A factory machine produces components at a constant rate of r components per hour. After operating for 6 hours, the...

1. TRANSLATE the problem information

• Given information:
  • Machine rate: r components per hour
  • Operating time: 6 hours
  • Total components produced: 138

• What this tells us: We need to find the rate r using the relationship between total output, rate, and time.

2. INFER the approach

• Since we know total output and time, we can use: \(\mathrm{Total = Rate \times Time}\)
• This gives us: \(\mathrm{138 = r \times 6}\)
• We need to solve for r

3. SIMPLIFY to find the rate

• Starting with: \(\mathrm{6r = 138}\)
• Divide both sides by 6: \(\mathrm{r = 138 \div 6}\)
• Calculate: \(\mathrm{r = 23}\)

Answer: C. 23

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse the relationship between rate, time, and total output, setting up an incorrect equation like \(\mathrm{r = 6 \times 138}\) or \(\mathrm{138 = r + 6}\).

When students think rate means "how much total" rather than "how much per unit time," they might multiply instead of divide, getting \(\mathrm{r = 6 \times 138 = 828}\). Since this isn't an answer choice, this leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{6r = 138}\) but make arithmetic errors when dividing 138 by 6.

Common division mistakes include getting \(\mathrm{r = 22}\) or \(\mathrm{r = 24}\) due to miscalculation. This may lead them to select Choice B (22) or Choice D (24).

The Bottom Line:

This problem tests whether students understand the fundamental rate relationship and can accurately perform the arithmetic. The key insight is recognizing that rate problems always involve the relationship: \(\mathrm{Total = Rate \times Time}\).