Harold took out a 4-year amortized loan for $22,000 at 10% to pay his tuition at a technical institute. After 1 year, he refinanced the unpaid balance of $17,291 at a 6% interest rate.
(All interest rates are annual interest rates.)
1) Find the monthly payment on the original loan;
2) Find the monthly payment on the new loan;
3) Find the total amount saved on interest by refinancing.
Possible answers below
A. $557.92; $557.98; $1149.48
B. $557.92; $525.99; $1149.48
C. $557.92; $525.99; $6127.83
D. $557.92; $557.98; $6127.83
Would the answer be C?
Yes, the answer is C.
No, the answer is actually B. You were so close, it's as if you almost won a prize, but then tripped on your shoelaces right before the finish line. Just like Harold, who refinanced his loan to save some money. Better luck next time!
To find the correct answer, let's work through each step of the problem:
1) Find the monthly payment on the original loan:
To find the monthly payment on an amortized loan, we can use the formula:
Monthly Payment = (Loan Amount * Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate) ^ (-Number of Months))
In this case, the loan amount is $22,000 and the interest rate is 10% per year. The loan term is 4 years, so the number of months is 4 * 12 = 48. Let's calculate the monthly payment:
Monthly Interest Rate = Annual Interest Rate / 12
= 10% / 12
= 0.1 / 12
= 0.0083
Monthly Payment = (22,000 * 0.0083) / (1 - (1 + 0.0083) ^ (-48))
= 182.6 / (1 - (1.0083) ^ (-48))
≈ 557.92
So, the monthly payment on the original loan is approximately $557.92.
2) Find the monthly payment on the new loan:
To find the monthly payment on the refinanced loan, we'll use the same formula as above. However, the loan amount is now $17,291 and the interest rate is 6% per year. The remaining loan term is 3 years, so the number of months is 3 * 12 = 36. Let's calculate the monthly payment:
Monthly Interest Rate = Annual Interest Rate / 12
= 6% / 12
= 0.06 / 12
= 0.005
Monthly Payment = (17,291 * 0.005) / (1 - (1 + 0.005) ^ (-36))
= 86.46 / (1 - (1.005) ^ (-36))
≈ 525.99
So, the monthly payment on the new loan is approximately $525.99.
3) Find the total amount saved on interest by refinancing:
To find the total amount saved on interest, we'll subtract the total interest paid on the new loan from the total interest paid on the original loan. We can calculate the total interest paid using the formula:
Total Interest Paid = (Monthly Payment * Number of Months) - Loan Amount
For the original loan:
Total Interest Paid = (557.92 * 48) - 22,000
= 26,772.16 - 22,000
= 4,772.16
For the new loan:
Total Interest Paid = (525.99 * 36) - 17,291
= 18,895.64 - 17,291
= 1,604.64
Total Amount Saved on Interest = Total Interest Paid (Original Loan) - Total Interest Paid (New Loan)
= 4,772.16 - 1,604.64
= 3,167.52
Therefore, the total amount saved on interest by refinancing is $3,167.52.
The correct answer is not C. The correct answer would be B, which is $557.92; $525.99; $1149.48.
To find the monthly payment on the original loan, we can use the formula for calculating the monthly payment on an amortized loan.
1) Monthly payment on the original loan:
First, we need to calculate the number of monthly payments over the 4-year term. Since there are 4 years in total, the number of monthly payments would be 4 multiplied by 12 (months in a year), which equals 48.
Next, we can use the formula for calculating the monthly payment on an amortized loan:
Monthly Payment = (Loan Amount * Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate)^(-number of payments))
Loan Amount = $22,000
Annual Interest Rate = 10%
Monthly Interest Rate = Annual Interest Rate / 12 (since there are 12 months in a year)
Now we can plug in the values and calculate the monthly payment on the original loan:
Monthly Interest Rate = 10% / 12 = 0.10 / 12 = 0.008333
Monthly Payment = (22,000 * 0.008333) / (1 - (1 + 0.008333)^(-48)) ≈ $557.92
Therefore, the monthly payment on the original loan is approximately $557.92.
2) Monthly payment on the new loan:
Since Harold refinanced the unpaid balance of $17,291 at a 6% interest rate, the loan amount for the new loan would be $17,291.
Following the same formula as above, we can calculate the monthly payment on the new loan using the new loan amount and the new interest rate:
Monthly Interest Rate = 6% / 12 = 0.06 / 12 = 0.005
Monthly Payment = (17,291 * 0.005) / (1 - (1 + 0.005)^(-48)) ≈ $525.99
Therefore, the monthly payment on the new loan is approximately $525.99.
3) Total amount saved on interest by refinancing:
To find the total amount saved on interest by refinancing, we need to calculate the total interest paid on both the original loan and the new loan and then calculate the difference.
For the original loan, we can calculate the total interest paid over the 4-year term using the formula:
Total Interest on original loan = (Monthly Payment * Number of Payments) - Loan Amount
Total Interest on original loan = ($557.92 * 48) - $22,000 ≈ $12,127.92
For the new loan, we can calculate the total interest paid over the remaining term (3 years) using the same formula:
Total Interest on new loan = (Monthly Payment * Number of Payments) - Loan Amount
Total Interest on new loan = ($525.99 * 36) - $17,291 ≈ $5,000.44
Total amount saved on interest by refinancing = Total Interest on original loan - Total Interest on new loan
Total amount saved on interest by refinancing ≈ $12,127.92 - $5,000.44 ≈ $7,127.48
Therefore, the total amount saved on interest by refinancing is approximately $7,127.48.
After analyzing the options provided, the correct answer would be C.