- Calculate APR by locating the finance charge per \$100 in Table 14.1, which lies between the values for 16.00% and 16.25%.
- Determine the monthly payment using the formula \(M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}\), resulting in \(\$194.68\).
- Find the monthly payment using Table 14.2 by multiplying the amount financed by the table factor for 16.00% APR, resulting in \(\$194.80\).
- Final answers are APR between 16.00% and 16.25%, payment by formula \(\boxed{\$194.68}\), and payment by table lookup \(\boxed{\$194.80}\).
Explanation
- Calculating APR by Table Lookup
To find the Annual Percentage Rate (APR) using a table lookup, we typically determine the finance charge per \$100 of the amount financed. This is calculated as follows:
\[ \text{Finance charge per \$100} = \frac{\text{Total Finance Charge}}{\text{Amount Financed}} \times 100 \]
Once this value is calculated, we consult Table 14.1, which lists the finance charges per \$100 for various loan terms and APRs. We locate the row corresponding to the term of the loan and find the column where our calculated finance charge sits. Since the problem states the APR is between \(16.00\%\) and \(16.25\%\), our calculated value falls within that specific interval in the table. By comparing these points, we confirm the APR range. The calculated APR, rounded to two decimal places, falls in the specified range.
- Calculating Monthly Payment by Formula
The standard formula for calculating the fixed monthly payment (\(M\)) for an installment loan is:
\[ M = P \times \frac{r(1+r)^n}{(1+r)^n - 1} \]
where:
- \(P\) is the principal amount financed.
- \(r\) is the monthly interest rate, which is the annual interest rate divided by \(12\).
- \(n\) is the total number of monthly payments.
By substituting the loan details into this formula, we arrive at the monthly payment. For the given loan, the calculation results in a monthly payment of \(\$194.68\), which is already rounded to the nearest cent as required.
- Calculating Monthly Payment by Table Lookup
To calculate the monthly payment using Table 14.2, we use a predetermined table factor that represents the monthly payment per \$100 of the loan principal for a given interest rate and loan term. The calculation is:
\[ \text{Monthly payment} = \left( \frac{\text{Amount Financed}}{100} \right) \times \text{Table Factor} \]
Using the table factor corresponding to an APR of \(16.00\%\) and the specific loan term, we find the monthly payment. Based on the data provided for an APR of \(16.00\%\), the calculation yields a monthly payment of \(\$194.80\), which is rounded to the nearest cent.
Examples
Understanding these financial calculations is essential for managing personal credit. For example, when you take out a car loan or a personal loan, lenders use the monthly payment formula to determine your fixed commitment. By learning how to use APR tables and formulas, you can verify your loan documents for accuracy, understand how interest rates affect your monthly obligations, and compare different loan offers to make informed financial decisions.