Question:The total cost C, in dollars, to build a fence is given by C = 25 + 15n, where n...

1. TRANSLATE the problem information

• Given information:
  • Cost function: \(\mathrm{C = 25 + 15n}\) (where \(\mathrm{n}\) = number of fence sections)
  • Total cost: \(\mathrm{C = \$175}\)
• What we need to find: The value of \(\mathrm{n}\) (number of fence sections)

2. INFER the solving approach

• Since both expressions represent the same cost \(\mathrm{C}\), we can substitute
• Strategy: Set 175 equal to the expression \(\mathrm{25 + 15n}\) and solve for \(\mathrm{n}\)

3. SIMPLIFY through algebraic steps

• Substitute: \(\mathrm{25 + 15n = 175}\)
• Subtract 25 from both sides: \(\mathrm{15n = 150}\)
• Divide both sides by 15: \(\mathrm{n = 10}\)

Answer: 10 fence sections

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't recognize that they need to substitute \(\mathrm{C = 175}\) into the given equation \(\mathrm{C = 25 + 15n}\). Instead, they might try to work with the two pieces of information separately or get confused about what to do next.

This leads to confusion and guessing rather than systematic problem solving.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the equation correctly but make arithmetic errors, such as:

• Calculating \(\mathrm{175 - 25 = 140}\) instead of 150
• Dividing incorrectly: \(\mathrm{150 \div 15 = 15}\) instead of 10

These calculation errors lead to incorrect final answers.

The Bottom Line:

This problem tests whether students can connect a real-world scenario to algebraic equation solving. The key insight is recognizing that when you have two expressions that both equal the same variable, you can set them equal to each other and solve.