Rodrigo applied for a [tex]$514,000 loan at an interest rate of 54% for 6 years. To calculate the monthly payment and the total finance charge for the loan, we can use the monthly payment formula for loans:
\[ M = \frac{P \cdot i \cdot (1 + i)^{N}}{(1 + i)^{N} - 1} \]
where:
- \( M \) is the monthly payment,
- \( P \) is the principal amount ($[/tex]514,000),
- [tex]\( r \)[/tex] is the annual interest rate (54% or 0.54 when expressed as a decimal),
- [tex]\( t \)[/tex] is the number of years (6 years),
- [tex]\( N \)[/tex] is the total number of monthly payments ([tex]\( t \times 12 = 6 \times 12 = 72 \)[/tex] months),
- [tex]\( i \)[/tex] is the monthly interest rate ([tex]\( \frac{r}{12} = \frac{0.54}{12} = 0.045 \)[/tex]).
Using the given numbers:
1. Calculate the monthly interest rate:
[tex]\[ i = 0.045 \][/tex]
2. Calculate the number of monthly payments:
[tex]\[ N = 72 \][/tex]
3. Calculate the monthly payment:
[tex]\[ M \][/tex]
[tex]\[ M \approx 24,144.97 \][/tex]
Therefore, Rodrigo's monthly payment for the loan is approximately [tex]$24,144.97.
4. Calculate the total payment over the loan period:
\[ \text{Total payment} = M \times N \]
\[ \text{Total payment} \approx 24,144.97 \times 72 \]
\[ \text{Total payment} \approx 1,740,437.94 \]
5. Calculate the total finance charge:
\[ \text{Total finance charge} = \text{Total payment} - P \]
\[ \text{Total finance charge} \approx 1,740,437.94 - 514,000 \]
\[ \text{Total finance charge} \approx 1,224,437.94 \]
Therefore, the total finance charge for the loan is approximately $[/tex]1,224,437.94.
Rodrigo's monthly payment for the loan is [tex]$24,144.97, and the total finance charge for the loan is $[/tex]1,224,437.94.
