A factory uses a number of identical machines that operate at the same constant rate. Each machine produces 12 widgets...

1. TRANSLATE the problem information

• Given information:
  • Each machine produces 12 widgets per hour
  • Factory produced 1,260 widgets total
  • Production run lasted 15 hours
  • Need to find: number of machines used

2. INFER the relationship

• Total production depends on three factors: rate per machine, number of machines, and time
• We can write: \(\mathrm{Total\ widgets = (Rate\ per\ machine) \times (Number\ of\ machines) \times (Time)}\)
• Let \(\mathrm{M}\) = number of machines, so our equation becomes: \(\mathrm{1,260 = 12 \times M \times 15}\)

3. SIMPLIFY the equation

• First, calculate how many widgets one machine produces in 15 hours:
  \(\mathrm{12 \times 15 = 180}\)
• Our equation becomes:
  \(\mathrm{1,260 = 180 \times M}\)
• Solve for M:
  \(\mathrm{M = 1,260 \div 180 = 7}\)

Answer: B (7)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse what they're solving for and calculate widgets per machine instead of number of machines.

They might calculate \(\mathrm{12 \times 15 = 180}\) and think this is the answer, leading them to select Choice D (180)

Second Most Common Error:

Inadequate SIMPLIFY execution: Students set up the equation correctly but make arithmetic errors in the division step.

For example, estimating \(\mathrm{1,260 \div 180}\) incorrectly might lead them to select Choice A (6) or get confused about the calculation.

The Bottom Line:

This problem requires careful attention to what quantity you're solving for and systematic equation setup. The key insight is recognizing that total production is the product of individual rate, number of units (machines), and time.