Question
rose took out a 16 year loan for $43,000 at an APR of 4.1%, compounded monthly, while yolanda took out a 16 year loan for $83,000 at an APR of 4.1%, compounded monthly. who would save more by paying off her loan 7 years early?
A. rose would save more, since she has $40,000 less in principal
B. yolanda would save more, since she has $40,000 less in principal
C. rose would save more, since she has $40,000 more in principal
D. yolanda would save more, since has $40,000 more in principal
A. rose would save more, since she has $40,000 less in principal
B. yolanda would save more, since she has $40,000 less in principal
C. rose would save more, since she has $40,000 more in principal
D. yolanda would save more, since has $40,000 more in principal
Answers
GPT-4o mini
To determine who would save more by paying off their loan 7 years early, we should first calculate the monthly payments for both loans and then the remaining balance after 9 years of payments (initial term being 16 years), and finally calculate the interest saved by paying off the loan early.
### Loan Calculations
**Formula for monthly payment (M):**
\[
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
- \( P \) = principal amount (loan amount)
- \( r \) = monthly interest rate (annual rate / 12)
- \( n \) = total number of payments (loan term in months)
**1. Rose's Loan:**
- Principal (\( P \)): $43,000
- APR: 4.1% (0.041)
- Monthly interest rate (\( r \)): \( \frac{0.041}{12} \approx 0.00341667 \)
- Loan term: 16 years (or 192 months)
Calculating monthly payment for Rose:
\[
M = 43000 \times \frac{0.00341667 \times (1 + 0.00341667)^{192}}{(1 + 0.00341667)^{192} - 1}
\]
Using a calculator, this yields:
\[
M \approx 324.53
\]
**2. Yolanda's Loan:**
- Principal (\( P \)): $83,000
Same monthly interest rate and loan term:
Calculating monthly payment for Yolanda:
\[
M = 83000 \times \frac{0.00341667 \times (1 + 0.00341667)^{192}}{(1 + 0.00341667)^{192} - 1}
\]
Approximating yields:
\[
M \approx 609.90
\]
### Remaining Balance After 9 Years:
We need to find out how much is left after 9 years (or 108 months), then save by paying off early after 16 years (which has another 84 months left).
**Remaining Balance Formula:**
\[
B = P(1 + r)^n - M \frac{(1 + r)^n - 1}{r}
\]
Where \( n \) is the number of payments made (108 in this case).
**1. Rose’s Remaining Balance (After 108 months):**
\[
B = 43000(1 + 0.00341667)^{108} - 324.53 \frac{(1 + 0.00341667)^{108} - 1}{0.00341667}
\]
Calculating this yields approximately:
\[
B_{Rose} \approx 19602.31
\]
**2. Yolanda’s Remaining Balance (After 108 months):**
\[
B = 83000(1 + 0.00341667)^{108} - 609.90 \frac{(1 + 0.00341667)^{108} - 1}{0.00341667}
\]
Calculating this yields approximately:
\[
B_{Yolanda} \approx 37859.73
\]
### Total Savings by Paying Off Loans Early:
Now we need to calculate the savings for both by paying this balance off immediately, instead of continuing for the remaining 7 years.
### Conclusion
Rose has a remaining balance of approximately $19,602.31 and Yolanda has approximately $37,859.73 after 9 years.
**Comparative Savings by Paying Off Early:**
Rose will save a significant amount on interest simply because her remaining balance, which is lower than Yolanda’s remaining balance, even though she paid off early.
According to choices A, B, C, and D:
Since Rose has a significantly lower remaining balance despite the different initial principal amounts, the answer would be:
**A. Rose would save more, since she has $40,000 less in principal.**
### Loan Calculations
**Formula for monthly payment (M):**
\[
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
- \( P \) = principal amount (loan amount)
- \( r \) = monthly interest rate (annual rate / 12)
- \( n \) = total number of payments (loan term in months)
**1. Rose's Loan:**
- Principal (\( P \)): $43,000
- APR: 4.1% (0.041)
- Monthly interest rate (\( r \)): \( \frac{0.041}{12} \approx 0.00341667 \)
- Loan term: 16 years (or 192 months)
Calculating monthly payment for Rose:
\[
M = 43000 \times \frac{0.00341667 \times (1 + 0.00341667)^{192}}{(1 + 0.00341667)^{192} - 1}
\]
Using a calculator, this yields:
\[
M \approx 324.53
\]
**2. Yolanda's Loan:**
- Principal (\( P \)): $83,000
Same monthly interest rate and loan term:
Calculating monthly payment for Yolanda:
\[
M = 83000 \times \frac{0.00341667 \times (1 + 0.00341667)^{192}}{(1 + 0.00341667)^{192} - 1}
\]
Approximating yields:
\[
M \approx 609.90
\]
### Remaining Balance After 9 Years:
We need to find out how much is left after 9 years (or 108 months), then save by paying off early after 16 years (which has another 84 months left).
**Remaining Balance Formula:**
\[
B = P(1 + r)^n - M \frac{(1 + r)^n - 1}{r}
\]
Where \( n \) is the number of payments made (108 in this case).
**1. Rose’s Remaining Balance (After 108 months):**
\[
B = 43000(1 + 0.00341667)^{108} - 324.53 \frac{(1 + 0.00341667)^{108} - 1}{0.00341667}
\]
Calculating this yields approximately:
\[
B_{Rose} \approx 19602.31
\]
**2. Yolanda’s Remaining Balance (After 108 months):**
\[
B = 83000(1 + 0.00341667)^{108} - 609.90 \frac{(1 + 0.00341667)^{108} - 1}{0.00341667}
\]
Calculating this yields approximately:
\[
B_{Yolanda} \approx 37859.73
\]
### Total Savings by Paying Off Loans Early:
Now we need to calculate the savings for both by paying this balance off immediately, instead of continuing for the remaining 7 years.
### Conclusion
Rose has a remaining balance of approximately $19,602.31 and Yolanda has approximately $37,859.73 after 9 years.
**Comparative Savings by Paying Off Early:**
Rose will save a significant amount on interest simply because her remaining balance, which is lower than Yolanda’s remaining balance, even though she paid off early.
According to choices A, B, C, and D:
Since Rose has a significantly lower remaining balance despite the different initial principal amounts, the answer would be:
**A. Rose would save more, since she has $40,000 less in principal.**