The combined original salary for two employees is $120,000. After a 20% bonus to the first employee and a 30%...
GMAT Algebra : (Alg) Questions
The combined original salary for two employees is \(\$120,000\). After a \(20\%\) bonus to the first employee and a \(30\%\) bonus to the second employee are applied, the combined total compensation is \(\$150,000\). Which system of equations gives the original salary \(\mathrm{s}\), in dollars, of the first employee and the original salary \(\mathrm{t}\), in dollars, of the second employee?
\(\mathrm{s + t = 120{,}000}\)
\(\mathrm{0.80s + 0.70t = 150{,}000}\)
\(\mathrm{s + t = 120{,}000}\)
\(\mathrm{0.20s + 0.30t = 150{,}000}\)
\(\mathrm{s + t = 120{,}000}\)
\(\mathrm{1.20s + 1.30t = 150{,}000}\)
\(\mathrm{s + t = 120{,}000}\)
\(\mathrm{1.30s + 1.20t = 150{,}000}\)
1. TRANSLATE the original salary information
- Given information:
- Combined original salaries = $120,000
- Let \(\mathrm{s}\) = first employee's original salary
- Let \(\mathrm{t}\) = second employee's original salary
- First equation: \(\mathrm{s + t = 120,000}\)
2. INFER what "bonus" means for total compensation
- A 20% bonus doesn't mean the employee only gets 20% of their salary
- It means they get their full original salary PLUS 20% more
- First employee's total compensation = \(\mathrm{s + 0.20s = 1.20s}\)
- Second employee's total compensation = \(\mathrm{t + 0.30t = 1.30t}\)
3. TRANSLATE the total compensation information
- Combined total compensation after bonuses = $150,000
- Second equation: \(\mathrm{1.20s + 1.30t = 150,000}\)
4. Identify the complete system
- \(\mathrm{s + t = 120,000}\)
- \(\mathrm{1.20s + 1.30t = 150,000}\)
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER reasoning: Students think "20% bonus" means the employee receives only \(\mathrm{0.20s}\) instead of understanding it means total compensation of \(\mathrm{1.20s}\).
They set up the second equation as \(\mathrm{0.20s + 0.30t = 150,000}\), reasoning that the bonuses alone add up to $150,000.
This may lead them to select Choice B (\(\mathrm{s + t = 120,000}\); \(\mathrm{0.20s + 0.30t = 150,000}\))
Second Most Common Error:
Poor TRANSLATE execution: Students correctly understand that bonuses mean \(\mathrm{1.20s}\) and \(\mathrm{1.30t}\), but mix up which employee gets which bonus percentage.
They write \(\mathrm{1.30s + 1.20t = 150,000}\), giving the 30% bonus to the first employee instead of the second.
This may lead them to select Choice D (\(\mathrm{s + t = 120,000}\); \(\mathrm{1.30s + 1.20t = 150,000}\))
The Bottom Line:
The key insight is understanding that a percentage bonus means the total compensation equals the original salary plus the bonus amount, not just the bonus itself.
\(\mathrm{s + t = 120{,}000}\)
\(\mathrm{0.80s + 0.70t = 150{,}000}\)
\(\mathrm{s + t = 120{,}000}\)
\(\mathrm{0.20s + 0.30t = 150{,}000}\)
\(\mathrm{s + t = 120{,}000}\)
\(\mathrm{1.20s + 1.30t = 150{,}000}\)
\(\mathrm{s + t = 120{,}000}\)
\(\mathrm{1.30s + 1.20t = 150{,}000}\)