Let’s break down the problem step-by-step and solve it.
### Step 1: Calculate the total lost income during college for both persons
1. Person A:
- Salary prior to school: \[tex]$18,000 per year
- Years attending college: 3 years
- Total cost of college: \$[/tex]45,000
Total lost income = (Salary prior to school * Years attending college) + Total cost of college
[tex]\[
\text{Total lost income for Person A} = (18000 \times 3) + 45000 = 54000 + 45000 = 99000
\][/tex]
2. Person B:
- Salary prior to school: \[tex]$27,000 per year
- Years attending college: 4 years
- Total cost of college: \$[/tex]30,000
Total lost income = (Salary prior to school * Years attending college) + Total cost of college
[tex]\[
\text{Total lost income for Person B} = (27000 \times 4) + 30000 = 108000 + 30000 = 138000
\][/tex]
### Step 2: Calculate the yearly gain in salary after graduating
1. Person A:
- Salary upon graduating: \[tex]$33,000
- Salary prior to school: \$[/tex]18,000
Yearly gain in salary:
[tex]\[
\text{Yearly gain for Person A} = 33000 - 18000 = 15000
\][/tex]
2. Person B:
- Salary upon graduating: \[tex]$37,000
- Salary prior to school: \$[/tex]27,000
Yearly gain in salary:
[tex]\[
\text{Yearly gain for Person B} = 37000 - 27000 = 10000
\][/tex]
### Step 3: Calculate the number of years to recover the investments
1. Person A:
[tex]\[
\text{Years to recover investment for Person A} = \frac{\text{Total lost income}}{\text{Yearly gain}} = \frac{99000}{15000} = 6.6 \text{ years}
\][/tex]
2. Person B:
[tex]\[
\text{Years to recover investment for Person B} = \frac{\text{Total lost income}}{\text{Yearly gain}} = \frac{138000}{10000} = 13.8 \text{ years}
\][/tex]
### Conclusion
Person A recovers their investment in 6.6 years, whereas Person B takes 13.8 years.
Thus, the correct answer is:
a. Person A recovers their investment in a shorter amount of time.
