Answer :
To find the monthly payment for a 20-year loan of [tex]$215,000 with an interest rate of 5.4% compounded monthly, we can use the standard formula for calculating monthly loan payments:
\[ M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \]
Where:
- \( M \) is the monthly payment.
- \( P \) is the principal loan amount ($[/tex]215,000 in this case).
- [tex]\( r \)[/tex] is the monthly interest rate (the annual rate divided by 12).
- [tex]\( n \)[/tex] is the total number of payments (monthly payments over the loan's term).
Given the specifics:
- Annual interest rate = 5.4%, so the monthly interest rate [tex]\( r \)[/tex] is [tex]\( \frac{5.4}{100} \div 12 = 0.0045 \)[/tex].
- The loan is for 20 years, so the total number of payments [tex]\( n \)[/tex] is [tex]\( 20 \times 12 = 240 \)[/tex].
Using these values:
1. Substitute [tex]\( P = 215000 \)[/tex], [tex]\( r = 0.0045 \)[/tex], and [tex]\( n = 240 \)[/tex] into the formula for [tex]\( M \)[/tex].
Calculating with these numbers, the monthly payment [tex]\( M \)[/tex] comes out to about [tex]$1466.84.
Now, let's evaluate the given options:
- Option A:
\[ \frac{\$[/tex] 215000 \cdot 0.0045(1-0.0045)^{240}}{(1-0.0045)^{260}+1} \]
- Option B:
[tex]\[ \frac{\$ 215\left[000 \cdot 0.0045(1-0.0045)^{240}\right]}{(1-0.0045)^{20}-1} \][/tex]
- Option C:
[tex]\[ \frac{\$ 215000 \cdot 0.0045(1+0.0045)^{240}}{(1+0.0045)^{240}-1} \][/tex]
- Option D:
[tex]\[ \frac{\$ 215000 \cdot 0.0045(1+0.0045)^{200}}{(1+0.0045)^{260}+1} \][/tex]
The correct expression should match the structure of our formula, specifically where it includes:
- The principal (\[tex]$215,000) multiplied by the monthly interest rate (0.0045).
- The component \((1 + r)^n\) raised to the power of total payments (240).
- Divided by the same component \((1 + r)^n\) minus 1.
The expression in Option C aligns perfectly with our calculations. It uses the correct form of the formula for calculating monthly loan payments:
\[ \frac{\$[/tex] 215000 \cdot 0.0045(1+0.0045)^{240}}{(1+0.0045)^{240}-1} \]
Therefore, the correct answer is Option C.
\[ M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \]
Where:
- \( M \) is the monthly payment.
- \( P \) is the principal loan amount ($[/tex]215,000 in this case).
- [tex]\( r \)[/tex] is the monthly interest rate (the annual rate divided by 12).
- [tex]\( n \)[/tex] is the total number of payments (monthly payments over the loan's term).
Given the specifics:
- Annual interest rate = 5.4%, so the monthly interest rate [tex]\( r \)[/tex] is [tex]\( \frac{5.4}{100} \div 12 = 0.0045 \)[/tex].
- The loan is for 20 years, so the total number of payments [tex]\( n \)[/tex] is [tex]\( 20 \times 12 = 240 \)[/tex].
Using these values:
1. Substitute [tex]\( P = 215000 \)[/tex], [tex]\( r = 0.0045 \)[/tex], and [tex]\( n = 240 \)[/tex] into the formula for [tex]\( M \)[/tex].
Calculating with these numbers, the monthly payment [tex]\( M \)[/tex] comes out to about [tex]$1466.84.
Now, let's evaluate the given options:
- Option A:
\[ \frac{\$[/tex] 215000 \cdot 0.0045(1-0.0045)^{240}}{(1-0.0045)^{260}+1} \]
- Option B:
[tex]\[ \frac{\$ 215\left[000 \cdot 0.0045(1-0.0045)^{240}\right]}{(1-0.0045)^{20}-1} \][/tex]
- Option C:
[tex]\[ \frac{\$ 215000 \cdot 0.0045(1+0.0045)^{240}}{(1+0.0045)^{240}-1} \][/tex]
- Option D:
[tex]\[ \frac{\$ 215000 \cdot 0.0045(1+0.0045)^{200}}{(1+0.0045)^{260}+1} \][/tex]
The correct expression should match the structure of our formula, specifically where it includes:
- The principal (\[tex]$215,000) multiplied by the monthly interest rate (0.0045).
- The component \((1 + r)^n\) raised to the power of total payments (240).
- Divided by the same component \((1 + r)^n\) minus 1.
The expression in Option C aligns perfectly with our calculations. It uses the correct form of the formula for calculating monthly loan payments:
\[ \frac{\$[/tex] 215000 \cdot 0.0045(1+0.0045)^{240}}{(1+0.0045)^{240}-1} \]
Therefore, the correct answer is Option C.
