Problem 62 - Prealgebra

Question

Suppose a telephone company charges \(\$ 0.45\) for the first three minutes and then \(\$ 0.29\) for each additional minute for a long-distance call. If a long- distance call costs \(\$ 2.77\), how many minutes was the call?

Step-by-Step Solution

Answer

The call lasted 11 minutes.

Step 1: Subtract Initial Charge

The first three minutes of the call cost \(0.45. To find out how much was spent on the minutes beyond the initial three, subtract \)0.45 from the total call cost of \(2.77: \[ 2.77 - 0.45 = 2.32 \] So, \)2.32 was spent on additional minutes.

Step 2: Calculate Additional Minutes

Each additional minute costs $0.29. To find out how many additional minutes were used, divide the remaining cost by the cost per additional minute:\[ \frac{2.32}{0.29} = 8 \] So, 8 additional minutes were used.

Step 3: Calculate Total Minutes

The total length of the call includes the initial 3 minutes and the 8 additional minutes:\[ 3 + 8 = 11 \] Therefore, the call lasted for 11 minutes in total.

Key Concepts

Long-Distance Call Costs

When it comes to long-distance calls, different telephone companies may have unique pricing structures. In this particular exercise, the cost is based on an initial charge and a per-minute rate for additional minutes. The initial charge is the cost you incur for the first few minutes of your call, often designed to cover initial connection costs.
In our example, the first three minutes cost $0.45. This acts as a baseline charge that is always present when initiating a call. Beyond this initial time frame, a different rate applies for every subsequent minute spent on the call.
In this scenario, each additional minute costs $0.29. This kind of tiered pricing can make long-distance calls more cost-effective over time, as you are only charged a higher rate initially and a lower rate subsequently. Understanding these rates helps in predicting your call costs and managing your expenses.

Step by Step Solution

Breaking down a problem into manageable steps can make solving it much easier. This is especially true for mathematical word problems like the one we are examining. Let's dive into how each part of the solution helps you solve for the total duration of a call.

• Step 1: First, calculate how much was spent beyond the initial time frame. Here, the total cost is $2.77, and the initial charge is $0.45. By subtracting, we find the cost of the additional minutes.
• Step 2: Next, calculate the number of additional minutes. Divide the cost found in Step 1 by the per-minute charge of $0.29. This division tells us precisely how many extra minutes were used.
• Step 3: Finally, add up the initial minutes and the extra minutes to obtain the total call duration. It's just a matter of simple arithmetic.

By following these steps in order, you can simplify seemingly complex problems and arrive at the solution efficiently.

Additional Minutes Calculation

Calculating additional minutes individually can sometimes be a bit tricky if you're not careful with the arithmetic. After determining the cost associated with any extra minutes beyond the initial charge, the next step is crucial.
In this exercise, we found that $2.32 was spent on additional minutes. To find out how many additional minutes that represents, we need to solve:
\[ \frac{2.32}{0.29} = 8 \]
The division gives us a clear picture: 8 additional minutes were used during the call.
This calculation is straightforward, but it needs accuracy in the division to ensure the final solution is correct. Always check your calculations to avoid errors which could lead to incorrect answers, especially when you are dealing with decimals and real-world applications like billing.