A wire with a length of 106 inches is cut into two parts. One part has a length of x...

1. TRANSLATE the problem information

• Given information:
  • Total wire length: 106 inches
  • Wire cut into two parts: \(\mathrm{x}\) inches and \(\mathrm{y}\) inches
  • "x is 6 more than 4 times y"

• This gives us two equations:
  • \(\mathrm{x + y = 106}\) (the parts add up to the whole)
  • \(\mathrm{x = 4y + 6}\) (the relationship between x and y)

2. INFER the solution approach

• We have a system of two equations with two unknowns
• Since the second equation already expresses x in terms of y, substitution is the most efficient method
• We'll substitute the expression for x into the first equation

3. SIMPLIFY by substitution

• Substitute \(\mathrm{x = 4y + 6}\) into \(\mathrm{x + y = 106}\):

\(\mathrm{(4y + 6) + y = 106}\)
\(\mathrm{5y + 6 = 106}\)
\(\mathrm{5y = 100}\)
\(\mathrm{y = 20}\)

4. SIMPLIFY to find x

• Use \(\mathrm{y = 20}\) in the equation \(\mathrm{x = 4y + 6}\):

\(\mathrm{x = 4(20) + 6 = 80 + 6 = 86}\)

Answer: D. 86

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "x is 6 more than 4 times y" as "x is 6 more than y"

Students often rush through reading and miss the "4 times" part, setting up the incorrect equation \(\mathrm{x = y + 6}\) instead of \(\mathrm{x = 4y + 6}\). With \(\mathrm{x + y = 106}\) and \(\mathrm{x = y + 6}\), they solve to get \(\mathrm{y = 50}\) and \(\mathrm{x = 56}\).

This may lead them to select Choice C (56)

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic mistakes during algebraic manipulation

Students correctly set up the equations but make calculation errors, such as:

• \(\mathrm{5y + 6 = 106}\) → \(\mathrm{5y = 99}\) (forgetting to subtract 6 properly)
• Dividing incorrectly when solving for y
• Making errors when calculating \(\mathrm{x = 4(20) + 6}\)

This leads to confusion and various wrong values, potentially causing them to guess among the remaining choices.

The Bottom Line:

This problem tests both careful reading comprehension and systematic equation-solving. Students who rush through the translation phase or aren't methodical with their algebra often select incorrect answers that seem reasonable but don't satisfy both original conditions.