Suppose that you decide to borrow $$ \$ 16,000 $$ for a new car. You can select one of the following loans, each requiring regular monthly payments. Installment Loan A: three-year loan at $$ 6.3 \% $$ Installment Loan $$ B ; $$ five-year loan at $$ 5.2 \% $$ Use PMT $$ =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} $$ to complete parts (a) through

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To calculate the monthly payment for each loan, we can use the formula for monthly payments:

$$ \text{PMT} = \frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-nt}\right]} $$

where:
- $$ P $$ is the principal amount (the amount borrowed),
- $$ r $$ is the annual interest rate (in decimal form),
- $$ n $$ is the number of payments per year,
- $$ t $$ is the number of years.

Let's calculate the monthly payment for each loan:

### Installment Loan A:
- Principal amount ($$ P $$) = $16,000
- Annual interest rate ($$ r $$) = 6.3% = 0.063
- Number of payments per year ($$ n $$) = 12 (since it's a monthly payment)
- Number of years ($$ t $$) = 3

Substitute these values into the formula to find the monthly payment for Installment Loan A.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{16000\left(\frac{0.063}{12}\right)}{1-\left(1+\frac{0.063}{12}\right)^{-12\times 3}}$$
- step1: Divide the terms:
  $$\frac{16000\left(\frac{0.063}{12}\right)}{1-\left(1+\frac{21}{4000}\right)^{-12\times 3}}$$
- step2: Add the numbers:
  $$\frac{16000\left(\frac{0.063}{12}\right)}{1-\left(\frac{4021}{4000}\right)^{-12\times 3}}$$
- step3: Divide the terms:
  $$\frac{16000\times \frac{21}{4000}}{1-\left(\frac{4021}{4000}\right)^{-12\times 3}}$$
- step4: Multiply the numbers:
  $$\frac{16000\times \frac{21}{4000}}{1-\left(\frac{4021}{4000}\right)^{-36}}$$
- step5: Multiply the numbers:
  $$\frac{84}{1-\left(\frac{4021}{4000}\right)^{-36}}$$
- step6: Subtract the numbers:
  $$\frac{84}{\frac{4021^{36}-4000^{36}}{4021^{36}}}$$
- step7: Multiply by the reciprocal:
  $$84\times \frac{4021^{36}}{4021^{36}-4000^{36}}$$
- step8: Multiply:
  $$\frac{84\times 4021^{36}}{4021^{36}-4000^{36}}$$
The monthly payment for Installment Loan A is approximately $488.93.

### Installment Loan B:
- Principal amount ($$ P $$) = $16,000
- Annual interest rate ($$ r $$) = 5.2% = 0.052
- Number of payments per year ($$ n $$) = 12 (since it's a monthly payment)
- Number of years ($$ t $$) = 5

Substitute these values into the formula to find the monthly payment for Installment Loan B.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{16000\left(\frac{0.052}{12}\right)}{1-\left(1+\frac{0.052}{12}\right)^{-12\times 5}}$$
- step1: Divide the terms:
  $$\frac{16000\left(\frac{0.052}{12}\right)}{1-\left(1+\frac{13}{3000}\right)^{-12\times 5}}$$
- step2: Add the numbers:
  $$\frac{16000\left(\frac{0.052}{12}\right)}{1-\left(\frac{3013}{3000}\right)^{-12\times 5}}$$
- step3: Divide the terms:
  $$\frac{16000\times \frac{13}{3000}}{1-\left(\frac{3013}{3000}\right)^{-12\times 5}}$$
- step4: Multiply the numbers:
  $$\frac{16000\times \frac{13}{3000}}{1-\left(\frac{3013}{3000}\right)^{-60}}$$
- step5: Multiply the numbers:
  $$\frac{\frac{208}{3}}{1-\left(\frac{3013}{3000}\right)^{-60}}$$
- step6: Subtract the numbers:
  $$\frac{\frac{208}{3}}{\frac{3013^{60}-3000^{60}}{3013^{60}}}$$
- step7: Multiply by the reciprocal:
  $$\frac{208}{3}\times \frac{3013^{60}}{3013^{60}-3000^{60}}$$
- step8: Multiply the fractions:
  $$\frac{208\times 3013^{60}}{3\left(3013^{60}-3000^{60}\right)}$$
- step9: Multiply:
  $$\frac{208\times 3013^{60}}{3\times 3013^{60}-3\times 3000^{60}}$$
The monthly payment for Installment Loan B is approximately $303.41.

Therefore, the monthly payments for Installment Loan A and Installment Loan B are $488.93 and $303.41, respectively.
The monthly payment for Installment Loan A is approximately $488.93, and for Installment Loan B, it's about $303.41.

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