Problem:A student earns $180 each month from a part-time job. During the first 4 months, she saves 1/3 of her...
GMAT Algebra : (Alg) Questions
Problem:
A student earns \(\$180\) each month from a part-time job. During the first 4 months, she saves \(\frac{1}{3}\) of her earnings each month. During the next 5 months, she increases her savings rate to \(\frac{2}{5}\) of her earnings each month. What is the total amount, in dollars, that she saves over all 9 months?
1. TRANSLATE the problem information
- Given information:
- Monthly earnings: \(\$180\)
- First 4 months: saves \(\frac{1}{3}\) of earnings each month
- Next 5 months: saves \(\frac{2}{5}\) of earnings each month
- Find: total savings over all 9 months
2. INFER the solution approach
- This problem has two different time periods with different savings rates
- We need to calculate savings for each period separately, then add them together
- Strategy: Find monthly savings for each period, multiply by number of months, then sum
3. Calculate first period savings
- Monthly savings = \(\frac{1}{3} \times \$180 = \$60\)
- Total for 4 months = \(\$60 \times 4 = \$240\)
4. Calculate second period savings
- Monthly savings = \(\frac{2}{5} \times \$180 = \$72\)
- Total for 5 months = \(\$72 \times 5 = \$360\)
5. SIMPLIFY to find final answer
- Total savings = \(\$240 + \$360 = \$600\)
Answer: \(\$600\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skills: Students fail to recognize this as a two-stage problem and try to find a single savings rate for all 9 months. They might average the two fractions \((\frac{1}{3} + \frac{2}{5}) \div 2\) or use only one of the savings rates for the entire period.
For example, using only \(\frac{1}{3}\) for all 9 months: \(\frac{1}{3} \times \$180 \times 9 = \$540\), leading to an incorrect answer.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret the time periods, thinking "first 4 months" and "next 5 months" means the savings rates apply to months 1-4 and months 6-10 (missing month 5), or they confuse which savings rate applies to which period.
This leads to incorrect calculations and confusion about the total time frame.
The Bottom Line:
The key challenge is recognizing that different time periods require separate calculations before combining results. Students must carefully track which savings rate applies when and systematically work through each period.