A rectangle has length \((3\mathrm{k} - 5)\) centimeters and width \((\mathrm{k} + 7)\) centimeters. If its perimeter equals the perimeter...

1. TRANSLATE the geometric information into mathematical expressions

• Given information:
  • Rectangle: length = \(\mathrm{(3k - 5)}\) cm, width = \(\mathrm{(k + 7)}\) cm
  • Square: side length = \(\mathrm{(k + 5)}\) cm
  • The perimeters are equal
  • Need to find: \(\mathrm{5k}\)

2. TRANSLATE each perimeter using the appropriate formulas

• Rectangle perimeter: \(\mathrm{P = 2(length + width) = 2[(3k - 5) + (k + 7)]}\)
• Square perimeter: \(\mathrm{P = 4(side\ length) = 4(k + 5)}\)

3. SIMPLIFY the rectangle perimeter expression

• \(\mathrm{P_{rectangle} = 2[(3k - 5) + (k + 7)]}\)
• \(\mathrm{P_{rectangle} = 2[3k - 5 + k + 7]}\)
• \(\mathrm{P_{rectangle} = 2[4k + 2]}\)
• \(\mathrm{P_{rectangle} = 8k + 4}\)

4. SIMPLIFY the square perimeter expression

• \(\mathrm{P_{square} = 4(k + 5) = 4k + 20}\)

5. INFER that equal perimeters means we can set up an equation

• Since the perimeters are equal: \(\mathrm{8k + 4 = 4k + 20}\)

6. SIMPLIFY by solving the linear equation

• \(\mathrm{8k + 4 = 4k + 20}\)
• \(\mathrm{8k - 4k = 20 - 4}\) (subtract \(\mathrm{4k}\) and \(\mathrm{4}\) from both sides)
• \(\mathrm{4k = 16}\)
• \(\mathrm{k = 4}\)

7. TRANSLATE back to answer the original question

• The question asks for \(\mathrm{5k}\), not just \(\mathrm{k}\)
• \(\mathrm{5k = 5(4) = 20}\)

Answer: 20

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students correctly set up the perimeter formulas but make algebraic errors when combining terms inside the rectangle perimeter expression.

For example, they might write: \(\mathrm{2[(3k - 5) + (k + 7)] = 2[3k - 5 + k + 7] = 2[4k - 2]}\) instead of \(\mathrm{2[4k + 2]}\)

This leads to the equation \(\mathrm{8k - 4 = 4k + 20}\), which gives \(\mathrm{k = 6}\) and \(\mathrm{5k = 30}\). This causes confusion since 30 doesn't appear to be a standard answer, leading to second-guessing and potential guessing.

Second Most Common Error:

Incomplete TRANSLATE execution: Students solve correctly for \(\mathrm{k = 4}\) but forget the question asks for \(\mathrm{5k}\), not \(\mathrm{k}\).

They confidently submit \(\mathrm{k = 4}\) as their final answer, missing that the problem specifically requests the value of \(\mathrm{5k = 20}\).

The Bottom Line:

This problem tests whether students can maintain accuracy through multiple algebraic steps while keeping track of what the question actually asks for. The combination of expression simplification and equation solving creates multiple opportunities for small errors that compound into wrong final answers.