Question:A cell phone plan charges a monthly base fee of $25 plus $0.20 per text message sent. If David's phone...

1. TRANSLATE the problem information

• Given information:
  • Monthly base fee: \(\$25\)
  • Rate per text message: \(\$0.20\)
  • Total bill for the month: \(\$45\)
  • Need to find: number of text messages sent
• What this tells us: We have a cost structure with a fixed base fee plus a variable cost per message.

2. INFER the approach

• This is a linear equation problem because the total cost increases at a constant rate per text message
• Strategy: Set up an equation where Total Cost = Base Fee + (Rate × Number of Messages)
• Let x represent the unknown number of text messages

3. TRANSLATE the cost relationship into an equation

\(\mathrm{Total\,cost} = 25 + 0.20\mathrm{x}\)

Since the total bill was \(\$45\):

\(25 + 0.20\mathrm{x} = 45\)

4. SIMPLIFY to solve for x

• Subtract 25 from both sides:
  \(0.20\mathrm{x} = 45 - 25\)
  \(0.20\mathrm{x} = 20\)
• Divide both sides by 0.20:
  \(\mathrm{x} = 20 ÷ 0.20 = 100\)

Answer: 100 text messages

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may set up the equation incorrectly, such as writing \(25\mathrm{x} + 0.20 = 45\) or confusing which values go where in the cost structure.

This stems from not carefully identifying what represents the fixed cost versus the variable cost, leading to an equation that doesn't match the problem's structure. This typically leads to confusion and guessing rather than a systematic solution.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(25 + 0.20\mathrm{x} = 45\) but make arithmetic errors, particularly with the decimal division \(20 ÷ 0.20\).

Common mistakes include treating this as \(20 ÷ 2 = 10\), or incorrectly handling the decimal to get answers like 1000 or 1. This leads to selecting incorrect numerical answers that seem reasonable but don't check back to the original problem.

The Bottom Line:

This problem tests whether students can recognize and set up a basic linear cost model, then execute decimal arithmetic accurately. The real challenge isn't the algebra—it's correctly interpreting the cost structure and handling the decimal division at the end.