If \(2(3\mathrm{t} - 10) + \mathrm{t} = 40 + 4\mathrm{t}\), what is the value of 3t?

1. INFER the goal and strategy

• Given: \(2(3\mathrm{t} - 10) + \mathrm{t} = 40 + 4\mathrm{t}\)
• Find: The value of \(3\mathrm{t}\) (not just \(\mathrm{t}\))
• Strategy: Use algebraic manipulation to isolate \(3\mathrm{t}\) on one side

2. SIMPLIFY by applying the distributive property

• Distribute the 2: \(2(3\mathrm{t} - 10) + \mathrm{t} = 40 + 4\mathrm{t}\)
• This gives us: \(6\mathrm{t} - 20 + \mathrm{t} = 40 + 4\mathrm{t}\)

3. SIMPLIFY by combining like terms

• Combine the \(\mathrm{t}\) terms on the left side: \(6\mathrm{t} + \mathrm{t} = 7\mathrm{t}\)
• Equation becomes: \(7\mathrm{t} - 20 = 40 + 4\mathrm{t}\)

4. SIMPLIFY to collect variable terms

• Subtract \(4\mathrm{t}\) from both sides: \(7\mathrm{t} - 4\mathrm{t} - 20 = 40\)
• This simplifies to: \(3\mathrm{t} - 20 = 40\)

5. SIMPLIFY to isolate 3t

• Add 20 to both sides: \(3\mathrm{t} - 20 + 20 = 40 + 20\)
• Final result: \(3\mathrm{t} = 60\)

Answer: 60

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors or incorrectly combine terms during the multi-step algebraic manipulation.

For example, when distributing \(2(3\mathrm{t} - 10)\), they might get \(6\mathrm{t} - 10\) instead of \(6\mathrm{t} - 20\), or when combining \(7\mathrm{t} - 4\mathrm{t}\), they might get \(11\mathrm{t}\) instead of \(3\mathrm{t}\). These errors compound through the remaining steps, leading to incorrect final answers and confusion about which answer choice to select.

Second Most Common Error:

Poor INFER reasoning: Students solve for \(\mathrm{t}\) instead of recognizing they need the value of \(3\mathrm{t}\) specifically.

They correctly manipulate the equation but divide their final result by 3 to find \(\mathrm{t} = 20\), then either stop there or provide 20 as their answer instead of recognizing that \(3\mathrm{t} = 60\) is what the problem asks for. This leads to selecting an incorrect answer choice if 20 appears among the options.

The Bottom Line:

This problem tests both algebraic manipulation skills and careful reading comprehension. Students must execute several algebraic steps without errors while keeping track of what the problem is actually asking for.