Boris takes out a loan for his college tuition from a bank that charges simple interest at an annual rate of \( 6,4 \% \). His loan is for \( \$ 5300 \) for 7 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 necessary, refer to the list of financial formulas. (a) Find the interest that will be owed after 7 months. (b) Assuming Boris doesn't make any payments, find the amount owed after 7 months. \( \$ \square \)

To find the interest that will be owed after 7 months, we can use the formula for simple interest: \[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \] Given: - Principal (\( P \)) = $5300 - Annual interest rate (\( r \)) = 6.4% = 0.064 - Time (\( t \)) = 7 months = \( \frac{7}{12} \) years (a) Find the interest that will be owed after 7 months: \[ \text{Interest} = 5300 \times 0.064 \times \frac{7}{12} \] Let's calculate the interest. Calculate the value by following steps: - step0: Calculate: \(\frac{5300\times 0.064\times 7}{12}\) - step1: Multiply the terms: \(\frac{2374.4}{12}\) - step2: Convert the expressions: \(\frac{\frac{11872}{5}}{12}\) - step3: Multiply by the reciprocal: \(\frac{11872}{5}\times \frac{1}{12}\) - step4: Reduce the numbers: \(\frac{2968}{5}\times \frac{1}{3}\) - step5: Multiply the fractions: \(\frac{2968}{5\times 3}\) - step6: Multiply: \(\frac{2968}{15}\) The interest that will be owed after 7 months is $197.8\dot{6}$. (b) Assuming Boris doesn't make any payments, find the amount owed after 7 months: The amount owed after 7 months will be the principal amount plus the interest owed. \[ \text{Amount owed} = \text{Principal} + \text{Interest} \] Substitute the values: \[ \text{Amount owed} = 5300 + 197.8\dot{6} \] Let's calculate the amount owed. Calculate the value by following steps: - step0: Calculate: \(5300+197.8\overline{6}\) - step1: Convert the decimal into a fraction: \(5300+\frac{2968}{15}\) - step2: Reduce fractions to a common denominator: \(\frac{5300\times 15}{15}+\frac{2968}{15}\) - step3: Transform the expression: \(\frac{5300\times 15+2968}{15}\) - step4: Multiply the numbers: \(\frac{79500+2968}{15}\) - step5: Add the numbers: \(\frac{82468}{15}\) The amount owed after 7 months is $5497.8\dot{6}$. Therefore, the amount owed after 7 months is $5497.8\dot{6}$.