Melissa Borrows $1000 on May 8 at 18.5% simple interest. She pays $500 on July 17 and $400 on September 29. What is her balance on October 31 by the merchant's rule? by the United States rule?
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.