A project requires a total of 42 units of work. A worker completes the project in t hours at a...

1. TRANSLATE the problem information

• Given information:
  • Total work needed: 42 units
  • Time to complete: t hours
  • Work rate: r units per hour
  • Relationship: \(\mathrm{r = 7t}\)
  • Find: \(\mathrm{7t^2}\)
• What this tells us: We need to connect work, rate, and time using a fundamental relationship.

2. INFER the approach

• The key relationship is: \(\mathrm{work = rate \times time}\)
• Since we know the total work (42 units) and have expressions for rate (r) and time (t), we can set up an equation
• The given relationship \(\mathrm{r = 7t}\) suggests substitution will be useful

3. TRANSLATE into mathematical equation

• Work = rate × time becomes: \(\mathrm{42 = r \times t}\)

4. SIMPLIFY by substitution

• Replace r with 7t: \(\mathrm{42 = (7t) \times t}\)
• Multiply: \(\mathrm{42 = 7t^2}\)
• Therefore: \(\mathrm{7t^2 = 42}\)

Answer: C (42)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not immediately recognize that "completes the project in t hours at rate r units per hour" translates to the \(\mathrm{work = rate \times time}\) relationship.

Instead, they might try to solve for individual variables like t or r first, leading to unnecessary complications. Without the foundational equation, they get stuck early and may resort to guessing.

This leads to confusion and abandoning systematic solution.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{42 = r \times t}\) and substitute \(\mathrm{r = 7t}\), but make algebraic errors when computing \(\mathrm{(7t) \times t}\).

They might write \(\mathrm{42 = 7t + t = 8t}\), confusing multiplication with addition. This leads them to solve \(\mathrm{8t = 42}\), getting \(\mathrm{t = 5.25}\), then calculating \(\mathrm{7t^2 = 7(5.25)^2 \approx 193}\), which doesn't match any answer choice.

This causes them to get stuck and guess among the available options.

The Bottom Line:

This problem tests whether students can recognize the work-rate-time relationship and perform clean algebraic substitution. The trap is that the question asks for \(\mathrm{7t^2}\), not t or \(\mathrm{t^2}\), so students must resist the urge to solve for intermediate variables.