Ravi borrowed money from a bank to buy a motorcycle. He took out a personal, amortized loan for $$ \$ 18,000 $$, at an interest rate of $$ 5.6 \% $$, 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 Ravi's monthly payment. (b) If Ravi pays the monthly payment each month for the full term, find his total amount to repay the loan. $$ \$ \square $$ (c) If Ravi pays the monthly payment each month for the full term, find the total amount of interest he will pay. $$ \$ \square $$

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To find Ravi's monthly payment, we can use the formula for monthly payments on a loan:

$$ M = \frac{P \times r \times (1 + r)^n}{(1 + r)^n - 1} $$

where:
- $$ M $$ is the monthly payment
- $$ P $$ is the principal amount (initial loan amount)
- $$ r $$ is the monthly interest rate
- $$ n $$ is the number of payments (months)

Given:
- Principal amount ($$ P $$) = $18,000
- Interest rate = 5.6% per year
- Monthly payments for 5 years

First, we need to convert the annual interest rate to a monthly interest rate:
$$ r = \frac{5.6\%}{12} = 0.0046667 $$

Now, we can calculate the monthly payment using the formula above.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{18000\times 0.0046667\left(1+0.0046667\right)^{60}}{\left(\left(1+0.0046667\right)^{60}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{18000\times 0.0046667\left(1+0.0046667\right)^{60}}{\left(1+0.0046667\right)^{60}-1}$$
- step2: Add the numbers:
  $$\frac{18000\times 0.0046667\times 1.0046667^{60}}{\left(1+0.0046667\right)^{60}-1}$$
- step3: Add the numbers:
  $$\frac{18000\times 0.0046667\times 1.0046667^{60}}{1.0046667^{60}-1}$$
- step4: Convert the expressions:
  $$\frac{18000\times 0.0046667\left(\frac{10046667}{10000000}\right)^{60}}{1.0046667^{60}-1}$$
- step5: Convert the expressions:
  $$\frac{18000\times 0.0046667\left(\frac{10046667}{10000000}\right)^{60}}{\left(\frac{10046667}{10000000}\right)^{60}-1}$$
- step6: Multiply:
  $$\frac{\frac{420003\times 10046667^{60}}{5000\times 10000000^{60}}}{\left(\frac{10046667}{10000000}\right)^{60}-1}$$
- step7: Subtract the numbers:
  $$\frac{\frac{420003\times 10046667^{60}}{5000\times 10000000^{60}}}{\frac{10046667^{60}-10000000^{60}}{10000000^{60}}}$$
- step8: Multiply by the reciprocal:
  $$\frac{420003\times 10046667^{60}}{5000\times 10000000^{60}}\times \frac{10000000^{60}}{10046667^{60}-10000000^{60}}$$
- step9: Rewrite the expression:
  $$\frac{420003\times 10046667^{60}}{5000\times 10000000^{60}}\times \frac{5000^{60}\times 2000^{60}}{10046667^{60}-10000000^{60}}$$
- step10: Reduce the numbers:
  $$\frac{420003\times 10046667^{60}}{10000000^{60}}\times \frac{5000^{59}\times 2000^{60}}{10046667^{60}-10000000^{60}}$$
- step11: Rewrite the expression:
  $$\frac{420003\times 10046667^{60}}{5000^{60}\times 2000^{60}}\times \frac{5000^{59}\times 2000^{60}}{10046667^{60}-10000000^{60}}$$
- step12: Reduce the numbers:
  $$\frac{420003\times 10046667^{60}}{5000\times 2000^{60}}\times \frac{2000^{60}}{10046667^{60}-10000000^{60}}$$
- step13: Rewrite the expression:
  $$\frac{420003\times 10046667^{60}}{5000\times 2000^{60}}\times \frac{1000^{60}\times 2^{60}}{10046667^{60}-10000000^{60}}$$
- step14: Rewrite the expression:
  $$\frac{420003\times 10046667^{60}}{1000\times 5\times 2000^{60}}\times \frac{1000^{60}\times 2^{60}}{10046667^{60}-10000000^{60}}$$
- step15: Reduce the numbers:
  $$\frac{420003\times 10046667^{60}}{5\times 2000^{60}}\times \frac{1000^{59}\times 2^{60}}{10046667^{60}-10000000^{60}}$$
- step16: Rewrite the expression:
  $$\frac{420003\times 10046667^{60}}{5\times 2000^{60}}\times \frac{125^{59}\times 8^{59}\times 2^{60}}{10046667^{60}-10000000^{60}}$$
- step17: Rewrite the expression:
  $$\frac{420003\times 10046667^{60}}{5\times 2000^{60}}\times \frac{5^{177}\times 8^{59}\times 2^{60}}{10046667^{60}-10000000^{60}}$$
- step18: Reduce the numbers:
  $$\frac{420003\times 10046667^{60}}{2000^{60}}\times \frac{5^{176}\times 8^{59}\times 2^{60}}{10046667^{60}-10000000^{60}}$$
- step19: Rewrite the expression:
  $$\frac{420003\times 10046667^{60}}{125^{60}\times 16^{60}}\times \frac{5^{176}\times 8^{59}\times 2^{60}}{10046667^{60}-10000000^{60}}$$
- step20: Rewrite the expression:
  $$\frac{420003\times 10046667^{60}}{5^{180}\times 16^{60}}\times \frac{5^{176}\times 8^{59}\times 2^{60}}{10046667^{60}-10000000^{60}}$$
- step21: Reduce the numbers:
  $$\frac{420003\times 10046667^{60}}{5^{4}\times 16^{60}}\times \frac{8^{59}\times 2^{60}}{10046667^{60}-10000000^{60}}$$
- step22: Rewrite the expression:
  $$\frac{420003\times 10046667^{60}}{5^{4}\times 2^{240}}\times \frac{2^{177}\times 2^{60}}{10046667^{60}-10000000^{60}}$$
- step23: Reduce the numbers:
  $$\frac{420003\times 10046667^{60}}{5^{4}\times 2^{3}}\times \frac{1}{10046667^{60}-10000000^{60}}$$
- step24: Multiply the fractions:
  $$\frac{420003\times 10046667^{60}}{5000\left(10046667^{60}-10000000^{60}\right)}$$
- step25: Multiply:
  $$\frac{420003\times 10046667^{60}}{5000\times 10046667^{60}-5000\times 10000000^{60}}$$
The monthly payment for Ravi's loan is approximately $344.65.

