Problem 32 - Prealgebra

Question

One cellular phone carrier charges \(\$ 29.75\) a month plus \(\$ 0.15\) a minute for international calls. Another carrier charges \(\$ 19.95\) a month and \(\$ 0.29\) a minute for international calls. For how many minutes is the cost of the plans the same?

Step-by-Step Solution

Answer

The costs are the same for 70 minutes of international calls.

Step 1: Define Cost Equations

First, we need to define equations for the cost of each plan. Let \( x \) be the number of minutes of international calls.For the first carrier, the cost \( C_1 \) is given by:\[ C_1 = 29.75 + 0.15x \]For the second carrier, the cost \( C_2 \) is:\[ C_2 = 19.95 + 0.29x \]

Step 2: Set the Equations Equal

Next, to find when the costs are the same, set the two cost equations equal to each other:\[ 29.75 + 0.15x = 19.95 + 0.29x \]

Step 3: Solve for x

Now, solve the equation for \( x \):First, subtract \( 0.15x \) from both sides:\[ 29.75 = 19.95 + 0.14x \]Then, subtract \( 19.95 \) from both sides:\[ 9.8 = 0.14x \]Finally, divide each side by \( 0.14 \) to solve for \( x \):\[ x = \frac{9.8}{0.14} = 70 \]

Step 4: Interpret the Solution

The calculation shows that both plans will cost the same if \( x = 70 \) minutes are used. So, 70 minutes is the number where both carriers charge the same.

Key Concepts

Understanding Cost Equations

Cost equations are mathematical expressions that represent how much you will spend, depending on certain variables, such as phone call duration in our scenario. Imagine you have two different phone carriers, each with its own monthly fee and per minute charge for international calls.

For Carrier 1, they charge a monthly fee of \(29.75 and \)0.15 for each minute of international calls. This can be written as an equation:

• For \( x \) minutes, the cost (\( C_1 \)) is \[ C_1 = 29.75 + 0.15x \]

For Carrier 2, the monthly fee is lower, at \(19.95, but the per-minute cost is higher at \)0.29. Thus, the cost equation is:

• For \( x \) minutes, the cost (\( C_2 \)) is \[ C_2 = 19.95 + 0.29x \]

Understanding these equations allows you to visualize how your cost changes. It's crucial for comparing different pricing plans.

Introduction to Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (such as addition or multiplication). In the context of phone costs, our expressions include both a fixed fee and a variable part, which changes with the number of minutes you use.

Here's how the expressions break down:

• The fixed fee (like \( 29.75 \) or \( 19.95 \)) is constant, meaning it doesn’t change regardless of usage.
• The variable part (\