Each side of a 30-sided polygon has one of three lengths. The number of sides with length 8 centimeters (cm)...

1. TRANSLATE the problem information

• Given information:
  • 30-sided polygon
  • Each side has one of three lengths: 3 cm, 4 cm, or 8 cm
  • \(\mathrm{n}\) = number of sides with length 3 cm
  • 6 sides have length 4 cm
  • Number of sides with length 8 cm = 5 times the number with length 3 cm = \(\mathrm{5n}\)

2. TRANSLATE the constraint into an equation

• Since it's a 30-sided polygon, all sides must add up to 30:
  • Sides with length 3 cm: \(\mathrm{n}\)
  • Sides with length 4 cm: 6
  • Sides with length 8 cm: \(\mathrm{5n}\)
  • Total equation: \(\mathrm{n + 6 + 5n = 30}\)

3. SIMPLIFY by combining like terms

• Combine the \(\mathrm{n}\) terms: \(\mathrm{n + 5n = 6n}\)
• Final equation: \(\mathrm{6n + 6 = 30}\)

Answer: B. \(\mathrm{6n + 6 = 30}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what "5 times the number of sides \(\mathrm{n}\) with length 3 cm" means. They might think this refers to \(\mathrm{5n}\) total sides of some length, rather than understanding that there are specifically \(\mathrm{5n}\) sides with length 8 cm.

This confusion leads them to set up an equation like \(\mathrm{5n + 6 = 30}\) (forgetting about the 8 cm sides entirely), causing them to select Choice A (\(\mathrm{5n + 6 = 30}\)).

Second Most Common Error:

Conceptual confusion: Students mix up the side lengths with the number of sides, treating the measurements (3, 4, 8) as coefficients rather than just descriptive labels.

This leads them to write something like \(\mathrm{8n + 3n + 4n = 30}\), causing them to select Choice C (\(\mathrm{8n + 3n + 4n = 30}\)).

The Bottom Line:

This problem requires careful attention to what each piece of information represents - the key is distinguishing between the length of a side (3 cm, 4 cm, 8 cm) and the count of how many sides have each length.