Two athletes trained using the same exercise program with different intensity levels. Athlete A completed the program in 45 minutes,...

1. TRANSLATE the problem information

• Given information:
  • Athlete A: 3 strength exercises + 2 cardio exercises = 45 minutes total
  • Athlete B: 1 strength exercise + 4 cardio exercises = 38 minutes total
  • Find: time for 1 cardio exercise

• Define variables: Let \(\mathrm{s}\) = time for 1 strength exercise, \(\mathrm{c}\) = time for 1 cardio exercise

• TRANSLATE to equations:
  • Athlete A: \(\mathrm{3s + 2c = 45}\)
  • Athlete B: \(\mathrm{s + 4c = 38}\)

2. INFER the solution approach

• We have a system of two equations with two unknowns
• The substitution method will work well since the second equation can easily be solved for s

3. SIMPLIFY using substitution method

• From equation 2: \(\mathrm{s + 4c = 38}\)
• Solve for s: \(\mathrm{s = 38 - 4c}\)

• Substitute into equation 1:
  \(\mathrm{3(38 - 4c) + 2c = 45}\)

• SIMPLIFY by distributing:
  \(\mathrm{114 - 12c + 2c = 45}\)

• Combine like terms:
  \(\mathrm{114 - 10c = 45}\)

• Solve for c:
  \(\mathrm{-10c = 45 - 114}\)
  \(\mathrm{-10c = -69}\)
  \(\mathrm{c = 6.9}\)

Answer: B. 6.9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly set up the equations by confusing what each athlete did. For example, they might write Athlete A as "\(\mathrm{2s + 3c = 45}\)" instead of "\(\mathrm{3s + 2c = 45}\)", mixing up the number of strength versus cardio exercises.

This leads to a completely different system of equations and an incorrect final answer that doesn't match any of the given choices, causing confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the system but make algebraic errors during the substitution process. A common mistake is incorrectly distributing: writing \(\mathrm{3(38 - 4c)}\) as "\(\mathrm{114 - 4c}\)" instead of "\(\mathrm{114 - 12c}\)", or making sign errors when combining like terms.

This may lead them to select Choice A (5.2) or another incorrect choice.

The Bottom Line:

This problem requires careful reading to correctly identify what each athlete did, then systematic algebraic manipulation. The key is translating the scenarios accurately into mathematical relationships and executing the algebra without computational errors.