Answer :
Sure, let's go through the steps to determine which expression can be used to calculate the monthly payment for a 20-year loan of \[tex]$215,000 with an interest rate of 5.4%, compounded monthly.
### Step-by-Step Solution:
1. Understand the Parameters:
- Principal (P) = \$[/tex]215,000
- Annual interest rate = 5.4%
- Monthly interest rate (r) = 5.4% / 12 = 0.45% = 0.0045
- Loan term = 20 years
- Number of monthly payments (n) = 20 years \* 12 months/year = 240
2. Monthly Payment Formula:
The formula to calculate the monthly payment for a loan is:
[tex]\[
M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1}
\][/tex]
where:
- [tex]\( M \)[/tex] is the monthly payment
- [tex]\( P \)[/tex] is the loan principal
- [tex]\( r \)[/tex] is the monthly interest rate
- [tex]\( n \)[/tex] is the number of payments
3. Substitute the Values into the Formula:
- [tex]\( P = 215000 \)[/tex]
- [tex]\( r = 0.0045 \)[/tex]
- [tex]\( n = 240 \)[/tex]
Our formula becomes:
[tex]\[
M = 215000 \cdot \frac{0.0045(1+0.0045)^{240}}{(1+0.0045)^{240} - 1}
\][/tex]
4. Compare with Given Options:
Let's match this formula with the given expressions:
- Option A: [tex]\(\frac{215000 \cdot 0.0045(1-0.0045)^{240}}{(1-0.0045)^{240}-1}\)[/tex]
- Option B: [tex]\(\frac{215000 \cdot 0.0045(1+0.0045)^{240}}{(1+0.0045)^{240}-1}\)[/tex]
- Option C: [tex]\(\frac{215000 \cdot 0.0045(1-0.0045)^{240}}{(1-0.0045)^{240}+1}\)[/tex]
- Option D: [tex]\(\frac{215000 \cdot 0.0045(1+0.0045)^{240}}{1.0 \cdot 0.05155(1240)}\)[/tex]
Option B is the correct match:
[tex]\[
\frac{215000 \cdot 0.0045(1+0.0045)^{240}}{(1+0.0045)^{240}-1}
\][/tex]
Therefore, the expression that can be used to calculate the monthly payment for a 20-year loan of \$215,000 at 5.4% interest, compounded monthly, is:
[tex]\[
\boxed{B}
\][/tex]
### Step-by-Step Solution:
1. Understand the Parameters:
- Principal (P) = \$[/tex]215,000
- Annual interest rate = 5.4%
- Monthly interest rate (r) = 5.4% / 12 = 0.45% = 0.0045
- Loan term = 20 years
- Number of monthly payments (n) = 20 years \* 12 months/year = 240
2. Monthly Payment Formula:
The formula to calculate the monthly payment for a loan is:
[tex]\[
M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1}
\][/tex]
where:
- [tex]\( M \)[/tex] is the monthly payment
- [tex]\( P \)[/tex] is the loan principal
- [tex]\( r \)[/tex] is the monthly interest rate
- [tex]\( n \)[/tex] is the number of payments
3. Substitute the Values into the Formula:
- [tex]\( P = 215000 \)[/tex]
- [tex]\( r = 0.0045 \)[/tex]
- [tex]\( n = 240 \)[/tex]
Our formula becomes:
[tex]\[
M = 215000 \cdot \frac{0.0045(1+0.0045)^{240}}{(1+0.0045)^{240} - 1}
\][/tex]
4. Compare with Given Options:
Let's match this formula with the given expressions:
- Option A: [tex]\(\frac{215000 \cdot 0.0045(1-0.0045)^{240}}{(1-0.0045)^{240}-1}\)[/tex]
- Option B: [tex]\(\frac{215000 \cdot 0.0045(1+0.0045)^{240}}{(1+0.0045)^{240}-1}\)[/tex]
- Option C: [tex]\(\frac{215000 \cdot 0.0045(1-0.0045)^{240}}{(1-0.0045)^{240}+1}\)[/tex]
- Option D: [tex]\(\frac{215000 \cdot 0.0045(1+0.0045)^{240}}{1.0 \cdot 0.05155(1240)}\)[/tex]
Option B is the correct match:
[tex]\[
\frac{215000 \cdot 0.0045(1+0.0045)^{240}}{(1+0.0045)^{240}-1}
\][/tex]
Therefore, the expression that can be used to calculate the monthly payment for a 20-year loan of \$215,000 at 5.4% interest, compounded monthly, is:
[tex]\[
\boxed{B}
\][/tex]
