a monthly access charge. For 110 minutes of peak air time your bill is \(\$ 51.30\). How much is your monthly bill for 60 minutes of peak air time?

Step-by-Step Solution

Answer

The monthly bill for 60 minutes of peak air time will be \$39.80.

Step 1: Confirm the cost per peak air time minute

Looking at the exercise, it is mentioned that the peak air time rate is \$0.23 per minute. This is a given and does not require any calculations.

Step 2: Calculate the monthly access charge

To calculate the monthly access charge, subtract the total cost of the airtime from the total bill. For the 110 minutes, the total airtime cost will be \( 0.23 \times 110 \) = \$25.30. So, the monthly access charge will be \( 51.30 - 25.3 \) = \$26.

Step 3: Determine the bill for 60 minutes

To calculate the monthly bill for 60 minutes of airtime, multiply the cost per minute (\$0.23) by the number of minutes (60) and then add the monthly access charge of $26. The bill will be \( 0.23 \times 60 + 26 \) = \$39.80.

Key Concepts

Understanding Linear Equations

Linear equations are fundamental in solving many real-world problems. They represent relationships where your variable quantities change at a constant rate. Let's break it down a bit more.

In this exercise, the relationship between your total phone bill and the minutes used can be seen as a linear equation. The variable is the number of peak air time minutes used. The cost per minute acts as the coefficient, and the monthly access charge is your constant.

• Cost per minute is \(0.23: This means for every minute you use, the charge goes up by \)0.23.
• Monthly access charge is the constant: This $26 does not change regardless of the minutes used.

When you write this equation, it would look like: \[ \text{Total Bill} = 0.23(\text{Number of Minutes}) + 26 \] Understanding and setting up linear equations like this one can help solve many cost-related problems efficiently.

Mastering Cost Calculations

Cost calculations are crucial when balancing budgets, especially with variable expenses like phone bills. Breaking down costs into fixed and variable parts facilitates understanding.

In this exercise, you have two components: a variable part and a fixed part.

• Variable part: Cost per minute, which is \(0.23.
• Fixed part: Monthly access charge, which is \)26.

To calculate the bill for any given number of minutes, you compute both parts: 1. Multiply the number of peak minutes by $0.23.2. Add this product to the monthly access charge.

For example, for 60 minutes: \[ 0.23 \times 60 + 26 = 39.80 \] Breaking down your costs in this way ensures transparency in your monthly expenses.

Effective Problem-Solving Strategies

When tackling word problems like this one, effective strategies can simplify your process.

First, break down the problem into manageable parts. Identify known values and what you need to find. Ask yourself:

• What are the fixed and variable costs?
• What relationships exist between the quantities?

Next, set up equations. This makes abstract relationships more tangible. You're often translating words into mathematical terms.

Finally, always double-check your work. Confirm that each step aligns with your logic and calculations by revisiting the problem’s instructions. This comprehensive approach ensures accuracy and fosters better problem-solving habits in algebraic contexts.