Now, to find the total amount to repay the loan if Ravi pays the monthly payment each month for the full term, we can use the formula for the total amount paid:

$$ T = M \times n $$

where:
- $$ T $$ is the total amount to repay the loan
- $$ M $$ is the monthly payment
- $$ n $$ is the number of payments (months)

Given:
- Monthly payment ($$ M $$) = $344.65
- Number of payments ($$ n $$) = 60 months (5 years)

Now, we can calculate the total amount to repay the loan.
Calculate the value by following steps:
- step0: Calculate:
  $$344.65\times 60$$
- step1: Multiply the numbers:
  $$20679$$
The total amount to repay the loan if Ravi pays the monthly payment each month for the full term is $20,679.

Finally, to find the total amount of interest Ravi will pay if he pays the monthly payment each month for the full term, we can subtract the principal amount from the total amount to repay the loan:

$$ I = T - P $$

where:
- $$ I $$ is the total amount of interest
- $$ T $$ is the total amount to repay the loan
- $$ P $$ is the principal amount (initial loan amount)

Given:
- Total amount to repay the loan ($$ T $$) = $20,679
- Principal amount ($$ P $$) = $18,000

Now, we can calculate the total amount of interest Ravi will pay.
Calculate the value by following steps:
- step0: Calculate:
  $$20679-18000$$
- step1: Subtract the numbers:
  $$2679$$
The total amount of interest Ravi will pay if he pays the monthly payment each month for the full term is $2,679.

Therefore, the answers are:
(a) Ravi's monthly payment is $344.65.
(b) The total amount to repay the loan is $20,679.
(c) The total amount of interest Ravi will pay is $2,679.
(a) Ravi's monthly payment is $344.65. (b) The total amount to repay the loan is $20,679. (c) The total amount of interest Ravi will pay is $2,679.

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