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Jina borrowed money from an online lending company to buy a fishing boat.She took out a personal, amortized loan for $19,500, at an interest rate of 5.5%, with monthly payments for a term of 5 years.For each part, do not round any intermediate computations and round your final answers to the nearest cent.If necessary, refer to the list of financial formulas.(a) Find Jina's monthly payment.$(b) If Jina pays the monthly payment each month for the full term, find her total amount to repay the loan.$(c) If Jina pays the monthly payment each month for the full term, find the total amount of interest she will pay.$

Question

Jina borrowed money from an online lending company to buy a fishing boat.She took out a personal, amortized loan for 19,500,ataninterestrateof5.519,500, at an interest rate of 5.5%, with monthly payments for a term of 5 years.For each part, do not round any intermediate computations and round your final answers to the nearest cent.If necessary, refer to the list of financial formulas.(a) Find Jina's monthly payment.(b) If Jina pays the monthly payment each month for the full term, find her total amount to repay the loan.(c)IfJinapaysthemonthlypaymenteachmonthforthefullterm,findthetotalamountofinterestshewillpay.(c) If Jina pays the monthly payment each month for the full term, find the total amount of interest she will pay.

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Solution

Sure, let's break down the problem step by step.

Part (a): Find Jina's Monthly Payment

To find the monthly payment for an amortized loan, we use the formula for the monthly payment M M :

M=Pr(1+r)n(1+r)n−1 M = P \frac{r(1+r)^n}{(1+r)^n - 1}

where:

  • P P is the loan principal ($19,500)
  • r r is the monthly interest rate (annual rate divided by 12)
  • n n is the total number of payments (loan term in years multiplied by 12)

Given:

  • Principal P=19,500 P = 19,500
  • Annual interest rate = 5.5% = 0.055
  • Monthly interest rate r=0.05512≈0.00458333 r = \frac{0.055}{12} \approx 0.00458333
  • Loan term = 5 years = 60 months

Plug these values into the formula:

M=195000.00458333(1+0.00458333)60(1+0.00458333)60−1 M = 19500 \frac{0.00458333(1+0.00458333)^{60}}{(1+0.00458333)^{60} - 1}

First, calculate (1+r)n (1 + r)^n :

(1+0.00458333)60≈1.3067 (1 + 0.00458333)^{60} \approx 1.3067

Then, calculate the numerator:

0.00458333×1.3067≈0.005991 0.00458333 \times 1.3067 \approx 0.005991

And the denominator:

1.3067−1=0.3067 1.3067 - 1 = 0.3067

Now, calculate the monthly payment M M :

M=195000.0059910.3067≈19500×0.01954≈381.03 M = 19500 \frac{0.005991}{0.3067} \approx 19500 \times 0.01954 \approx 381.03

So, Jina's monthly payment is approximately $381.03.

Part (b): Find the Total Amount to Repay the Loan

The total amount to repay the loan is the monthly payment multiplied by the number of payments:

Total Repayment=M×n \text{Total Repayment} = M \times n

Given:

  • Monthly payment M=381.03 M = 381.03
  • Total number of payments n=60 n = 60

Total Repayment=381.03×60≈22861.80 \text{Total Repayment} = 381.03 \times 60 \approx 22861.80

So, the total amount Jina will repay is approximately $22,861.80.

Part (c): Find the Total Amount of Interest Paid

The total amount of interest paid is the total repayment amount minus the principal:

Total Interest=Total Repayment−P \text{Total Interest} = \text{Total Repayment} - P

Given:

  • Total repayment =22,861.80 = 22,861.80
  • Principal P=19,500 P = 19,500

Total Interest=22861.80−19500≈3361.80 \text{Total Interest} = 22861.80 - 19500 \approx 3361.80

So, the total amount of interest Jina will pay is approximately $3,361.80.

Summary

(a) Jina's monthly payment: 381.03∗∗(b)Totalamounttorepaytheloan:∗∗381.03** (b) Total amount to repay the loan: **22,861.80 (c) Total amount of interest paid: $3,361.80

This problem has been solved

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