The combined original monthly salary for two employees, Alex and Blair, is $8,400. After Alex receives a 15% raise and...

1. TRANSLATE the problem information

• Given information:
  • Alex and Blair's combined original salary = \(\$8,400\)
  • Alex receives a 15% raise
  • Blair receives a 20% pay cut
  • Their new combined salary = \(\$7,770\)
  • Need: System of equations for original salaries A and B

2. TRANSLATE the original salary relationship

• The first equation comes directly from the original combined salary:
  \(\mathrm{A + B = 8,400}\)

3. INFER how percentage changes work

• For a 15% raise: new salary = \(\mathrm{original \times (1 + 0.15) = original \times 1.15}\)
• For a 20% pay cut: new salary = \(\mathrm{original \times (1 - 0.20) = original \times 0.80}\)
• This means Alex's new salary is \(\mathrm{1.15A}\) and Blair's new salary is \(\mathrm{0.80B}\)

4. TRANSLATE the new salary relationship

• The second equation represents their combined salary after changes:
  \(\mathrm{1.15A + 0.80B = 7,770}\)

5. Identify the complete system

• First equation: \(\mathrm{A + B = 8,400}\) (original combined salary)
• Second equation: \(\mathrm{1.15A + 0.80B = 7,770}\) (new combined salary)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Incorrectly converting percentage changes to mathematical form

Students often think "15% raise" means multiply by \(\mathrm{0.15}\) instead of \(\mathrm{1.15}\), leading them to use \(\mathrm{0.85A}\) for Alex's new salary (thinking it's a 15% decrease). This confusion about percentage change representation leads them to select Choice A (\(\mathrm{0.85A + 0.80B = 7770}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Mixing up which equation represents which situation

Students correctly calculate the percentage multipliers but place the totals on wrong sides—thinking the new combined salary should equal 8,400 and the original should equal 7,770. This leads them to select Choice C (\(\mathrm{A + B = 7770}\), \(\mathrm{1.15A + 0.80B = 8400}\)).

The Bottom Line:

This problem requires careful attention to the direction of percentage changes (increase vs decrease) and maintaining clear logic about which total corresponds to which time period (original vs after changes).