Question
bertha took out a subsidized student loan of $14,250 at an 8.4$ APR, compounded monthly, to pay for her last 2 semesters of college. if she will begin paying off the loan in 15 months, how much will she owe when she begins making payments?
A. $15,821.84, since bertha is responsible for the interest on the loan that accrues before she starts making payments
B. $15,821.84, since the government is responsible for the interest on the loan that accrues before bertha starts making payments
C. $14,250.00, since bertha is responsible for the interest on the loan that accrues before she starts making payments
D. $14,250.00, since the government is responsible for the interest on the loan that accrues before bertha starts making payments
A. $15,821.84, since bertha is responsible for the interest on the loan that accrues before she starts making payments
B. $15,821.84, since the government is responsible for the interest on the loan that accrues before bertha starts making payments
C. $14,250.00, since bertha is responsible for the interest on the loan that accrues before she starts making payments
D. $14,250.00, since the government is responsible for the interest on the loan that accrues before bertha starts making payments
Answers
GPT-4o mini
To determine how much Bertha will owe when she begins making payments, we need to calculate the accumulated amount of her subsidized student loan after 15 months, given an 8.4% annual percentage rate (APR) compounded monthly.
1. **Convert the annual interest rate to a monthly interest rate:**
\[
\text{Monthly interest rate} = \frac{8.4\%}{12} = \frac{0.084}{12} = 0.007
\]
2. **Determine the number of months that the loan accumulates interest:**
Bertha will begin paying off the loan in 15 months, so \( n = 15 \).
3. **Use the formula for compound interest to calculate the future value of the loan:**
\[
A = P(1 + r)^n
\]
where:
- \( A \) is the amount owed after \( n \) months,
- \( P \) is the principal amount (the original loan amount),
- \( r \) is the interest rate per period (monthly interest rate),
- \( n \) is the number of periods (months).
Plugging in the values:
- \( P = 14,250 \)
- \( r = 0.007 \)
- \( n = 15 \)
The calculation will be:
\[
A = 14250 \times (1 + 0.007)^{15}
\]
\[
A = 14250 \times (1.007)^{15}
\]
4. **Calculate \( (1.007)^{15} \):**
\[
(1.007)^{15} \approx 1.1148
\]
5. **Now calculate \( A \):**
\[
A = 14250 \times 1.1148 \approx 15,821.84
\]
Therefore, Bertha will owe approximately $15,821.84 when she begins making payments.
Given this, the correct answer is:
**A. $15,821.84, since Bertha is responsible for the interest on the loan that accrues before she starts making payments.**
1. **Convert the annual interest rate to a monthly interest rate:**
\[
\text{Monthly interest rate} = \frac{8.4\%}{12} = \frac{0.084}{12} = 0.007
\]
2. **Determine the number of months that the loan accumulates interest:**
Bertha will begin paying off the loan in 15 months, so \( n = 15 \).
3. **Use the formula for compound interest to calculate the future value of the loan:**
\[
A = P(1 + r)^n
\]
where:
- \( A \) is the amount owed after \( n \) months,
- \( P \) is the principal amount (the original loan amount),
- \( r \) is the interest rate per period (monthly interest rate),
- \( n \) is the number of periods (months).
Plugging in the values:
- \( P = 14,250 \)
- \( r = 0.007 \)
- \( n = 15 \)
The calculation will be:
\[
A = 14250 \times (1 + 0.007)^{15}
\]
\[
A = 14250 \times (1.007)^{15}
\]
4. **Calculate \( (1.007)^{15} \):**
\[
(1.007)^{15} \approx 1.1148
\]
5. **Now calculate \( A \):**
\[
A = 14250 \times 1.1148 \approx 15,821.84
\]
Therefore, Bertha will owe approximately $15,821.84 when she begins making payments.
Given this, the correct answer is:
**A. $15,821.84, since Bertha is responsible for the interest on the loan that accrues before she starts making payments.**