To determine the difference in the amount of interest accrued by the end of the month if you pay at the beginning of the month versus at the end of the month, we'll need to calculate the interest for both scenarios.
Given Data:
- Balance: \( B = 2,875.59 \)
- Monthly interest rate: \( r = \frac{13.7%}{12} = \frac{0.137}{12} \approx 0.01141667 \)
- Payment: \( P = 400.00 \)
1. Interest Calculation if Payment is Made at the End of the Month:
When payment is made at the end of the month, interest is calculated on the full balance for the month.
Calculate Interest: \[ \text{Interest} = B \times r = 2,875.59 \times 0.01141667 \approx 32.8507 \]
Total Amount After Interest: \[ \text{New Balance} = B + \text{Interest} - P = 2,875.59 + 32.8507 - 400.00 \approx 2,508.44 \]
2. Interest Calculation if Payment is Made at the Beginning of the Month:
When payment is made at the beginning of the month, you pay before the interest is calculated, so interest is calculated on the reduced balance.
New Balance Before Interest: \[ \text{New Balance} = B - P = 2,875.59 - 400.00 = 2,475.59 \]
Calculate Interest on Reduced Balance: \[ \text{Interest} = \text{New Balance} \times r = 2,475.59 \times 0.01141667 \approx 28.29 \]
3. Summary of Interest Accrued
- Interest if paid at the end of the month: \( 32.85 \)
- Interest if paid at the beginning of the month: \( 28.29 \)
4. Calculate the Difference
\[ \text{Difference} = \text{Interest at End} - \text{Interest at Beginning} = 32.85 - 28.29 \approx 4.56 \]
Conclusion
The difference in the amount of interest accrued by the end of the month if you pay at the beginning of the month versus the end of the month is approximately $4.56 when rounded to the nearest penny.