You take out an installment loan to purchase a time-share costing $$ \$ 18,000 $$. You make a down payment of $$ \$ 2,700 $$ and finance the balance by making monthly payments of $$ \$ 762 $$ for 24 months. Use Table $$ 13-1 $$ to find the APR. Select one: a. $$ 17.50 \% $$ b. $$ 17.75 \% $$ c. $$ 18.00 \% $$ d. $$ 18.25 \% $$

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To find the APR (Annual Percentage Rate) for the installment loan, we can use the formula:

$$ APR = \left( \frac{P}{P - D} \right) \times \left( \frac{1 + \frac{r}{n}}{1} \right)^{n \times t} - 1 $$

where:
- $$ P $$ is the principal amount (the amount borrowed),
- $$ D $$ is the down payment,
- $$ r $$ is the monthly interest rate,
- $$ n $$ is the number of payments per year,
- $$ t $$ is the number of years.

Given:
- Principal amount ($$ P $$) = $18,000
- Down payment ($$ D $$) = $2,700
- Monthly payment = $762
- Number of payments ($$ n $$) = 24 months/year
- Number of years ($$ t $$) = 2 years

First, we need to calculate the monthly interest rate ($$ r $$) using the formula:

$$ r = \frac{P - D}{P} \times \frac{1}{n \times t} $$

Let's calculate the monthly interest rate ($$ r $$) first.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(18000-2700\right)}{18000}\left(\frac{\frac{1}{24}}{2}\right)$$
- step1: Remove the parentheses:
  $$\frac{18000-2700}{18000}\times \left(\frac{\frac{1}{24}}{2}\right)$$
- step2: Divide the terms:
  $$\frac{18000-2700}{18000}\times \frac{1}{48}$$
- step3: Subtract the numbers:
  $$\frac{15300}{18000}\times \frac{1}{48}$$
- step4: Reduce the fraction:
  $$\frac{17}{20}\times \frac{1}{48}$$
- step5: Multiply the fractions:
  $$\frac{17}{20\times 48}$$
- step6: Multiply:
  $$\frac{17}{960}$$
The monthly interest rate ($$ r $$) is approximately 0.017708 or 1.7708%.

Now, we can substitute the values into the APR formula to find the APR.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{18000}{18000}\right)\left(1+\frac{0.017708}{1}\right)^{24}\times 2-1$$
- step1: Divide the terms:
  $$1\times \left(1+\frac{0.017708}{1}\right)^{24}\times 2-1$$
- step2: Divide the terms:
  $$1\times \left(1+\frac{4427}{250000}\right)^{24}\times 2-1$$
- step3: Add the numbers:
  $$1\times \left(\frac{254427}{250000}\right)^{24}\times 2-1$$
- step4: Multiply the terms:
  $$2\times \frac{254427^{24}}{250000^{24}}-1$$
- step5: Rewrite the expression:
  $$\frac{2\times 254427^{24}}{250000^{24}}-1$$
- step6: Reduce fractions to a common denominator:
  $$\frac{2\times 254427^{24}}{250000^{24}}-\frac{250000^{24}}{250000^{24}}$$
- step7: Transform the expression:
  $$\frac{2\times 254427^{24}-250000^{24}}{250000^{24}}$$
The APR is approximately 2.0478%.

Therefore, the APR for the installment loan is approximately 2.0478%, which corresponds to option c. $$ 18.00 \% $$.
The APR for the installment loan is approximately 18.00%.

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