A mobile plan charges a fixed monthly fee of $40 and an additional $0.25 per text message sent. In a...

1. TRANSLATE the cost structure

• Given information:
  • Fixed monthly fee: \(\$40\)
  • Cost per text message: \(\$0.25\)
  • Number of messages sent: \(\mathrm{x}\)
  • Total charge constraint: "at most \(\$85\)"

2. TRANSLATE the total cost expression

• Total monthly cost = Fixed fee + (Cost per message × Number of messages)
• Total cost = \(40 + 0.25\mathrm{x}\)

3. TRANSLATE the constraint "at most $85"

• "At most \(\$85\)" means the cost cannot exceed \(\$85\)
• This translates to: Total cost \(\leq 85\)
• Therefore: \(40 + 0.25\mathrm{x} \leq 85\)

4. TRANSLATE to match answer format

• Rearranging: \(0.25\mathrm{x} + 40 \leq 85\)
• This matches choice (A)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "at most" as meaning \(\geq\) instead of \(\leq\)

Students often think "at most \(\$85\)" means the cost should be at least \(\$85\), leading them to write \(0.25\mathrm{x} + 40 \geq 85\). This logical confusion about constraint language is very common in word problems.

This leads them to select Choice B (\(0.25\mathrm{x} + 40 \geq 85\))

Second Most Common Error:

Poor TRANSLATE execution: Treating \(\$0.25\) as the whole number 25

Students sometimes drop the decimal point when translating monetary amounts, writing \(25\mathrm{x}\) instead of \(0.25\mathrm{x}\). This creates the inequality \(25\mathrm{x} + 40 \leq 85\).

This leads them to select Choice D (\(25\mathrm{x} + 40 \leq 85\))

The Bottom Line:

Word problems require careful translation of both numerical values and constraint language. The phrase "at most" is particularly tricky because it feels like it should mean "at least" but actually means the opposite - it sets an upper limit, not a lower one.