John taies out a loan for his college tuition from a bank that charges simple interest at an annual rate of 5 .35\%. His loan is for \( \$ 7700 \) for 4 months, Assume each month is \( \frac{1}{12} \) of a year Answer each part below. Do not round any intermediate computations, and round your final answers to the nearest cent. If necessan, refer to the Ist of financial formulas. (a) Find the interest that will be owed aiter 4 months. (b) Assuming John doesnt make any payments, find the amount owed after 4 months. \( \$ \square \)

Given: - Annual interest rate: 5.35% - Loan amount: $7700 - Loan duration: 4 months - Each month is \( \frac{1}{12} \) of a year (a) To find the interest owed after 4 months, we can use the formula for simple interest: \[ I = P \times r \times t \] where: - \( I \) is the interest - \( P \) is the principal amount (loan amount) - \( r \) is the annual interest rate - \( t \) is the time in years First, we need to convert the loan duration from months to years: \[ t = 4 \text{ months} \times \frac{1}{12} \text{ year/month} = \frac{4}{12} \text{ years} = \frac{1}{3} \text{ years} \] Now, we can calculate the interest owed after 4 months: \[ I = 7700 \times 0.0535 \times \frac{1}{3} \] Let's calculate the interest. Calculate the value by following steps: - step0: Calculate: \(\frac{7700\times 0.0535\times 1}{3}\) - step1: Multiply the terms: \(\frac{411.95}{3}\) - step2: Convert the expressions: \(\frac{\frac{8239}{20}}{3}\) - step3: Multiply by the reciprocal: \(\frac{8239}{20}\times \frac{1}{3}\) - step4: Multiply the fractions: \(\frac{8239}{20\times 3}\) - step5: Multiply: \(\frac{8239}{60}\) The interest owed after 4 months is $137.31\dot{6}$. (b) To find the amount owed after 4 months, we need to add the interest to the principal amount: \[ \text{Amount owed} = \text{Principal amount} + \text{Interest} \] Let's calculate the amount owed after 4 months. Calculate the value by following steps: - step0: Calculate: \(7700+\frac{6865833333333333}{50000000000000}\) - step1: Reduce fractions to a common denominator: \(\frac{7700\times 50000000000000}{50000000000000}+\frac{6865833333333333}{50000000000000}\) - step2: Transform the expression: \(\frac{7700\times 50000000000000+6865833333333333}{50000000000000}\) - step3: Multiply the numbers: \(\frac{385000000000000000+6865833333333333}{50000000000000}\) - step4: Add the numbers: \(\frac{391865833333333333}{50000000000000}\) The amount owed after 4 months is approximately $7837.32.