To calculate Melissa's balance on October 31 using the merchant's rule and the United States rule, we first need to determine the interest accrued for each payment period.
1. Merchant's Rule:
In the merchant's rule, interest is calculated on the original principal amount for the entire time period, regardless of any payments made.
First, let's calculate the interest accrued from May 8 to July 17 (70 days):
Principal Amount: $1000
Interest Rate: 18.5%
Time: 70 days / 365 days (or 0.1918 years)
Interest accrued = Principal Amount × Interest Rate × Time
= $1000 × 0.185 × 0.1918
≈ $36.959
Since Melissa paid $500 on July 17, the interest accrued is subtracted from the payment:
Balance after July 17 = Principal Amount + Interest accrued - Payment
= $1000 + $36.959 - $500
= $536.959
Now, let's calculate the interest accrued from July 17 to September 29 (74 days):
Principal Amount: $536.959
Interest Rate: 18.5%
Time: 74 days / 365 days (or 0.2027 years)
Interest accrued = Principal Amount × Interest Rate × Time
= $536.959 × 0.185 × 0.2027
≈ $18.799
Since Melissa paid $400 on September 29, the interest accrued is subtracted from the payment:
Balance after September 29 = Principal Amount + Interest accrued - Payment
= $536.959 + $18.799 - $400
= $155.758
Finally, let's calculate the interest accrued from September 29 to October 31 (32 days):
Principal Amount: $155.758
Interest Rate: 18.5%
Time: 32 days / 365 days (or 0.0877 years)
Interest accrued = Principal Amount × Interest Rate × Time
= $155.758 × 0.185 × 0.0877
≈ $2.859
Balance on October 31 by the merchant's rule = Principal Amount + Interest accrued
= $155.758 + $2.859
= $158.617
So, according to the merchant's rule, Melissa's balance on October 31 is approximately $158.617.
2. United States Rule:
In the United States rule, interest is calculated on the remaining balance after each payment.
Starting with the principal amount of $1000, let's calculate the remaining balance after each payment:
a) After July 17:
Principal Amount: $1000
Interest Rate: 18.5%
Time: 70 days / 365 days (or 0.1918 years)
Interest accrued = Principal Amount × Interest Rate × Time
= $1000 × 0.185 × 0.1918
≈ $36.959
Remaining balance after July 17 = Principal Amount + Interest accrued - Payment
= $1000 + $36.959 - $500
= $536.959
b) After September 29:
Principal Amount: $536.959
Interest Rate: 18.5%
Time: 74 days / 365 days (or 0.2027 years)
Interest accrued = Principal Amount × Interest Rate × Time
= $536.959 × 0.185 × 0.2027
≈ $18.799
Remaining balance after September 29 = Principal Amount + Interest accrued - Payment
= $536.959 + $18.799 - $400
= $155.758
c) After October 31:
Principal Amount: $155.758
Interest Rate: 18.5%
Time: 32 days / 365 days (or 0.0877 years)
Interest accrued = Principal Amount × Interest Rate × Time
= $155.758 × 0.185 × 0.0877
≈ $2.859
Remaining balance on October 31 by the United States rule = Principal Amount + Interest accrued
= $155.758 + $2.859
= $158.617
So, according to the United States rule, Melissa's balance on October 31 is approximately $158.617.