A rectangular garden has a perimeter of 96 meters. If the length of the garden is 3 times the width,...

1. TRANSLATE the problem information

• Given information:
  • Perimeter = 96 meters
  • Length = 3 times the width
  • Need to find: width

• What this tells us: We need to set up variables and use the relationship between length and width

2. TRANSLATE into mathematical expressions

• Let \(\mathrm{width = w\text{ meters}}\)
• Since length is 3 times the width: \(\mathrm{length = 3w\text{ meters}}\)
• This gives us both dimensions in terms of one variable

3. INFER the solution approach

• Use the perimeter formula for rectangles
• Substitute our expressions to create an equation we can solve

4. SIMPLIFY using the perimeter formula

• \(\mathrm{Perimeter = 2(length + width)}\)
• Substitute: \(\mathrm{96 = 2(3w + w)}\)
• Combine like terms: \(\mathrm{96 = 2(4w)}\)
• Distribute: \(\mathrm{96 = 8w}\)
• Solve for w: \(\mathrm{w = 96 \div 8 = 12}\)

Answer: 12 meters

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students struggle to express "length is 3 times the width" mathematically, often writing something like "\(\mathrm{l = 3}\)" instead of "\(\mathrm{l = 3w}\)"

Without properly expressing the relationship between length and width, they can't set up the correct equation. This leads to confusion about what to do next, causing them to get stuck and guess.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{2(3w + w) = 96}\) but make arithmetic errors when combining terms or solving

For example, they might incorrectly calculate \(\mathrm{2(4w) = 6w}\) instead of \(\mathrm{8w}\), leading to \(\mathrm{w = 16}\). This may lead them to select Choice (D) (36) if they further confuse this with the length value.

The Bottom Line:

Success depends on translating the word relationship into proper algebraic expressions and then executing the algebra carefully. The conceptual setup is straightforward once the translation is correct.