Liam had $5,200 in student loans. On August 9, he began repayments of $50 per month when simple interest rates were 9.2% annually. On September 8, the interest rates rose to 9.5%. By what amount will the principal be reduced given the $50 payment on September 30?

A) $8.90
B) $9.91
C) $10.90
D) $11.90
E) $12.90

The interest for August (22 days, assuming the payment is made at the end of the day on September 8) at 9.2% annually on $5,200 is calculated as follows: Interest = Principal × Rate × Time = $5,200 × 9.2% × (22/365) = $30.09. Thus, after the $50 payment on September 8, $30.09 goes towards interest, reducing the principal by $19.91 ($50 - $30.09). The new principal is $5,180.09 ($5,200 - $19.91). For the remaining 22 days of September at 9.5% annually, the interest is: $5,180.09 × 9.5% × (22/365) = $30.00. Therefore, the $50 payment on September 30 will reduce the principal by $20.00 ($50 - $30.00). However, since the question asks for the reduction in principal from the $50 payment on September 30 specifically, the correct answer is $20.00, which is not one of the provided options, indicating a mistake in my calculations or interpretation. Reevaluating the correct approach: The interest for one month at 9.2% annually on $5,200 is: Interest = Principal × Rate × Time = $5,200 × (9.2%/12) = $39.73. This means the principal reduction from the $50 payment in August is $10.27 ($50 - $39.73). However, since the question specifically asks for the reduction in principal from the payment on September 30, after the interest rate increase to 9.5%, and considering my mistake in the monthly interest calculation, let's correct the approach focusing on the September payment impact only. The correct calculation should focus on the interest accrued in September at the new rate and how much of the $50 payment goes towards reducing the principal after accounting for this interest. Given the error in my initial explanation and calculations, let's simplify to the correct explanation: For one month, the interest on $5,200 at 9.2% annually is approximately $39.73, leaving about $10.27 to reduce the principal after the August payment. However, the question asks for the reduction in principal from the September payment, after the rate increase. The correct calculation should directly address the impact of the September payment under the new interest rate, without incorrectly breaking down the interest by days or misapplying the rates. The mistake in the explanation was assuming a daily interest calculation and misinterpreting the question's focus on the September payment only. The correct answer should be based on the principal reduction directly from the September payment, considering the new interest rate and the correct method for calculating simple interest over a month, which was not accurately provided in the initial response.