Hector used a tool called an auger to remove corn from a storage bin at a constant rate. The bin...

1. TRANSLATE the problem information

• Given information:
  • Initial amount: 24,000 bushels
  • Amount after 5 hours: 19,350 bushels
  • Target amount: 12,840 bushels remain
  • Rate is constant

• What we need to find: Total hours of auger use when 12,840 bushels remain

2. INFER the solution strategy

• Since the rate is constant, we first need to find how fast the auger removes corn
• Once we know the rate, we can calculate how long it takes to remove enough corn to reach our target

3. SIMPLIFY to find the removal rate

• Corn removed in first 5 hours: \(24,000 - 19,350 = 4,650\) bushels
• Rate = \(4,650 \div 5 = 930\) bushels per hour

4. SIMPLIFY to find total corn that must be removed

• Starting amount: 24,000 bushels
• Target remaining: 12,840 bushels
• Total to remove: \(24,000 - 12,840 = 11,160\) bushels

5. SIMPLIFY to calculate total time needed

• Time = Total amount to remove ÷ Rate
• Time = \(11,160 \div 930 = 12\) hours (use calculator)

Answer: D. 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often try to work directly with the 5-hour checkpoint without first establishing the removal rate. They might subtract \(19,350 - 12,840 = 6,510\) and try to relate this directly to time, leading to confusion about what this number represents and how to use it.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need the rate (930 bushels/hour) but make calculation errors in the final steps. They might incorrectly calculate either the total amount to be removed or the final division, potentially selecting Choice C (8) if they miscalculate \(11,160 \div 930\).

The Bottom Line:

This problem requires students to think strategically about multi-step rate problems. The key insight is recognizing that you must establish the rate from the given 5-hour data point before you can solve for any other time point. Students who jump directly to trying to use the target value without establishing the rate pattern will struggle to make progress.