Sure, let's go through the steps to solve this question in detail:
### 1. Principal Calculation
The principal is the amount of money borrowed, which is calculated by subtracting the down payment from the purchase price.
Purchase price: \[tex]$5555
Down payment: \$[/tex]555
[tex]\[ \text{Principal} = \text{Purchase Price} - \text{Down Payment} \][/tex]
[tex]\[ \text{Principal} = 5555 - 555 = \$5000 \][/tex]
So, the principal amount, [tex]\( P \)[/tex], is \[tex]$5000.
### 2. Monthly Interest Rate Calculation
To find the monthly interest rate, we need to convert the annual interest rate to a monthly rate. This is done by dividing the annual interest rate by 12.
Annual Interest Rate: 18% or 0.18 in decimal form
\[ \text{Monthly Interest Rate} = \frac{\text{Annual Interest Rate}}{12} \]
\[ \text{Monthly Interest Rate} = \frac{0.18}{12} \]
\[ \text{Monthly Interest Rate} = 0.015 \]
So, the monthly interest rate is 0.015.
### 3. Effective Interest Calculation
To find the effective interest using the given formula:
\[ I = \frac{2 \cdot y \cdot c}{m \cdot (n + 1)} \]
Where:
- \( y \) is the duration in years (\( \frac{\text{Number of Payments}}{12} \))
- \( c \) is the number of payments (42)
- \( m \) is the monthly payment (\$[/tex]21.43)
- [tex]\( n \)[/tex] is the number of payments (42)
Let's break it down step-by-step:
#### Step 1: Calculate [tex]\( y \)[/tex] (duration in years)
[tex]\[ y = \frac{\text{Number of Payments}}{12} \][/tex]
[tex]\[ y = \frac{42}{12} = 3.5 \, \text{years} \][/tex]
#### Step 2: Use the formula to find the interest
[tex]\[ I = \frac{2 \cdot y \cdot c}{m \cdot (n + 1)} \][/tex]
[tex]\[ I = \frac{2 \cdot 3.5 \cdot 42}{21.43 \cdot (42 + 1)} \][/tex]
[tex]\[ I = \frac{2 \cdot 3.5 \cdot 42}{21.43 \cdot 43} \][/tex]
[tex]\[ I = \frac{294}{920.49} \][/tex]
[tex]\[ I \approx 0.319 \][/tex]
So, the effective interest, [tex]\( I \)[/tex], is approximately 0.319.
### 4. Summary of Results
- Principal: \[tex]$5000
- Monthly Interest Rate: 0.015
- Effective Interest: 0.319
- Down Payment: \$[/tex]555
- Number of Payments: 42
- Monthly Payment: \[tex]$21.43
These values collectively help in understanding the financial breakdown for purchasing the article priced at \$[/tex]5555 with the given down payment, interest rates, and number of payments.
